January 01, 1970
This paper investigates the nonlinear asymptotic stability and optimal decay rates of entropy waves for the Landau equation with physically realistic Coulomb interactions under general perturbations. We consider the infinite channel domain \(\mathbb{R} \times \mathbb{T}^2\) in three dimensions, which possesses both one-dimensional and high-dimensional characteristics, thereby posing two primary analytical challenges: (i) for the one-dimensional Landau equation with Coulomb potentials, the absence of a spectral gap in the linearized operator has obstructed the derivation of wave pattern stability results with explicit time decay rates; (ii) in the study of contact discontinuities, the multidimensional case fundamentally differs from the one-dimensional setting due to lack of a key structural condition. We develop effective analytical approaches to treat those difficulties. To overcome the weak dissipation caused by the spectral gap deficiency, we implement a time-velocity interpolation technique to enhance dissipation and simultaneously construct coupled diffusion waves to compensate for the loss of time decay. To address the missing structural condition in higher dimensions, a novel transformation is introduced to recover the two-sided structural condition within the perturbation system. By developing a derivative-level transformation and a refined energy framework, we restore the necessary structural condition for derivatives, establish the optimal decay of the solution, and prove the stretched exponential decay of its non-zero modes. In contrast to previous methods that rely on artificial viscosity or the Navier–Stokes approximation, our approach directly leverages the intrinsic physical dissipation of the equation and its coupling with the microscopic kinetic component, ensuring broader applicability.
The Landau equation in three spatial dimensions reads \[\begin{align} \label{equ-landau} &\partial_t{f} + {\xi }\cdot\nabla_x{f} = Q\left( {f,f}\right) , \end{align}\tag{1}\] where \(f\left( {t,x,\xi }\right) \geq 0\) is the density distribution function for the particles with velocity \(\xi = \left( {{\xi }_{1},{\xi }_{2},{\xi }_{3}}\right) \in {\mathbb{R}}^{3}\) at time \(t \geq 0\) and position \(x=(x_1,x_2,x_3) \in \Omega:= \mathbb{R}\times\mathbb{T}^2\), where \(\mathbb{T}:=\mathbb{R}/\mathbb{Z}\). Through the paper, we only consider the Landau collision operator for physically most relevant Coulomb potentials: \[\begin{align} \label{landau-operator} Q\left( {f,g}\right) \left( \xi \right) = \mathop{\sum }\limits_{{i,j = 1}}^{3}{\partial }_{\xi_i}{\int }_{{\mathbb{R}}^{3}}\tilde{\phi}^{ij}\left( {\xi - {\xi }^{\prime }}\right) \left\{ {f\left( {\xi }^{\prime }\right) {\partial }_{\xi_j}g\left( \xi \right) - {\partial }_{\xi_j}f\left( {\xi }^{\prime }\right) g\left( \xi \right) }\right\} d{\xi }^{\prime }, \end{align}\tag{2}\] where the collision kernel is given by the nonnegative matrix-valued function \(\tilde{\phi}(z)=(\tilde{\phi}(z))_{1\leq i,j\leq 3}\) with \(z=\xi - {\xi }^{\prime }\) taking the form \[\begin{align} \notag \tilde{\phi}^{ij}\left( z\right) = \left\{ {{\delta }^{ij} - \frac{{z }_{i}{z }_{j}}{{\left| z \right| }^{2}}}\right\} {\left| z \right| }^{\gamma + 2},\;\gamma = - 3. \end{align}\] We also call very soft potentials in case \(-3\leq \gamma<-2\) including the Coulomb potentials given above. Since their derivation, the Boltzmann and Landau equations have been closely linked to fluid dynamics, especially the Euler and Navier-Stokes equations. Early attempts to derive fluid equations from kinetic theory trace back to Maxwell and Boltzmann, who used intuitive, somewhat ad hoc arguments. To formalize this, Hilbert introduced a systematic expansion method in 1912, later refined independently by Enskog (1916) and Chapman (1917); see the recent breakthrough [1] by Deng-Hani-Ma.
The Riemann problem, first studied in the 1860s for one-dimensional isentropic flow, is foundational to hyperbolic conservation laws. Its solution captures the local and global nonlinear structure of such systems. The Euler equations feature three basic waves: compressive shocks, expansive rarefactions, and contact discontinuities. The last one splits into entropy waves (discontinuous entropy) and vortex sheets (discontinuous shear velocity), both with a diffusive structure. Consequently, analyzing the hydrodynamic limit of the Boltzmann or Landau equation for a full Riemann solution — the superposition of all three wave patterns — remains a major mathematical challenge in kinetic theory; see [2] and references therein.
In this paper, we are interested in the Cauchy problem related to the basic wave patterns for 1 supplemented with the following initial condition \[\label{ini} f(0,x,\xi) = f_0(x,\xi).\tag{3}\] We assume that \(f_0(x,\xi)\) connects two different global Maxwellians at the far fields of the first spatial direction \(x_1 = \pm\infty\): \[\notag f_0(x,\xi) \to \frac{\rho_\pm}{(2\pi R \theta_\pm)^{3/2}} \exp\left( -\frac{|\xi - u_\pm|^2}{2 R \theta_\pm} \right) \quad \text{as}\quad x_1 \to \pm\infty,\] where \(\rho_\pm > 0\), \(\theta_\pm > 0\), and \(u_\pm = (u_{1\pm}, 0, 0)\) with \(u_{1\pm}\) are constants independent of \((x_2,x_3)\in \mathbb{T}^2\), and \(R>0\) is the gas constant. To the end, we study only the entropy wave for which the far-field data above satisfy \[\begin{align} \label{eqs1146se1461} \rho_-\neq\rho_+\qquad\qquad u_{1-}=u_{1+},\qquad\quad R\rho_+\theta_+=p_+=p_-=R\rho_-\theta_-. \end{align}\tag{4}\] Next, we shall start from the micro-macro decomposition of 1 .
It is well known that the collision invariants \({\tilde{\psi}}_{\alpha }\left( \xi \right)\) for the Landau collision operator 2 are given by \[\begin{align} \notag \tilde{\psi }_{0}\left( \xi \right) \equiv 1, \quad\quad\tilde{\psi }_{i}\left( \xi \right) \equiv {\xi }_{i},\;\text{ for }i = 1,2,3, \quad\quad \tilde{\psi }_{4}\left( \xi \right) \equiv \frac{1}{2}{\left| \xi \right| }^{2}, \end{align}\] satisfying \[\begin{align} \notag {\int }_{{\mathbb{R}}^{3}}\tilde{\psi }_{i}\left( \xi \right) Q\left( {f,f}\right) {d\xi } = 0,\text{ for }i = 0,1,2,3,4. \end{align}\] Motivated by [3], we decompose the solution of Landau equation 1 as \[\begin{align} \label{assumption-1} f\left( {t,x,\xi }\right) = M_{[\rho,u, \theta]}\left( {t,x,\xi }\right) + G\left( {t,x,\xi }\right) , \end{align}\tag{5}\] where the local Maxwellian \(M_{[\rho,u,\theta]}\) and \(G\) represent the macroscopic and microscopic component in the solution, respectively. Precisely, the local Maxwellian \(M_{[\rho,u,\theta]}\) is defined as \[\begin{align} \label{20254664614-1} M \equiv {M}_{\left[ \rho , u, \theta \right] }\left( t,x,\xi \right)\equiv \frac{\rho \left( t,x \right) }{\sqrt{\left( 2\pi R \theta \left( t,x\right) \right) ^{3}}} \exp{ \left( -\frac{\left| \xi - u \left( t,x \right) \right| ^{2}}{2R \theta \left( t,x\right)}\right)}, \end{align}\tag{6}\] in terms of five conserved quantities mass density \(\rho \left( {t,x}\right)\), momentum \(m\left( {t,x}\right) =\) \((\rho u) \left( {t,x}\right)\), and energy density \(\mathbb{E}\left( {t,x}\right) = \rho\left( {t,x}\right) \left( {e\left( {t,x}\right) + \frac{1}{2}{\left| u\left( t,x\right) \right| }^{2}}\right)\) given by \[\begin{align} \left\{ \begin{aligned} &\rho \left( {t,x}\right) \equiv {\int }_{{\mathbb{R}}^{3}}f\left( {t,x,\xi }\right) {d\xi }, \\ &{m}_{i}\left( {t,x}\right) \equiv {\int }_{{\mathbb{R}}^{3}}{\tilde{\psi} }_{i}\left( \xi \right) f\left( {t,x,\xi }\right) {d\xi },\text{ for }i = 1,2,3, \\ &\mathbb{E}\left( {t,x}\right) = \left[ \rho \left( e \left( {t,x}\right) + \frac{1}{2}{\left| u \left( t,x\right) \right| }^{2} \right) \right] \equiv {\int }_{{\mathbb{R}}^{3}}{\tilde{\psi} }_{4}\left( \xi \right) f\left( {t,x,\xi }\right) {d\xi }. \end{aligned} \right. \end{align}\] Here \(\theta \left( {t,x}\right)\) is the temperature which is related to the internal energy \(e\) by \(e = \frac{3}{2}{R\theta }\), and \(u \left( {t,x}\right)\) is the fluid velocity. In the sequel, the inner product of two functions \(h\) and \(g\) in \({L}^{2}( {\mathbb{R}}^{3}_\xi)\) or \({L}^{2}( \Omega\times \mathbb{R}^{3}_{\xi })\) with respect to an arbitrary Maxwellian \(\widetilde{M}\) is defined by \[\begin{align} \notag \langle h,g \rangle_{\widetilde{M}} \equiv {\int }_{{\mathbb{R}}^{3}}\frac{1}{\widetilde{M}}h\left( \xi \right) g\left( \xi \right) {d\xi },\quad {\left( h,g\right) _{\widetilde{M}} \equiv \int_{\Omega}\int_{\mathbb{R}^{3}}\frac{1}{\widetilde{M}}h\left( \xi \right) g\left( \xi \right) {d\xi dx}.} \end{align}\] If \(\widetilde{M}\) is taken as the local Maxwellian \(M\) associated with \(f(t,x,\xi)\) as in 6 , then the macroscopic \(L^2_\xi\) function space with respect to the corresponding inner product is spanned by the following five pairwise orthonormal functions \[\begin{align} \label{chi-1-1-1} \left\{ \begin{array}{l} {\chi }_{0}\left( \xi \right) \equiv \frac{1}{\sqrt{\rho }}M, \qquad\qquad\qquad\qquad {\chi }_{i}\left( \xi \right) \equiv \frac{{\xi }_{i} - {u}_{i}}{\sqrt{R\theta \rho }}M,\text{ for }i = 1,2,3, \\ {\chi }_{4}\left( \xi \right) \equiv \frac{1}{\sqrt{6\rho} }\left( {\frac{{\left| \xi -u\right| }^{2}}{R\theta } - 3}\right) M, \qquad \left\langle {{\chi }_{i},{\chi }_{j}}\right\rangle = {\delta }_{ij},\;i,j = 0,1,2,3,4. \end{array}\right. \end{align}\tag{7}\] Using these five base functions, we define the macroscopic projection \({P}_{0}\) and microscopic projection \({P}_{1}\) as follows: \[\begin{align} \label{2025-2025} {P}_{0}h \equiv \mathop{\sum }\limits_{{j = 0}}^{4}\left\langle {h,{\chi }_{j}}\right\rangle {\chi }_{j},\qquad{P}_{1}h \equiv h - {P}_{0}h. \end{align}\tag{8}\] The projections \({P}_{0}\) and \({P}_{1}\) are orthogonal and satisfy \[\begin{align} \notag {P}_{0}{P}_{0} = {P}_{0},\;{P}_{1}{P}_{1} = {P}_{1},\;{P}_{1}{P}_{0} = {P}_{0}{P}_{1} = 0. \end{align}\] A function \(h\left( \xi \right)\) is called microscopic if \[\begin{align} \notag {\int }_{{\mathbb{R}}^{3}}h\left( \xi \right) \tilde{\psi }_{i}\left( \xi \right) {d\xi } = 0,\text{ for }i = 0,1,2,3,4. \end{align}\] Under the decomposition 8 , the solution \(f\left( {t,x,\xi }\right)\) of the Landau equation 1 satisfies \[\begin{align} \notag {P}_{0}f = M,\qquad{P}_{1}f = G. \end{align}\] Moreover, the Landau equation 1 can be re-written as \[\begin{align} \notag \partial_t{\left(M + G\right) } + \xi \cdot \nabla_x{\left( M + G\right) } = Q\left( {M,G}\right) + Q\left( {G,M}\right) + Q\left( {G,G}\right) , \end{align}\] which is equivalent to the following fluid-type system for the macroscopic quantities of \(f\): \[\begin{align} \label{landau-1} \left\{\begin{array}{l} \partial_t \rho +\operatorname{div}_x(\rho u)=0, \\[2mm] \partial_t (\rho u)+\operatorname{div}_{x}(\rho u \otimes u)+\nabla_x p=-\int_{\mathbb{R}^3} \xi \otimes \xi \cdot \nabla_x G\, d \xi, \\[2mm] {\partial_t \big[\rho\big(e+\frac{|u|^{2}}{2}\big)\big]+\operatorname{div}_x\big[\rho u\big(e+\frac{|u|^{2}}{2}\big)+p u\big]=-\int_{\mathbb{R}^3} \frac{1}{2}|\xi|^{2} \xi \cdot \nabla_x G\, d \xi}, \end{array}\right. \end{align}\tag{9}\] coupled with the equation for the microscopic component \(G\): \[\begin{align} \label{equ-G} \partial_t{ G} + {P}_{1}\left( \xi \cdot\nabla_x G \right) + {P}_{1}\left(\xi \cdot\nabla_x M \right) = {L}_{M}G + Q\left( {G,G}\right) , \end{align}\tag{10}\] where \(p = {R\rho \theta }\), \[\begin{align} \label{G} G = {L}_{M}^{-1}\left( {{P}_{1}\left( \xi \cdot \nabla_x M\right) }\right) + {L}_{M}^{-1}\Pi, \end{align}\tag{11}\] and \[\begin{align} \label{Theta} \Pi = \partial_t {G} + {P}_{1}\left( \xi \cdot \nabla_x G\right) - Q\left( {G,G}\right) . \end{align}\tag{12}\] Here \({L}_{M}\) is the linearized collision operator with respect to the local Maxwellian \(M\): \[\begin{align} \label{2025466461446-2} {L}_{M}h = Q\left( {M,h}\right) + Q\left( {h,M}\right), \end{align}\tag{13}\] and the null space \(\mathcal{N}\) of \({L}_{M}\) is spanned by the macroscopic variables: \({\chi }_{j},\;j =\) \(0,1,2,3,4\) 7 . Plugging 11 into 9 , we have \[\label{landau-3} \begin{cases} \partial_t \rho +\operatorname{div}_{x}(\rho u)=0, \\[1mm] \partial_t (\rho u)+\operatorname{div}_{x}(\rho u \otimes u)+\nabla_{x} p \\ \qquad\qquad\qquad =-\int_{\mathbb{R}^3} \xi \otimes \xi \cdot \nabla_{x}\left(L_{M}^{-1}\left[P_{1}\left(\xi \cdot \nabla_{x} \mathbf{M}\right)\right]\right) \,d \xi-\int_{\mathbb{R}^3} \xi \otimes \xi \cdot \nabla_{x}\left( L_M^{-1}\Pi\right)\, d \xi, \\[1mm] {\partial_t \big[\rho\big(e+\frac{|u|^{2}}{2}\big)\big]+\operatorname{div}_{x}\big[\rho u\big(e+\frac{|u|^{2}}{2}\big)+p u\big]} \\ \qquad\qquad\qquad =-\int_{\mathbb{R}^3} \frac{1}{2}|\xi|^{2} \xi \cdot \nabla_{x}\left(L_{M}^{-1}\left[P_{1}\left(\xi \cdot \nabla_{x} \mathbf{M}\right)\right]\right)\,\, d \xi-\int_{\mathbb{R}^3} \frac{1}{2}|\xi|^{2} \xi \cdot\nabla_{x}\left( L_M^{-1}\Pi\right)\, d \xi. \end{cases}\tag{14}\] Furthermore, a direct calculation yields \[\begin{align} &-\int_{\mathbb{R}^3} \xi_{i} \xi \cdot \nabla_{x}\left(L_{M}^{-1}\left[P_{1}\left(\xi \cdot \nabla_{x} \mathbf{M}\right)\right]\right) d \xi=\sum_{j=1}^{3}\partial_{x_j}\left[\mu(\theta)\left(\partial_{x_j}u_{i }+\partial_{x_i}u_{j}-\frac{2}{3} \delta_{i j} \operatorname{div}_{x} u\right)\right]=:\sum_{j=1}^3\partial_{x_j}[\mu(\theta)S_{ij}],\tag{15}\\ &-\int_{\mathbb{R}^3} \frac{1}{2}|\xi|^{2} \xi \cdot \nabla_{x}\left(L_{M}^{-1}\left[P_{1}\left(\xi \cdot \nabla_{x} \mathbf{M}\right)\right]\right) d \xi =\text{div}_x\left(\kappa(\theta) \nabla_x \theta\right)+\text{div}_x\left\{\mu(\theta) u\cdot \mathbf{S}\right\},\tag{16} \end{align}\] where \(\mathbf{S}:=(S_{ij})_{1\leq i,j\leq 3}\in\mathbb{R}^{3\times3}\) in 16 is defined in 15 . Here, \(\mu(\theta)\) and \(\kappa(\theta)\) are to be introduced in subsection 7.1 using the Burnett functions in the Appendix. Therefore, 14 can also be written as \[\begin{align} \label{landau-4} \left\{ \begin{aligned} &\partial_t \rho +u \cdot \nabla_x \rho+\rho \operatorname{div}_x u=0, \\[-1mm] &\rho \partial_tu+\rho u \cdot \nabla_x u+\frac{2 }{3 }\theta \nabla_x \rho+\frac{2}{3}\rho \nabla_x \theta=\mu(\theta)\big(\Delta_x u+\frac{1}{3} \nabla_x \operatorname{div}_x u\big) \\[-1mm] &\qquad\;+\mu^{\prime}(\theta) \nabla_x \theta \cdot\big(\nabla_x u+(\nabla_x u)^{t}-\frac{2}{3}\mathbb{I}_1 \operatorname{div}_x u\big)-\int_{\mathbb{R}^3} \xi \otimes \xi \cdot \nabla_{x} (L_M^{-1}\Pi) d \xi, \\[-1mm] &\rho\partial_t\theta+\rho u \cdot \nabla_x \theta+\frac{2}{3} \rho\theta \operatorname{div}_x u=\kappa(\theta) \Delta_x \theta+{\mu(\theta)}\Big[\frac{\left(\nabla_x u+(\nabla_x u)^{t}\right)^{2}}{2}-\frac{2}{3}(\operatorname{div}_x u)^{2}\Big] \\[-1mm] &\qquad\;+{\kappa^{\prime}(\theta)}|\nabla_x \theta|^{2}-\int_{\mathbb{R}^3} \frac{1}{2}|\xi|^{2} \xi \cdot \nabla_{x} (L_M^{-1}\Pi) d \xi+u \cdot \int_{\mathbb{R}^3} \xi \otimes \xi \cdot \nabla_{x} (L_M^{-1}\Pi) d \xi . \end{aligned}\right. \end{align}\tag{17}\] To the end we have normalized the gas constant \(R\) to be \(\frac{2}{3}\) for convenience, so \(e=\theta\) and \(p=\frac{2}{3}\rho\theta.\)
The macroscopic system 14 can be regarded as the corresponding compressible Navier-Stokes equations with some source terms coming from non-fluid components. We construct the ansatz for the entropy wave profile of the Landau equation as follows. It is worth giving an explicit expression of the entropy wave for the hyperbolic conservation laws. Corresponding to 9 for \(x=(x_1,x_2,x_3) \in \Omega= \mathbb{R}\times\mathbb{T}^2\), the 1-D Euler equations depending only one the first spatial direction \(x_1\in \mathbb{R}\) read as \[\begin{align} \label{eqs36} \left\{\begin{aligned} &\partial_t\rho+\partial_{x_1}(\rho u_1)=0,\\ &\partial_t(\rho u_1)+\partial_{x_1}(\rho u_1^2+p)=0,\;\;\;x_1\in\mathbb{R},\;t>0,\\ & \partial_t\mathbb{E}+\partial_{x_1}\left( \mathbb{E}u_1+pu_1 \right)=0, \end{aligned}\right. \end{align}\tag{18}\] with the initial data: \[\begin{align} (\rho, u_1,\theta)(0,x_1)= \begin{cases} (\rho_-,0,\theta_-),\;\;x_1<0,\\ (\rho_+,0,\theta_+),\;\;x_1>0. \end{cases} \end{align}\] Using the conserved quantities, 18 can be re-written as \[\begin{align} \label{eq-rmns} &\left\{\begin{array}{l} \partial_t\rho+\partial_{x_1} m_1=0,\\ \partial_t{m_1}+\frac{2}{3}\partial_{x_1}\left({\mathbb{E}}+\frac{{m}_{1}^2}{\rho}\right)=0, \\ \partial_t{\mathbb{E}}+\partial_{x_1}\left(\frac{5{m}_1 {\mathbb{E}}}{3\rho}-\frac{{m}_1^{3}}{3 \rho^{2}}\right)=0. \end{array}\right. \end{align}\tag{19}\] Studies for entropy waves are somehow very subtle because of their linear degeneracy. Accordingly, there are some structural conditions that need to be considered when studying entropy waves. The Jacobi matrix of the flux for 19 is \[\begin{align} A\left(\rho, m_1, \mathbb{E}\right)=\left(\begin{array}{ccc} 0 & 1 & 0 \\ -\frac{2}{3}\frac{m_1^2}{\rho^2} & \frac{4}{3}\frac{m_1}{ \rho} & \frac{2}{3} \\ -\frac{5 m_1 \mathbb{E}}{ 3\rho^2}+\frac{2m_1^3}{ 3\rho^3} & \frac{5 \mathbb{E}}{3 \rho}-\frac{m_1^2}{ \rho^2} & \frac{5 m_1}{ 3\rho} \end{array}\right), \end{align}\] which has the second eigenvalue \(\lambda_2=\frac{m_1}{\rho}\) with the corresponding left and right eigenvectors given by \[\begin{align} \label{NSlr} r_2=\left(1,\frac{m_1}{\rho},\frac{m^2_1}{2\rho^2}\right)^t,\qquad\qquad l_2=\left(\frac{5\mathbb{E}}{3\rho}-\frac{4m_1^2}{3\rho^2},\frac{2m_1}{3\rho},-1\right). \end{align}\tag{20}\] From direct computations, 20 further gives that \[\begin{align} \label{NSlr46veri} \nabla l_2\cdot r_2\neq0,\qquad\nabla r_2\cdot r_2=0. \end{align}\tag{21}\] However, the classical left and right structural conditions as in [4] are given as \[\begin{align} \label{SC} \nabla l_2\cdot r_2=0, \quad \text{and} \quad \nabla r_2\cdot r_2=0,\quad\text{ respectively. } \end{align}\tag{22}\] Therefore, from 21 the left structural condition is violated for the Euler equations 19 in Eulerian coordinates. The presence of two-sides structural conditions has enhanced the decay of lower-order terms [4]–[7]. For more details, see difficulties and strategies to be specified in subsection 1.6 later on.
As in 4 , the entropy wave for 19 satisfies \[\begin{align} \label{eqs33} \frac{2}{3}\rho_+\theta_+=p_+=p_-=\frac{2}{3}\rho_-\theta_-,\qquad u_{1+}=u_{1-}. \end{align}\tag{23}\] We assume that the strength of wave \(\delta:=|\theta_+-\theta_-|\) is small. As in [6], [8], let \(\rho^{cd}(\frac{x_1}{\sqrt{1+t}})\) be the self-similar solution of the following nonlinear diffusion equation \[\begin{align} \label{eqs1446cdw} \left\{\begin{aligned} &\partial_t\rho^{cd}=\partial_{x_1}\left(\frac{3\tilde{\kappa}(\rho^{cd})}{5}\frac{\partial_{x_1}\rho^{cd}}{\rho^{cd}}\right), \;\;\tilde{\kappa}(\rho^{cd})=\kappa(\frac{3p_{+}}{2 \rho^{cd}}) ,\\ &\rho^{cd}(t,+\infty)=\rho_{+},\qquad\qquad \rho^{cd}(t,-\infty)=\rho_{-}. \end{aligned}\right. \end{align}\tag{24}\] It then holds \[\begin{align} \label{eqs27} \left\{\begin{aligned} &\left|(1+t)^{\frac{1}{2}}\partial_{x_1}\rho^{cd}\right|=O(\delta)e^{-\frac{d x_1^2}{1+t}},\;x_1\in \mathbb{R}, \\ &|\rho^{cd}-\rho_{+}|=O(\delta) e^{-\frac{d x_1^2}{1+t}},\;x_1\geq 0; \quad |\rho^{cd}-\rho_{-}|=O(\delta) e^{-\frac{d x_1^2}{1+t}},\;x_1\leq 0, \end{aligned}\right. \end{align}\tag{25}\] where \(d>0\) is a constant.
In terms of the self-similar diffusion profile \(\rho^{cd}(\cdot)\) 24 , we can further construct the viscous entropy wave profiles \(\check{\rho},\check{u},\check{\theta}\) corresponding to the compressible Euler fluid part of 14 for the Landau equation: \[\label{efvew} \begin{align} \check{\rho}=\rho^{cd},\quad \check{u}=\left(-\frac{3\tilde{\kappa}(\check{\rho})\partial_{x_1}\check{\rho}}{5\check{\rho}^2},0,0\right)^t, \quad \check{\theta}=\frac{3p_+}{2\check{\rho}}- \frac{\check{u}_1^2}{2},\quad\frac{2}{3}\check{\rho}\check\theta=\check p=p_+-\frac{1}{3} \check{\rho}\check{u}_1^2. \end{align}\tag{26}\] Moreover, the associated 1-D local Maxwellian corresponding to 26 above is defined as \[\begin{align} \label{2026-5-9-1} {M}_{[ \check{\rho}, \check{u}, \check\theta ] }\left( t,x_1,\xi \right)\equiv \frac{\check{\rho}\left( t,x_1 \right) }{\sqrt{\left( 2\pi R \check\theta \left( t,x_1\right) \right) ^{3}}} \exp{ \left( -\frac{\left| \xi - \check{u}\left( t,x_1 \right) \right| ^{2}}{2R \check\theta \left( t,x_1\right)}\right)} . \end{align}\tag{27}\]
Remark 1. For convenience of the proof regarding the dynamical stability of the planar entropy wave 27 , throughout the paper we fix a normalized global Maxwellian with the fluid constant state \((1,0,\frac{3}{2})\): \[\begin{align} \label{2026-5-9-2} \mu=M_{[1,0,\frac{3}{2}]}=(2\pi)^{-\frac{3}{2}}e^{-\frac{\left\lvert\xi\right\rvert^2}{2}} \end{align}\qquad{(1)}\] as a reference equilibrium state, and choose both the far-field data 23 to be close enough to the constant state \((1,0,\frac{3}{2})\) such that the viscous contact wave in 26 further satisfies that \[\begin{align} \left\{ \begin{aligned} &\sup_{t\ge0,x_1\in\mathbb{R}}\{ \left\lvert\check{\rho}(t,x_1)-1\right\rvert+\left\lvert\check{u}(t,x_1)\right\rvert+|\check \theta(t,x_1)-\frac{3}{2}| \} \le3\delta,\\ &\frac{1}{2} \sup_{t\ge0,x_1\in\mathbb{R}}\check\theta(t,x_1)\le \frac{3}{2} \le \inf_{t\ge0,x_1\in\mathbb{R}} \check\theta(t,x_1), \end{aligned}\right. \end{align}\] where \(\delta>0\) is the wave strength small enough.
Setting \(\check{m}_i=\check{\rho}\check{u}_i\) and \(\check{\mathbb{E}}=\check{\rho}\left(\check{\theta}+\frac{\left\lvert\check{u}\right\rvert^2}{2}\right)\), by direct calculations, it follows from 26 that \[\begin{align} \label{contact-wave-check} \left\{ \begin{aligned} &\partial_t{\check{\rho}} + \partial_{x_1}{\check{m}}_{1 } = 0 \\ &\partial_t{\check{m}}_{1} + \partial_{x_1}{\left( \frac{{\check{m}}_{1}^{2}}{\check{\rho}} + \check{p}\right) } = \frac{4}{3} \partial_{x_1}{\left( \mu \left( \check{\theta} \right) \partial_{x_1} \check{u}_{1}\right) } +\partial_{x_1}\mathcal{R}_{1 }, \\ &\partial_t{\check{m}}_{i} + \partial_{x_1}{\left( \frac{{\check{m}}_{1}{\check{m}}_{i}}{\check{\rho}}\right) } = \partial_{x_1}{\left( \mu \left( \check{\theta} \right) {\partial_{x_1}\check{u}}_{i}\right) } ,\;i = 2,3, \\ &\partial_t{\check{\mathbb{E}}} + \partial_{x_1}{\left( \frac{{\check{m}}_{1}\check{\mathbb{E}}}{\check{\rho}} + \frac{{\check{m}}_{1}}{\check{\rho}}\check{p}\right) } = \partial_{x_1}{\left( \kappa \left( \check{\theta} \right) {\partial_{x_1}\check{\theta} }\right) } + \frac{4}{3}\partial_{x_1}{\left( \mu \left( \check{\theta} \right) {\check{u}}_{1}\partial_{x_1} \check{u}_{1}\right) }+ \mathop{\sum }\limits_{{i = 2}}^{3} \partial_{x_1}{ \left( \mu \left( \check{\theta} \right) {\check{u}}_{i}\partial_{x_1} \check{u}_{i}\right) }+\partial_{x_1}\mathcal{R}_{4} , \end{aligned}\right. \end{align}\tag{28}\] where \[\begin{align} &\mathcal{R}_1=-\frac{3\tilde{\kappa}(\check{\rho})\partial_t\check{\rho}}{5\check{\rho}}+\frac{2}{3}\check{\rho}\check{u}_1^2-\frac{4}{3}\mu(\check \theta) \partial_{x_1}\check{u}_{1}=O(\delta)(1+t)^{-1}e^{-\frac{c x_1^2}{1+t}},\\ &\mathcal{R}_4=\left[\kappa(\check{\theta})-\kappa(\frac{3p_{+}}{2 \check{\rho}}) \right]\frac{3p_{+}\partial_{x_1}\check{\rho}}{2 \check{\rho}^2}-\frac{1}{3}\check{\rho}\check{u}_1^3-\left(\frac{4}{3}\mu(\check{\theta})+\kappa(\check\theta)\right)\check{u}_1\partial_{x_1}\check{u}_{1}=O(\delta)(1+t)^{-\frac{3}{2}}e^{-\frac{c x_1^2}{1+t}}, \end{align}\] with some constant \(c>0\). For convenience, we denote the conserved quantities by \[\begin{align} \label{conserved-law} \mathbf{U}=(\rho,m_1,m_2,m_3,\mathbb{E})^{t},\;\;\;\check{\mathbf{U}}=(\check{\rho},\check{m}_1, \check{m}_2,\check{m}_3,\check{\mathbb{E}})^{t},\;\;\; \mathbf{U}^{\mathcal{\#}}=(\rho,m_1,\mathbb{E})^{t}, \;\;\;\check{\mathbf{U}}^{\#}=(\check{\rho},\check{m}_1, \check{\mathbb{E}})^{t}. \end{align}\tag{29}\]
The Landau equation with Coulomb interactions is central to plasma physics and has been widely studied. For the spatially inhomogeneous case, key results include: global weak solutions (with defect measure) by Lions [9] and Villani [10], [11]; the grazing collision limit from the non-cutoff Boltzmann equation by Desvillettes [12] and Alexandre–Villani [13]; spectral analysis of the linearized equation by Degond–Lemou [14]; and a derivation from particle systems by Bobylev–Pulvirenti–Saffirio [15].
Directly related work concerns solutions near global Maxwellians: Guo [16] established global existence on the torus, with decay rates given by Strain–Guo [17], [18]. For the Vlasov–Poisson–Landau system, global solutions near Maxwellians were obtained by Guo [19] on the torus and Strain–Zhu [20] on \(\mathbb{R}^3\) (see also [21]–[23]). Other contributions include [24]–[31].
A fundamental open problem is the global existence and behavior of solutions to the Cauchy problem 1 with initial data of small total variation connecting two different Maxwellians at the far fields (cf. [32], [33]). Based on fluid dynamic limits, the long-time profile is expected to be a nonlinear wave pattern—shock, rarefaction, or contact wave—or their superposition (cf. [34]), an expectation motivated by pointwise estimates via the Green’s function method [35].
For the cutoff Boltzmann equation, wave patterns are well-studied: shock profiles [36]–[38]; contact wave stability and hydrodynamics [6], [8], [39]; rarefaction wave stability [40]–[42]; composite wave stability [43]; and patterns with self-consistent forces [44], [45]. For viscous conservation laws (e.g., compressible Navier-Stokes), contact wave stability was established earlier [4], [5], [46]–[51] (see survey [52]).
In contrast, the known results for the non-cutoff Boltzmann and Landau equations are quite few, with all previous ones concentrated on only the one-dimensional case over \(\mathbb{R}\). Recent progress includes the stability of rarefaction waves for the Landau equation [53] and the analysis of its small-Knudsen limit [54]. The stability of viscous contact waves was established in [7], with its corresponding small-Knudsen limit further investigated in [55]. Meanwhile, the existence of shock profiles has been proven in [56], although their nonlinear stability and behavior in the small-Knudsen limit remain open questions.
Several important issues merit further investigation. The result in [7], while establishing stability, was only obtained in 1-d Lagrangian coordinates and does not provide an explicit decay rate—a property that is well-understood for the corresponding Navier-Stokes equations. This problem is particularly challenging for the Landau equation with very soft potentials (\(\gamma<-2\)), where the linearized operator lacks a spectral gap, distinguishing it fundamentally from the fluid dynamic case. Furthermore, it should be noted that the known decay-rate results for the Landau equation (e.g., [22], [23]) are obtained in a multi-dimensional whole-space setting. In such multi-dimensional frameworks, the analytical difficulty associated with the spectral gap is generally less severe.
The current work aims at studying the nonlinear stability of viscous entropy waves for the Landau equation with Coulomb interactions in the three-dimensional infinite channel domain \(\Omega=\mathbb{R}\times\mathbb{T}^2\). We are able to obtain (i) the existence of a unique global solution near a local Maxwellian whose fluid components are viscous entropy wave profiles, and (ii) the time-asymptotic stability and optimal decay rate of this solution. This is the first result of such kind for the Landau equation. The rough version of the main result will be given in Theorem 2; we also refer to Remarks 3, 4 and 5 for more discussions on the result. The precise statement will be given in Theorem 8 together with the more detailed version Theorem 9 in the next section.
Corresponding to the global Maxwellian \(\mu\) in ?? , the Landau collision frequency is \[\label{14634} \sigma^{ij}(\xi):=\tilde{\phi}^{ij}\ast \mu(\xi)=\int_{{\mathbb{R}}^3}\tilde{\phi}^{ij}(\xi-\xi')\mu(\xi')\,d\xi'.\tag{30}\] We remark that \(\sigma^{ij}(\xi)\) is a positive definite symmetric matrix. We first denote the weight function \[\begin{align} \label{14632} \omega=\omega(\beta)(\xi):=\langle\xi\rangle^{(l-|\beta|)}e^{q\langle\xi\rangle^2},\quad l\geq|\beta|,\quad\langle\xi\rangle=\sqrt{1+|\xi|^2},\quad 0<q<1. \end{align}\tag{31}\] We denote the weighted \(L^2\) norms as \[|\omega g|^2_{2}:= \int_{{\mathbb{R}}^3}\omega^2 g^2\,d\xi,\;\;\;\|\omega g\|_2^2:={ \int_{\mathbb{R}\times\mathbb{T}^2}\int_{{\mathbb{R}}^3}}\omega^2 g^2\,d\xi dx.\] In terms of linearization of the nonlinear Landau operator around \(\mu\) ?? (cf. [16]), with 30 we define the weighted dissipative norms: \[| g|^2_{\sigma,\omega}:=\sum_{i,j=1}^3\int_{{\mathbb{R}}^3}\omega^{2}[\sigma^{ij}\partial_{\xi_i} g\partial_{\xi_j} g+\sigma^{ij}\frac{\xi_i}{2}\frac{\xi_j}{2} g^2]\,d\xi,\quad and\quad \;\| g\|_{\sigma,\omega}:= \|| g|_{\sigma,\omega} \|_{L_x^2}.\] And let \(| g|_{\sigma}=| g|_{\sigma,1}\) and \(\| g\|_{\sigma}=\| g\|_{\sigma,1}\). From [16] and [18], one has \[\label{14635} | g|_\sigma\approx |\langle \xi\rangle^{-\frac{1}{2}} g|_2+\Big|\langle \xi\rangle^{-\frac{3}{2}}\frac{\xi}{|\xi|}\cdot\nabla_\xi g\Big|_2+\Big|\langle \xi\rangle^{-\frac{1}{2}}\frac{\xi}{|\xi|}\times \nabla_\xi g\Big|_2.\tag{32}\] We also denote \[\|\partial^\alpha_\beta g\|^2_{2,\omega}:=\int_{\mathbb{R}\times\mathbb{T}^2}\int_{{\mathbb{R}}^3}\omega^{2}(\beta)|\partial^\alpha_\beta g(x,\xi)|^2\, d\xi dx,\] and \[\|\partial^\alpha_\beta g\|^2_{\sigma,\omega}:=\sum_{i,j=1}^3\int_{\mathbb{R}\times\mathbb{T}^2}\int_{{\mathbb{R}}^3}\omega^{2}(\beta)[\sigma^{ij}\partial_{\xi_i}\partial^\alpha_\beta g(x,\xi)\partial_{\xi_j}\partial^\alpha_\beta g(x,\xi)+\sigma^{ij}\frac{\xi_i}{2}\frac{\xi_j}{2}|\partial^\alpha_\beta g(x,\xi)|^2]\, d\xi dx.\]
Theorem 2 (Rough version). Let \(\Omega=\mathbb{R}\times\mathbb{T}^2\). The Cauchy problem 1 and 3 admits a unique global-in-time solution \(f=f(t,x,\xi)\geq 0\) satisfying \[\begin{align} \left\lVert\frac{f-{M}_{[\check{\rho},\check{u},\check{\theta}]}}{\sqrt{\mu}}\right\rVert_{L^{\infty}_{x}(\Omega)L^2_\xi(\mathbb{R}^3)}\leq C\tilde{\delta} (1+t)^{-\frac{1}{2}}, \label{decay} \qquad \left\lVert\frac{(f-{M}_{[\check{\rho},\check{u},\check{\theta}]})_{\neq}}{\sqrt{\mu}}\right\rVert_{L_x^\infty(\Omega) L_\xi^2(\mathbb{R}^3)}^2 \leq C \tilde{\delta} e^{-c t^{\frac{2}{3}}}. \end{align}\qquad{(2)}\] where the non-zero mode \(f_{\neq}\) is defined by \(f_{\neq}:=f-\int_{\mathbb{T}^2}fdx_2d_3\), \({M}_{[\check{\rho},\check{u},\check{\theta}]}\) and \(\mu\) are defined in 27 and ?? , respectively, and \(\tilde{\delta}\) is a small constant associated with the wave strength and the appropriate initial data \(f_0(x,\xi)\) near \(\mu\).
Remark 3. To the best of our knowledge, there have been no existing results addressing time-decay rate of solutions around entropy waves for the Landau equation in the spatial domain either the infinite 3-D channel \(\Omega=\mathbb{R}\times\mathbb{T}^2\) or 1-D whole line \(\Omega=\mathbb{R}\) with slab symmetry. Indeed, even the stability result on high-dimensional planar waves for the Landau equation is not yet known so far. Therefore, Theorem 2 above provides the first result of such kind and also resolves the question on rate of convergence left open in [7]. Moreover, our proof of Theorem 2 furnishes a robust framework for analyzing the time-decay behavior of the Landau equation near a local Maxwellian.
Remark 4. In this paper, we consider the spatial domain \(\mathbb{R}\times\mathbb{T}^2\). The analysis in this setting presents two main challenges: (i) It is highly nontrivial to establish stability results with an explicit time-decay rate for the Landau equation with very soft potentials \(\gamma<-2\). Indeed, the methods used in existing works, such as [22], [23], may not be directly adapted to the one-dimensional problem in an effective manner. (ii) Studying the planar entropy wave in \(\mathbb{R}\times\mathbb{T}^2\) introduces fundamental analytical difficulties that are distinct from those arising in the purely one-dimensional setting, cf. [7].
Remark 5. Regarding the rate of convergence of solutions toward entropy waves in ?? , the polynomial decay rate \((1+t)^{-\frac{1}{2}}\) is optimal under generic perturbations, due to the presence of diffusion waves propagating along the one-dimensional unbounded direction \(x_1\in\mathbb{R}\). Moreover, for non-zero modes we obtain the stretched exponential decay \(\exp(-c t^{\frac{2}{3}})\) corresponding to Coulomb interaction potentials \(\gamma=-3\). As far as we know, this is the first result of exponential decay of perturbations in non-zero modes near a local Maxwellian.
The main difficulties of this work lie in the following aspects. First, we consider the feature of the entropy waves. Previous studies often relied on the fact that entropy waves satisfy certain structural conditions 22 at the macroscopic level, cf. [4]–[7]. The structural conditions lead to cancellation in lower order terms, resulting in the following form \[\begin{align} \notag \int_{\Omega}(1+t)^{-1}e^{-\frac{cx^2_1}{1+t}}|(\rho-\check{\rho},u-\check{u},\theta-\check{\theta})|^2dx. \end{align}\] However, those structural conditions are derived from one-dimensional models formulated in Lagrangian coordinates, and they generally do not hold in higher dimensions. Actually, only the following form with the slower time decay coefficient can be expected \[\begin{align} \notag \int_{\Omega}(1+t)^{-\frac{1}{2}}e^{-\frac{cx^2_1}{1+t}}|(\rho-\check{\rho},u-\check{u},\theta-\check{\theta})|^2dx. \end{align}\] In this work, we introduce a novel transformation under which the perturbed equations in multiple dimensions can satisfy the structural condition in a suitable sense.
Second, for the Landau operator with Coulomb interactions, the lack of spectral gap leads to very weak dissipation, i.e., \(\left\lvert g\right\rvert_{\sigma}\) 32 . Specifically, the dissipation induced by the linearized collision operator is not strong enough to control the \(L^2\)-norm \(\left\lvert g\right\rvert_{L^2}\). Moreover, it is important to note that the local Maxwellian \(M_{[\check{\rho},\check{u},\check \theta]}\) 27 associated with the entropy wave is not an exact solution to the underlying equation, and the corresponding error \(\bar{\mathcal{R}}\) 38 is not well-controlled. The combination of weak dissipation and slowly decaying errors makes it extremely challenging to establish asymptotic stability of planar entropy wave, and it has remained an outstanding open problem for obtaining optimal decay rates, cf. [7], [54], [57].
To overcome these challenges, we develop the following strategies.
Macro-micro decomposition for Landau equation around the entropy wave.
By decomposing the solution of the Landau equation 1 into its macroscopic and microscopic components, we are able to take advantage of the desirable properties of the entropy wave in the macroscopic equations.
The time-velocity interpolation technique sacrifices time decay in exchange for stronger dissipation.
In this paper, due to lack of spectral gap, we use a time-velocity interpolation technique as in Lemma 30 introduced by Strain-Guo [17], [18] to obtain stronger dissipation, i.e. \[\begin{align} \left\lvert\partial^{\alpha} f\right\rvert_2^2 \leq \upsilon(1+t)^{\epsilon} \left\lvert\partial^{\alpha} f\right\rvert_{\sigma}^2+e^{-\frac{q}{4}\upsilon^2( 1+t )^{{2\epsilon}}} \left\lvert e^{\frac{q}{8}\langle\xi\rangle^2} \partial^{\alpha} f\right\rvert_2^2. \end{align}\] but at the cost of a \((1+t)^{\epsilon}\) loss in the decay rate. Unfortunately, previous results [6] imply that the growth of energy for anti-derivatives is \((1+t)^{1/2}\) due to the error terms \(\bar{\mathcal{R}}\) 38 , and thus the decay of the original energy should be at least \((1+t)^{-1/2}\) to close the a priori assumptions. However, heat dissipation indicates that the original energy decays faster than its anti-derivative counterpart by at most a factor of \((1+t)^{-1}\). Therefore, to afford the additional \((1+t)^{\epsilon}\) decay loss introduced by the time-velocity interpolation technique, we need to remove these error terms \(\bar{\mathcal{R}}\) 38 .
Construct diffusion waves and coupled diffusion waves.
Under generic perturbations, solutions to the macroscopic equations 14 generate diffusion waves \(\Theta_i\) 34 , which correspond to the first-order expansion of the macroscopic fluid system; see [58], [59]. In general, the background solution \(\bar{\mathbf{U}}:=(\bar{\rho},\bar{m},\bar{\mathbb{E}})=(\check{\rho},\check{m},\check{\mathbb{E}})+\Theta_i\), coupled to the entropy wave \((\check{\rho},\check{m},\check{\mathbb{E}})\) 28 and diffusion wave \(\Theta_i\) 34 , satisfies the system (see 37 and 43 for details) \[\begin{align} \partial_t\bar\mathbf{U}+ \partial_{x_1}F(\bar \mathbf{U})=\partial_{x_1}\big(\mathcal{B}(\bar{\mathbf{U}})\partial_{x_1}\bar{\mathbf{U}}\big)+\partial_{x_1}\bar{\mathcal{R}}. \end{align}\] These diffusion waves \(\Theta_i\) exhibit insufficient time decay of the spatial \(L^2\)-norm, making it difficult to control the induced error terms \(\bar{\mathcal{R}}\) 38 (see the reason for strategy ([2026-5-14])). In this paper, we construct a family of coupled diffusion waves \(\Xi_i\) by exploiting specific features of the entropy wave to cancel the slowly decaying error terms \(\bar{\mathcal{R}}\) (see ?? for details) \[\begin{align} \notag \quad\qquad\left\{ \begin{aligned} &\partial_t\Xi+\partial_{x_1}(F'(\bar{\mathbf{U}})\Xi)=\partial_{x_1}\big(\mathcal{B}(\bar{\mathbf{U}})\partial_{x_1}\Xi\big)-\frac{1}{2}\partial_{x_1}(\Xi^{t}F''(\bar{\mathbf{U}})\Xi)+\partial_{x_1}\big(\Xi^{t}\mathcal{B}{'}(\bar{\mathbf{U}})\partial_{x_1}\bar{\mathbf{U}}\big) -\partial_{x_1}\bar{\mathcal{R}}-\partial_{x_1}\mathcal{R}_{G}, \\ &\Xi|_{t=0}=0, \end{aligned}\right. \end{align}\] where \(\mathcal{R}_G\) is the slowly decaying term induced by the microscopic parts. To obtain optimal decay rates for derivatives of all orders of the coupled diffusion waves \(\Xi_i\), we diagonalize the system 78 , which is derived from ?? . In the diagonalized system 79 , we observe that different components propagate at distinct speeds. Based on this feature, we find out a new dissipation mechanism 85 . This enables us to effectively control the slowly decaying terms in 79 and achieve the optimal decay rates.
The transformations to ensure structural conditions hold.
For the new ansatz \((\tilde{\rho},\tilde{m},\tilde{\mathbb{E}})\) (see 50 ) formed by coupling \(\bar \mathbf{U}\) and the coupled diffusion wave \(\Xi_i\), we study the perturbation in an integrated system with \[\begin{align} (\Phi,\Psi,H)=\int_{-\infty}^{x_1}\int_{\mathbb{T}^2}(\rho-\tilde{\rho},m-\tilde{m},\mathbb{E}-\tilde{\mathbb{E}})(t,y)dy'dy_1, \end{align}\] which presents a possibility to obtain the decay rate. Next, we apply the following transformation\[\begin{align} \notag &\check{\Phi}=\tilde{\theta}\Phi,\quad \check{\Psi}=\Psi,\quad \check{H}=H-\check{\Phi}, \end{align}\] where \(\tilde{\theta}\) is a non-conserved quantity of temperature for the new ansatz defined in 53 . Through this transformation, the new system is simplified a lot and satisfies both the two structural conditions, see 144 , i.e., neither \[\int_{\mathbb{R}} (1+t)^{-\frac{1}{2}}e^{-\frac{cx_1^2}{1+t}}(\check{\Phi}^2+|\check{\Psi}|^2+\check{H}^2)dx_1\] nor \[\int_{\mathbb{R}} (1+t)^{-\frac{1}{2}}e^{-\frac{cx_1^2}{1+t}}[(\partial_{x_1}\check{\Phi})^2+|\partial_{x_1}\check{\Psi}|^2+(\partial_{x_1}\check{H})^2]dx_1\] appears in the \(L^2\)-estimates of the anti-derivative itself and its first derivative, respectively. Unfortunately, the structural conditions 22 are once again not met in the derivative system \[(\partial_{x_1}^2\check{\Phi},\partial_{x_1}^2\check{\Psi},\partial_{x_1}^2\check{H}).\] This is also why it is so difficult to obtain the optimal decay rate for contact discontinuities. In this paper, we find a new transformation in the derivative level, i.e., we study the following quantities \[\begin{align} \notag \left(\frac{2}{3}\tilde{\theta}\partial_{x_1}^2\check{\Phi},\partial_{x_1}\left(\tilde{\theta}\partial_{x_1}\check{\Psi}_{1}\right), \partial_{x_1}^2\check{\Psi}_{2},\partial_{x_1}^2\check{\Psi}_{3}, \; \tilde{\theta}\partial_{x_1}^2\check{H}\right). \end{align}\] Several cancellations are achieved under this transformation and thus the optimal decay rates are available, see Remark 19.
The stretched exponential decay for the non-zero mode.
For non-zero modes, we observe that the Poincaré’s inequality is applicable. By utilizing this inequality and combining the small strength of the entropy wave, the low-order terms arising in the flux can be effectively controlled through dissipation. Furthermore, for the non-zero modes, we observe that \[\frac{(f-M_{[\check{\rho},\check{u},\check{\theta}]})_{\neq}}{\sqrt{\mu}}=\frac{(f-\mu)_{\neq}}{\sqrt{\mu}}.\] To obtain the stretched exponential decay for these modes, we perturb the Landau equation around the global Maxwellian \(\mu\), thereby deriving dissipation estimates for the non-zero modes of the macroscopic part. Then, combining these estimates with the \(\xi\)-derivative and the exponentially weighted estimates for the microscopic part, the Poincaré’s inequality, and the time-velocity interpolation technique Lemma 30 establishes the stretched exponential decay for the non-zero modes, see Theorem 31.
The delicate energy structure to obtain the optimal polynomial decay rate.
Since the slowly decaying error terms are eliminated by constructing coupled diffusion waves, and the structural conditions are restored, we obtain the optimal decay rate by deriving the following differential inequalities \[\begin{align} &\frac{d}{dt}\mathcal{E}_1 + \mathcal{D}_1\leq C\check{\delta}(1+t)^{-1}\mathcal{E}_1 + C\bar{\delta}(1+t)^{-\frac{3}{2}}, \\ &\frac{d}{dt}\mathcal{E}_2 + \mathcal{D}_2\leq C\check{\delta}\left[(1+t)^{-1}\mathcal{D}_1+(1+t)^{-2}\mathcal{E}_1 \right]+ C\bar{\delta}(1+t)^{-\frac{3}{2}}, \\ &\frac{d}{dt}\tilde{\mathcal{E}}_2 +\tilde{\mathcal{D}}_2 \leq C\check{\delta}\left[(1+t)^{-1}\mathcal{D}_1+(1+t)^{-2}\mathcal{E}_1\right]+C\bar{\delta}(1+t)^{-\frac{5}{2}},\\ &\frac{d}{dt}\mathcal{E}_3 + \mathcal{D}_3\leq C\check{\delta}\left[(1+t)^{-1}\mathcal{D}_2+(1+t)^{-2}\mathcal{D}_1+(1+t)^{-3}\mathcal{E}_1 \right]+ C\bar{\delta}(1+t)^{-\frac{5}{2}}, \end{align}\] where \(\mathcal{E}_{(1,2),\omega}\), \(\mathcal{E}_{1,2,3}\) and \(\tilde{\mathcal{E}}_{2}\) are transient energies (see 56 –61 ), and \(\mathcal{D}_{(1,2,3),\omega}\), \(\mathcal{D}_{1,2,3}\) and \(\tilde{\mathcal{D}}_{2}\) are dissipative energies (see 62 –68 ), and \(\bar{\delta},\check{\delta}\) are small constants associated with wave strength and initial perturbation. It is noteworthy that the introduction of \(\tilde{\mathcal{E}}_{2}\) 60 and \(\tilde{\mathcal{D}}_{2}\) 67 effectively circumvents the impact of the time loss induced by the time-velocity interpolation technique on the optimal decay rate; specifically, we need to use the relation \[\begin{align} &\int_0^t(1+\tau) \tilde{\mathcal{D}}_2 d\tau\le C \bar{\delta}(1+t)^{\frac{1}{5}},\\ &\int_0^t (1+\tau)\mathcal{E}_3 d \tau \le C(1+t)^{\frac{1}{10}} \int_0^t(1+\tau) \tilde{\mathcal{D}}_2 d\tau +C\bar{\delta}. \end{align}\] For the case of \(\gamma\ge-2\), it is unnecessary to introduce \(\tilde{\mathcal{E}}_2\) and \(\tilde{\mathcal{D}}_2\). See Section 5 for all the details.
Throughout the paper we shall use \(\langle \cdot , \cdot \rangle\) to denote the standard \(L^{2}\) inner product in \(\mathbb{R}_{\xi}^{3}\) with its corresponding \(L^{2}\) norm \(|\cdot|_2\). We also use \(( \cdot , \cdot )\) to denote \(L^{2}\) inner product in \(\mathbb{R}_{x}\) or \(\mathbb{R}_{x}\times \mathbb{R}_{\xi}^{3}\) with its corresponding \(L^{2}\) norm \(\|\cdot\|_2\). Let nonnegative integer \(\alpha\) and \(\beta\) be multi indices \(\alpha=[\alpha_{0},\alpha_{1},\alpha_{2},\alpha_{3}]\) and \(\beta=[\beta_{1},\beta_{2},\beta_{3}]\), respectively. Denote \(\partial_{\beta}^{\alpha}=\partial_{t}^{\alpha_{0}}\partial_{x_1}^{\alpha_{1}}\partial_{x_2}^{\alpha_{2}}\partial_{x_3}^{\alpha_{3}} \partial_{\xi_{1}}^{\beta_{1}}\partial_{\xi_{2}}^{\beta_{2}}\partial_{\xi_{3}}^{\beta_{3}}\). If each component of \(\beta\) is not greater than the corresponding one of \(\overline{\beta}\), we use the standard notation \(\beta\leq\overline{\beta}\). And \(\beta<\overline{\beta}\) means that \(\beta\leq\overline{\beta}\) and \(|\beta|<|\overline{\beta}|\). \(C^{\bar\beta}_{\beta}\) is the usual binomial coefficient. Throughout the paper, generic positive constants are denoted by either \(c\) or \(C\), and \(c_{1}\), \(c_{2}\) or \(C_{1}\), \(C_{2}\) etc. are some given constants. The notation \(A\lesssim B\) is used to denote that there exists constant \(c_{0}>1\) such that \(A\leq c_{0}B\) and \(A\approx B\) is used to denote \(c_{0}^{-1}B\leq A\leq c_{0}B\). We define the zero and non-zero modes for an integrable function \(f\in \mathbb{R}\times \mathbb{T}^2\) \[\begin{align} \label{2025-11-10-4} \mathbf{D}_0f:=\mathring{f}:=\int_{\mathbb{T}^2}fdx_2dx_3,\qquad \mathbf{D}_{\neq}f:={f}_{\neq}:=f-\mathbf{D}_0f. \end{align}\tag{33}\]
The rest of the present paper is organized as follows. In Section 2, we construct the diffusion waves, the coupled diffusion waves, and the new ansatz, and state the main theorem. Then, in Section 3, we give the estimate of the coupled diffusion wave. In Section 4, we study the stability of the new ansatz. The optimal decay rate and the stretched exponential decay are obtained in Sections 5 and 6, respectively. Finally, in Section 7, we present some necessary technical lemmas.
Recall the definition of \(\mathbf{U},\check\mathbf{U},\mathbf{U}^{\#},\check\mathbf{U}^{\#}\) in 29 . In this paper, we are concerned with the general non-zero initial perturbation in \(\Omega:=\mathbb{R}\times\mathbb{T}^2\) with \(\mathbb{T}:={\mathbb{R}}/{\mathbb{Z}}\), that is, the integral \(\int_{\Omega}(\mathbf{U}-\check \mathbf{U})(0,x)dx\neq0\). Note that the extra initial mass is distributed along the \(x_1\)-direction. We consider the far-field Jacobi matrices for the flux of 1-d Navier-Stokes equations in Eulerian coordinates after taking \(\int_{\mathbb{T}^2}\)14 \(dx_2dx_3\) is \[\begin{align} A_\pm=\left(\begin{array}{ccc} 0&1&0\\ 0&0&\frac{2}{3}\\ 0&\frac{5}{3}\frac{\mathbb{E}_\pm}{\rho_\pm}&0 \end{array}\right), \end{align}\] corresponding to \(\mathbf{U}^{\#}_{+}\) and \(\mathbf{U}^{\#}_{-}\), respectively. It is easy to see that \(\lambda_1^-=-\sqrt{\frac{10\mathbb{E}_-}{9\rho_-}}\) is the first eigenvalue of \(A_-\) corresponding to \(r_1^-=\left(1,\lambda_1^-,\frac{3(\lambda_1^-)^2}{2}\right)^t\) and \(\lambda_3^+=\sqrt{\frac{10\mathbb{E}_+}{9\rho_+}}\) is the third eigenvalue of \(A_+\) corresponding to \(r_3^+=\left(1,\lambda_3^+,\frac{3(\lambda_3^+)^2}{2}\right)^t\). Since the three vectors \(r_1^-,\mathbf{U}^{\#}_{-}-\mathbf{U}^{\#}_{+}\) and \(r_3^+\) are linearly independent by strict hyperbolicity, we have the following identity in case of non-zero initial mass: \[\begin{align} \notag \int_{\Omega} (\mathbf{U}^{\#}-\check \mathbf{U}^{\#})(0,x)dx=\bar\Theta_1r_1^-+\bar\Theta_2(\mathbf{U}^{\#}_{-}-\mathbf{U}^{\#}_{+})+\bar\Theta_3r_3^+, \end{align}\] with unique constants \(\bar\Theta_i,i=1,2,3.\) Without loss of generality, we can assume that \(\bar\Theta_2=0\) as in [6]. Then, we construct two diffusion waves to carry the extra initial mass as follows \[\begin{align} \label{eqs19} \Theta_1(t,x_1)=\frac{1}{\sqrt{4 \pi(1+t)}} e^{-\frac{\left(x_1-\lambda_1^{-}(1+t)\right)^2}{4(1+t)}}, \quad \Theta_3(t,x_1)=\frac{1}{\sqrt{4 \pi(1+t)}} e^{-\frac{\left(x_1-\lambda_3^{+}(1+t)\right)^2}{4(1+t)}}, \end{align}\tag{34}\] satisfying \[\begin{align} \notag \partial_t\Theta_{1 }+\lambda_1^{-} \partial_{x_1}\Theta_{1 }=\partial_{x_1}^2\Theta_{1}, \quad \partial_t \Theta_{3 }+\lambda_3^{+} \partial_{x_1}\Theta_{3 }=\partial_{x_1}^2\Theta_{3 }, \end{align}\] and \(\int_{-\infty}^{+\infty}\Theta_i(t,x_1)dx_1=1\) for \(i=1,3,\) and \(t\ge0.\) Let \[\begin{align} \notag \bar\Theta_{i+2}:=\int_{\Omega} m_i(0,x)-\bar m_i(0,x_1)dx,\;\;\;i=2,3. \end{align}\] We define the new ansatz \((\bar{\rho},\bar{m},\bar{\mathbb{E}})\) as \[\begin{align} \label{new-ansatz-new} \begin{aligned} &\bar{\rho}(t,x_1)={\check{\rho}}\left(t,x_1\right)+\bar{\Theta}_{1} \Theta_{1}+\bar{\Theta}_{3} \Theta_{3}, \\ &\bar{m}_{1}(t,x_1)={\check{m}}_{1}\left(t,x_1\right)+\lambda_{1-} \bar{\Theta}_{1} \Theta_{1}+\lambda_{3+} \bar{\Theta}_{3} \Theta_{3}, \\ &\bar{m}_{i}(t,x_1)=\frac{\bar{\Theta}_{i+2}}{\sqrt{4 \pi(1+t)}} e^{-\frac{x_1^{2}}{4(1+t)}}, \quad i=2,3, \\ &\bar{\mathbb{E}}(t,x_1)=\check{\mathbb{E}}\left(t,x_1\right)+\left(\frac{3(\lambda_{1-})^2}{2}\bar{\Theta}_{1} \Theta_{1}+\frac{3(\lambda_{3+})^2}{2}\bar{\Theta}_{3} \Theta_{3}\right). \end{aligned} \end{align}\tag{35}\] Then the initial extra mass under the new ansatz above becomes \[\begin{align} \label{eq-initial32mass} &\int_{\Omega}\left(\mathbf{U}-\bar{\mathbf{U}}\right)(0,x)dx=\int_{\Omega}\left(\mathbf{U}^{\#}-\check{\mathbf{U}}^{\#}\right)(0,x)dx+\sum_{i=2}^3\bar{\Theta}_{i+2}+\int_{\Omega}\left(\check{\mathbf{U}}-\bar{\mathbf{U}}\right)(0,x)dx=0. \end{align}\tag{36}\] For \(i=1,2,3\), we also define non-conserved quantities: velocity \(\bar{u}_i\), temperature \(\bar{\theta}\) and pressure \(\bar{p}\) as follows \[\begin{align} \notag \bar{u}_i:=\frac{\bar{m}_i}{\bar{\rho}},\qquad \qquad\bar{\theta}:=\frac{\bar{\mathbb{E}}}{\bar{\rho}}-\frac{\left\lvert\bar{m}\right\rvert^2}{2\bar{\rho}^2},\qquad\qquad \bar{p}:=\frac{2}{3}\bar{\mathbb{E}}-\frac{\left\lvert\bar{m}\right\rvert^2}{3\bar{\rho}}. \end{align}\] By 28 and 35 , we find that the ansatz 35 satisfies the following approximate Navier-Stokes equations \[\begin{align} \label{landau-2} \left\{ \begin{aligned} &\partial_t{\bar{\rho}} + \partial_{x_1}{\bar{m}}_{1} = \partial_{x_1}\bar{\mathcal{R}}_{0}, \\ &\partial_t{\bar{m}}_{1} + \partial_{x_1}{\left( \frac{{\bar{m}}_{1}^{2}}{\bar{\rho}} + \bar{p}\right) } = \frac{4}{3}\partial_{x_1} {\left( \mu \left( \bar{\theta}\right)\partial_{x_1} {\bar{u}}_{1 }\right) } +\partial_{x_1}\bar{\mathcal{R}}_{1 }, \\ &\partial_t{\bar{m}}_{i} + \partial_{x_1}{\left( \frac{{\bar{m}}_{1}{\bar{m}}_{i}}{\bar{\rho}}\right) } = \partial_{x_1}{\left( \mu \left( \bar{\theta}\right) \partial_{x_1}{\bar{u}}_{i }\right) } + \partial_{x_1}\bar{\mathcal{R}}_{i },\;i = 2,3, \\ &\partial_t{\bar{\mathbb{E}}} +\partial_{x_1} {\left( \frac{{\bar{m}}_{1}\bar{\mathbb{E}}}{\bar{\rho}} + \frac{{\bar{m}}_{1}}{\bar{\rho}}\bar{p}\right) } = \partial_{x_1}{\left( \kappa \left( \bar{\theta}\right) \partial_{x_1}{\bar{\theta}}\right) } \\ &\qquad\qquad\qquad+ \frac{4}{3}\partial_{x_1}{\left( \mu \left( \bar{\theta}\right) {\bar{u}}_{1}\partial_{x_1}{\bar{u}}_{1 }\right) }+ \mathop{\sum }\limits_{{i = 2}}^{3}\partial_{x_1} { \left( \mu \left( \bar{\theta}\right) {\bar{u}}_{i}\partial_{x_1}{\bar{u}}_{i }\right) }+\partial_{x_1}\bar{\mathcal{R}}_{4} , \end{aligned}\right. \end{align}\tag{37}\] where \[\begin{align} & \bar{\mathcal{R}}_0:=\bar{\Theta}_1\partial_{x_1}\Theta_{1}+\bar{\Theta}_3\partial_{x_1}\Theta_{3},\\ &\bar{\mathcal{R}}_1:=\lambda_{1-}\bar{\Theta}_1\partial_{x_1}\Theta_{1}+\lambda_{3+}\bar{\Theta}_3\partial_{x_1}\Theta_{3}-\frac{4}{3}\left( \mu(\bar{\theta})\partial_{x_1} \bar{u}_{1}-\mu(\check{\theta})\partial_{x_1} \check{u}_{1} \right)+\mathcal{R}_{1 } \nonumber\\ &\qquad\;\;+\partial_{x_1}\left(\frac{\bar{m}_1^2}{\bar{\rho}}-\frac{ \left\lvert\bar{m}\right\rvert^2}{3\bar{\rho}} -\frac{\check{m}^2_1}{\check{\rho}}+\frac{ \left\lvert\check{m}\right\rvert^2}{3\check{\rho}}\right),\nonumber\\ &\bar{\mathcal{R}}_i:=\bar{\Theta}_{i+2}\partial_{x_1}\Theta_{i+2}-\left( \mu(\bar{\theta})\partial_{x_1} \bar{u}_{i}-\mu(\check{\theta})\partial_{x_1} \check{u}_{i} \right)\;+\partial_{x_1}\left(\frac{\bar{m}_1\bar{m}_i}{\bar{\rho}} -\frac{\check{m}_1\check{m}_i}{\check{\rho}}\right),\quad i=2,3,\nonumber\\ &\bar{\mathcal{R}}_4:=\frac{3(\lambda_{1-})^2}{2}\bar{\Theta}_1\partial_{x_1}\Theta_{1}+\frac{3(\lambda_{3+})^2}{2}\bar{\Theta}_3\partial_{x_1}\Theta_{3} - (\kappa(\bar{\theta})\partial_{x_1}\bar{\theta}- \kappa(\check{\theta})\partial_{x_1}\check{\theta}) + \mathcal{R}_{4} \nonumber\\ &\qquad\;\;+\frac{4}{3}\partial_{x_1}\left( \mu \left( \bar{\theta}\right) {\bar{u}}_{1}\partial_{x_1} \bar{u}_{1}-\mu \left( \check{\theta} \right) {\check{u}}_{1}\partial_{x_1} \check{u}_{1} \right) + \mathop{\sum }\limits_{{i = 2}}^{3} \partial_{x_1}{ \left( \mu \left( \bar{\theta}\right) {\bar{u}}_{i}\partial_{x_1} \bar{u}_{i}-\left( \mu \left( \check{\theta} \right) {\check{u}}_{i}\partial_{x_1} \check{u}_{i}\right)\right) } \nonumber\\ &\qquad\;\;+\frac{5}{3} \left( \frac{\bar{m}_1 \bar{\mathbb{E}}}{\bar{\rho}}-\frac{\check{m}_1 \check{\mathbb{E}}}{\check{\rho}} -\lambda_1^{-} \frac{\mathbb{E}_{-}}{\rho_{-}}\bar{\Theta}_1\Theta_1 -\lambda_3^{+}\frac{\mathbb{E}_{+}}{\rho_{+}}\bar{\Theta}_3\Theta_3 \right)+\frac{2}{3} \left( \frac{\left\lvert\bar{m}\right\rvert^2\bar{m}_1}{2\bar{\rho}^2} - \frac{\left\lvert\check{m}\right\rvert^2\check{m}_1}{2\check{\rho}^2} \right) .\nonumber \end{align}\] We set \(\bar{\mathcal{R}}=(\bar{\mathcal{R}}_0,\cdots,\bar{\mathcal{R}}_4)^{t}\) and \(\bar{\mathcal{R}}^{*}=(\bar{\mathcal{R}}_0,\bar{\mathcal{R}}_1,\bar{\mathcal{R}}_4)^{t}\). For \(i=0,\cdots,4\), simple calculations yield \[\begin{align} \label{tildeE} \big|\bar{\mathcal{R}}_i\big|\leq O\left(\delta+\sum_{j=1}^{5}\left|\bar{\Theta}_{j}\right|\right) \frac{1}{1+t}\left(e^{-\frac{c x_1^{2}}{1+t}}+e^{-\frac{c\left(x_1-\lambda_{1-}(1+t)\right)^{2}}{1+t}}+e^{-\frac{c\left(x_1-\lambda_{3+}(1+t)\right)^{2}}{1+t}}\right), \end{align}\tag{38}\] where \(c>0\) is a constant independent of any small parameters throughout the paper. We use the following notation for convenience \[\begin{align} \label{errors} D_{-\alpha}= \frac{1}{(1+t)^{\alpha}}\left(e^{-\frac{c x_1^{2}}{1+t}}+e^{-\frac{c\left(x_1-\lambda_{1-}(1+t)\right)^{2}}{1+t}}+e^{-\frac{c\left(x_1-\lambda_{3+}(1+t)\right)^{2}}{1+t}}\right), \qquad \Upsilon_{-\alpha}={(1+t)^{-\alpha}}e^{-\frac{\tilde{d} x_1^{2}}{1+t}}. \end{align}\tag{39}\] Compared with 25 , the constant \(\tilde{d}>0\) in \(\Upsilon_{- \alpha}\) above is less than \(d\). We also always use \(\tilde{D}_{-\alpha}\) to denote quantities that have the same decay rate as \(D_{-\alpha}\) in the sense of the \(L^p\)-norm, namely, for \(1\leq p \le +\infty\), \[\begin{align} \label{tilde-d} \left\lVert\tilde{D}_{-\alpha}\right\rVert_{L^p}\thickapprox \left\lVert D_{-\alpha}\right\rVert_{L^p}. \end{align}\tag{40}\]
In order to write system 37 in the form of conservation laws with respect to \(\bar{\rho},\bar{m}_i,\bar{\mathbb{E}}\), we introduce the following notation: \[\begin{align} &F(\mathbf{U}):=\left(m_1,\frac{2m_1^2}{3\rho}+\frac{2}{3}\mathbb{E}-\frac{m_2^2+m_3^2}{3\rho},\frac{m_1m_2}{\rho},\frac{m_1m_3}{\rho},\frac{5m_1\mathbb{E}}{3\rho}-\frac{m_1 \left\lvert m\right\rvert^2}{3\rho^2}\right)^t,\tag{41}\\ &\mathcal{B}(\mathbf{U}):=\mathcal{B}_1(\mathbf{U})+\mathcal{B}_1^{*}(\mathbf{U}),\tag{42} \end{align}\] where \[\begin{align} &\mathcal{B}_1(\mathbf{U})= \left(\begin{array}{ccccc} 0&0&0&0&0\\ - \frac{4\mu(\theta)m_1}{3\rho^2}& \frac{4\mu(\theta)}{3\rho}&0&0&0\\ 0&0&\frac{\mu(\theta)}{\rho}&0&0\\ 0&0&0&\frac{\mu(\theta)}{\rho}&0\\ -\frac{\theta\kappa(\theta)}{\rho}&0&0&0&\frac{\kappa(\theta)}{\rho} \end{array}\right),\quad \mathcal{B}_{1}^{*}(\mathbf{U})= \left(\begin{array}{ccccc} 0&0&0&0&0\\ 0&0&0&0&0\\ \mathcal{B}_{31}&0&0&0&0\\ \mathcal{B}_{41}&0&0&0&0\\ \mathcal{B}_{51}&\mathcal{B}_{52}&\mathcal{B}_{53}&\mathcal{B}_{54}&0 \end{array}\right),\\ &\mathcal{B}_{31}=-\frac{\mu(\theta)m_2}{\rho^2},\quad \mathcal{B}_{41}= -\frac{\mu(\theta)m_3}{\rho^2}, \quad \mathcal{B}_{51}=-\frac{\mu(\theta)}{\rho^3}\left( \left\lvert m\right\rvert^2+\frac{m_1^2}{3} \right)+\frac{\left\lvert m\right\rvert^2}{2\rho^3}\kappa(\theta),\\ &\mathcal{B}_{52}=\frac{4 \mu(\theta)m_1-3\kappa(\theta)m_1}{3\rho^2},\quad \mathcal{B}_{53}=\frac{(\mu(\theta)-\kappa(\theta))m_2}{\rho^2},\quad \mathcal{B}_{54}=\frac{(\mu(\theta)-\kappa(\theta))m_3}{\rho^2}. \end{align}\] Since \(\theta=\frac{\mathbb{E}}{\rho}-\frac{|m|^2}{2\rho^2},\;\;p=\frac{2}{3}\rho\theta=\frac{2}{3}\mathbb{E}-\frac{|m|^2}{3\rho}\), combining 41 and 42 , system 37 can be written as \[\begin{align} \label{U-bar} \partial_t\bar\mathbf{U}+ \partial_{x_1}F(\bar \mathbf{U})=\partial_{x_1}\big(\mathcal{B}(\bar{\mathbf{U}})\partial_{x_1}\bar{\mathbf{U}}\big)+\partial_{x_1}\bar{\mathcal{R}}. \end{align}\tag{43}\] For the errors generated by macroscopic quantities, we expect to construct the coupled diffusion wave \(\Xi=(\Xi_1,\Xi_2,\Xi_3,\Xi_4,\Xi_5)^t\) such that the error terms of the equation for \(\tilde{\mathbf{U}}:=\bar{\mathbf{U}}+\Xi\) are improved from \(\bar{\mathcal{R}}\) to \(\tilde{\mathcal{R}}\), that is, \(\tilde{\mathbf{U}}\) satisfies \[\begin{align} \partial_t\tilde{\mathbf{U}}+\partial_{x_1}F(\tilde{\mathcal{\mathbf{U}}})=\partial_{x_1}\big(\mathcal{B}(\tilde{\mathbf{U}})\partial_{x_1}\tilde{\mathbf{U}}\big)+\partial_{x_1}\tilde{\mathcal{R}}, \end{align}\] with the improved time-decay \(\tilde{\mathcal{R}}\approx \tilde{\delta}\tilde{D}_{-\frac{3}{2}}\), where \(\tilde{\delta}\) is a small constant associated with the wave strength and the initial perturbation, and \(\tilde{D}_{-\alpha}\) is defined in 40 . To achieve this goal, we need to calculate the following quantities: \[\begin{align} &F'(\mathbf{U})=\mathcal{A}_1(\mathbf{U})+\mathcal{A}_1^{*}(\mathbf{U}) ,\tag{44}\\ &\Xi^{t}F''(\mathbf{U})= \left(\begin{array}{ccccc} 0&0&0&0&0\\ \mathcal{F}^{\Xi}_{21}&\mathcal{F}^{\Xi}_{22}&\mathcal{F}^{\Xi}_{23}&\mathcal{F}^{\Xi}_{24}&0\\ \mathcal{F}^{\Xi}_{31}&\mathcal{F}^{\Xi}_{32}&\mathcal{F}^{\Xi}_{33}&0&0\\ \mathcal{F}^{\Xi}_{41}&\mathcal{F}^{\Xi}_{42}&0&\mathcal{F}^{\Xi}_{44}&0\\ \mathcal{F}^{\Xi}_{51}&\mathcal{F}^{\Xi}_{52}&\mathcal{F}^{\Xi}_{53}&\mathcal{F}^{\Xi}_{54}&\mathcal{F}^{\Xi}_{55} \end{array}\right),\quad \Xi^{t} \mathcal{B}'(\mathbf{U})= \left(\begin{array}{ccccc} 0&0&0&0&0\\ \mathcal{B}^{\Xi}_{21}&\mathcal{B}^{\Xi}_{22}&0&0&0\\ \mathcal{B}^{\Xi}_{31}&0&\mathcal{B}^{\Xi}_{33}&0&0\\ \mathcal{B}^{\Xi}_{41}&0&0&\mathcal{B}^{\Xi}_{44}&0\\ \mathcal{B}^{\Xi}_{51}&\mathcal{B}^{\Xi}_{52}&\mathcal{B}^{\Xi}_{53}&\mathcal{B}^{\Xi}_{54}&\mathcal{B}^{\Xi}_{55} \end{array}\right), \tag{45} \end{align}\] where \[\begin{align} &\mathcal{A}_1(\mathbf{U})=\left(\begin{array}{ccccc} 0&1&0&0&0\\ 0&0&0&0&\frac{2}{3}\\ 0&0&0&0&0\\ 0&0&0&0&0\\ 0&\frac{5\theta}{3}&0&0&0 \end{array}\right),\quad \mathcal{A}_1^{*}(\mathbf{U})=\left(\begin{array}{ccccc} 0&0&0&0&0\\ \mathcal{A}_{21}&\mathcal{A}_{22}&\mathcal{A}_{23}&\mathcal{A}_{24}&0\\ \mathcal{A}_{31}&\mathcal{A}_{32}&\mathcal{A}_{33}&0&0\\ \mathcal{A}_{41}&\mathcal{A}_{42}&0&\mathcal{A}_{44}&0\\ \mathcal{A}_{51}&\mathcal{A}_{52}&\mathcal{A}_{53}&\mathcal{A}_{54}&\mathcal{A}_{55} \end{array}\right), \\ &\mathcal{A}_{21}=-\frac{2m_1^2}{3\rho^2}+\frac{m_2^2}{3\rho^2}+\frac{m_3^2}{3\rho^2},\quad \mathcal{A}_{22}=\frac{4m_1}{3\rho}, \quad \mathcal{A}_{23}=-\frac{2m_2}{3\rho}, \quad\mathcal{A}_{24}=-\frac{2m_3}{3\rho},\quad \mathcal{A}_{31}=-\frac{m_1 m_2}{\rho^2}, \\ & \mathcal{A}_{32}=\frac{m_2}{\rho},\quad \mathcal{A}_{33}=\frac{m_1}{\rho},\quad \mathcal{A}_{41}=-\frac{m_1 m_3}{\rho^2},\quad \mathcal{A}_{42}=\frac{m_3}{\rho},\quad \mathcal{A}_{44}=\frac{m_1}{\rho},\quad \mathcal{A}_{51}=\frac{2m_1 \left\lvert m\right\rvert^2}{3\rho^3}-\frac{5m_1 \mathbb{E}}{3\rho^2},\\ &\mathcal{A}_{52}=\frac{ \left\lvert m\right\rvert^2}{2\rho^2}-\frac{2m_1^2 }{3\rho^2},\quad \mathcal{A}_{53}=-\frac{2m_1 m_2 }{3\rho^2},\quad \mathcal{A}_{54}=-\frac{2m_1 m_3 }{3\rho^2},\quad \mathcal{A}_{55}=\frac{5m_1 }{3\rho}, \end{align}\] and \[\begin{align} &\Xi^{t} \cdot \nabla_{*} f:=\Xi_1 \partial_{\rho} f + \Xi_2 \partial_{m_1} f+ \Xi_3 \partial_{m_2} f+ \Xi_4 \partial_{m_3} f + \Xi_5 \partial_{\mathbb{E}} f,\\[1.5mm] &\mathcal{F}^{\Xi}_{21}=\Xi^{t} \cdot \nabla_{*} \mathcal{A}_{21},\quad\mathcal{F}^{\Xi}_{22}=\Xi^{t} \cdot \nabla_{*} \mathcal{A}_{22}, \quad \mathcal{F}^{\Xi}_{23}=\Xi^{t} \cdot \nabla_{*} \mathcal{A}_{23}, \quad\mathcal{F}^{\Xi}_{24}=\Xi^{t} \cdot \nabla_{*} \mathcal{A}_{24}, \quad\mathcal{F}^{\Xi}_{31}=\Xi^{t} \cdot \nabla_{*} \mathcal{A}_{31}, \\[1.5mm] & \mathcal{F}^{\Xi}_{32}=\Xi^{t} \cdot \nabla_{*} \mathcal{A}_{32},\quad \mathcal{F}^{\Xi}_{33}=\Xi^{t} \cdot \nabla_{*} \mathcal{A}_{33}, \quad\mathcal{F}^{\Xi}_{41}=\Xi^{t} \cdot \nabla_{*} \mathcal{A}_{41}, \quad\mathcal{F}^{\Xi}_{42}=\Xi^{t} \cdot \nabla_{*} \mathcal{A}_{42}, \quad\mathcal{F}^{\Xi}_{44}=\Xi^{t} \cdot \nabla_{*} \mathcal{A}_{44}, \\[1.5mm] & \mathcal{F}^{\Xi}_{51}=\Xi^{t} \cdot \nabla_{*} \mathcal{A}_{51},\quad\mathcal{F}^{\Xi}_{52}=\Xi^{t} \cdot \nabla_{*} \left(\mathcal{A}_{52} +\frac{5\theta}{3}\right), \quad \mathcal{F}^{\Xi}_{53}=\Xi^{t} \cdot \nabla_{*} \mathcal{A}_{53}, \quad \mathcal{F}^{\Xi}_{54}=\Xi^{t} \cdot \nabla_{*} \mathcal{A}_{54}, \\ & \mathcal{F}^{\Xi}_{55}=\Xi^{t} \cdot \nabla_{*} \mathcal{A}_{55},\quad \mathcal{B}^{\Xi}_{21}=-\Xi^{t} \cdot \nabla_{*} \left(\frac{4 \mu(\theta) m_1}{3 \rho^2}\right),\quad \mathcal{B}^{\Xi}_{22}=\Xi^{t} \cdot \nabla_{*} \left(\frac{4 \mu(\theta) }{3 \rho}\right), \quad \mathcal{B}^{\Xi}_{31}=\Xi^{t} \cdot \nabla_{*} \mathcal{B}_{31},\\ & \mathcal{B}^{\Xi}_{33}=\Xi^{t} \cdot \nabla_{*} \left(\frac{ \mu(\theta) }{ \rho}\right),\;\;\mathcal{B}^{\Xi}_{41}=\Xi^{t} \cdot \nabla_{*} \mathcal{B}_{41}, \;\; \mathcal{B}^{\Xi}_{44}=\Xi^{t} \cdot \nabla_{*} \left(\frac{ \mu(\theta) }{ \rho}\right), \;\; \mathcal{B}^{\Xi}_{51}=\Xi^{t} \cdot \nabla_{*} \left(\mathcal{B}_{51}-\frac{\theta \mu(\theta)}{\rho} \right), \\ & \mathcal{B}^{\Xi}_{52}=\Xi^{t} \cdot \nabla_{*} \mathcal{B}_{52},\quad\mathcal{B}^{\Xi}_{53}=\Xi^{t} \cdot \nabla_{*} \mathcal{B}_{53},\quad \mathcal{B}^{\Xi}_{54}=\Xi^{t} \cdot \nabla_{*} \mathcal{B}_{54},\quad \mathcal{B}^{\Xi}_{55}=\Xi^{t} \cdot \nabla_{*} \left(\frac{ \kappa(\theta) }{ \rho}\right). \end{align}\]
With the preparation of notations above, we now introduce the coupled diffusion wave which plays a vital role in obtaining the optimal time-decay for convergence of the kinetic Landau solution toward the planar entropy wave.
Definition 6. Corresponding to the diffusion wave \(\bar{\mathbf{U}}\) which satisfies 43 , the coupled diffusion wave \(\Xi\) is defined to satisfy the following equation with zero initial data: \[\begin{align} \label{coupled-diffusion-wave-1} \left\{ \begin{aligned} &\partial_t\Xi+\partial_{x_1}(F'(\bar{\mathbf{U}})\Xi)=\partial_{x_1}\big(\mathcal{B}(\bar{\mathbf{U}})\partial_{x_1}\Xi\big)-\frac{1}{2}\partial_{x_1}(\Xi^{t}F''(\bar{\mathbf{U}})\Xi)+\partial_{x_1}\big(\Xi^{t}\mathcal{B}{'}(\bar{\mathbf{U}})\partial_{x_1}\bar{\mathbf{U}}\big) -\partial_{x_1}\bar{\mathcal{R}}-\partial_{x_1}\mathcal{R}_{G}, \\ &\Xi|_{t=0}=0. \end{aligned}\right. \end{align}\qquad{(3)}\] Here, \(\mathcal{R}_{G}\) in the source term is to be specified in 49 later on. The purpose of introducing the coupled diffusion wave is to cancel out the slowly decaying terms in the macroscopic part corresponding to 14 or 17 .
In order to construct \(\mathcal{R}_{G}\) in the desired way, we define the linearized operator around the local Maxwellian \(\bar{M}:=M_{[\bar{\rho},\bar{u},\bar{\theta}]}\) as \[\begin{align} \notag L_{\bar{M}}h=Q(\bar{M},h)+Q(h,\bar{M}), \end{align}\] and the base functions for the kernel space of \(L_{\bar{M}}\) are given as \[\begin{align} \label{chi-1-1} \left\{ \begin{array}{ll} \bar{\chi}_{0}\left( \xi \right) \equiv \frac{1}{\sqrt{\bar{\rho}}}\bar{M}, \qquad\quad &\bar{\chi}_{i}\left( \xi \right) \equiv \frac{{\xi }_{i} - \bar{u}_{i}}{\sqrt{R\bar{\theta} \bar{\rho} }}\bar{M},\text{ for }i = 1,2,3, \\ [2mm] \bar{\chi}_{4}\left( \xi \right) \equiv \frac{1}{\sqrt{6\bar{\rho}} }\left( {\frac{{\left| \xi -\bar{u}\right| }^{2}}{R\bar{\theta} } - 3}\right) \bar{M}, \qquad\quad &\left\langle {\bar{\chi}_{i},\bar{\chi }_{j}}\right\rangle = {\delta}_{ij},\;i,j = 0,1,2,3,4. \end{array}\right. \end{align}\tag{46}\] Using these five base functions in 46 above, we define the macroscopic projection \(\bar{P}_{0}\) and the microscopic projection \(\bar{P}_{1}\) as follows: \[\begin{align} \notag \bar{P}_{0}h \equiv \mathop{\sum }\limits_{{j = 0}}^{4}\left\langle {h,\bar{\chi}_{j}}\right\rangle \bar{\chi}_{j},\qquad\bar{P}_{1}h \equiv h - \bar{P}_{0}h. \end{align}\] According to \(G\) in 11 , we then introduce two correction functions by \[\begin{align} \bar{G}_0&=\frac{3}{2\theta}{\boldsymbol{L}}_{M}^{-1}\left\{P_1 \left[\xi_1\left(\frac{|\xi-u|^2}{2\theta}\partial_{x_1} \bar{\theta}+(\xi-u)\cdot\partial_{x_1}\bar u\right){M}\right]\right\}\notag\\ &=\frac{\sqrt{R} \partial_{x_1} \bar{\theta} }{\sqrt{\theta} } {A}_1\left( \frac{\xi-u}{\sqrt{R \theta}} \right) + \sum_{j=1}^3 \partial_{x_1}\bar u_{j}{B}_{1 j}\left( \frac{\xi-u}{\sqrt{R \theta}} \right),\tag{47} \\ \bar{G}&=\frac{3}{2\bar{\theta}}{\boldsymbol{L}}_{\bar{M}}^{-1}\left\{{\bar{P}}_1\left[\xi_1\left(\frac{|\xi-\bar{u}|^2}{2\bar{\theta}}\partial_{x_1} \bar{\theta}+(\xi-\bar{u})\cdot\partial_{x_1}\bar{u}\right){\bar{M}}\right]\right\}\notag\\ &=\frac{\sqrt{R} \partial_{x_1} \bar{\theta} }{\sqrt{\bar{\theta}} } \bar{A}_1\left( \frac{\xi-\bar{u}}{\sqrt{R \bar{\theta}}} \right) + \sum_{j=1}^3 \partial_{x_1}\bar{u}_{j}\bar{B}_{1 j}\left( \frac{\xi-\bar{u}}{\sqrt{R \bar{\theta}}} \right).\tag{48} \end{align}\] By 12 and 48 , we define \(\bar{\Pi}_1:=\bar{P}_1(\xi_1\partial_{x_1} \bar{G})-Q(\bar{G},\bar{G}).\) Thus, the errors of the microscopic components can be written as \[\begin{align} \label{2025-11-10-1} \mathcal{R}_{G}:=\left(0,\;\int_{\mathbb{R}^3}\xi_1^2L^{-1}_{\bar{M}}\bar{\Pi}_1d\xi,\;\int_{\mathbb{R}^3}\xi_1\xi_2L^{-1}_{\bar{M}}\bar{\Pi}_1d\xi,\;\int_{\mathbb{R}^3}\xi_1\xi_3L^{-1}_{\bar{M}}\bar{\Pi}_1d\xi,\;\frac{1}{2}\int_{\mathbb{R}^3}\xi_1|\xi|^2L^{-1}_{\bar{M}}\bar{\Pi}_1d\xi\right)^t. \end{align}\tag{49}\] For \(k\geq1\) with \(k\in \mathbb{N}\), using 258 , 259 and the property of \(L^{-1}_{\bar{M}}\), we have \[\begin{align} &\partial_{x_1}^k\int_{\mathbb{R}^{3}}\xi_{1}\xi_{i}L^{-1}_{\bar{M}}\bar{\Pi}_{1} \,d\xi= \partial_{x_1}^k\int_{\mathbb{R}^{3}} L^{-1}_{\bar{M}}\{\bar{P}_{1}( \xi_{1}\xi_{i}\bar{M})\}\frac{\bar{\Pi}_{1}}{\bar{M}} \,d\xi \nonumber\\ =&\partial_{x_1}^k\int_{\mathbb{R}^{3}} L^{-1}_{\bar{M}}\{R\bar{\theta}\hat{B}_{1i}(\frac{\xi-\bar{u}}{\sqrt{R\bar{\theta}}})\bar{M}\}\frac{\bar{\Pi}_{1}}{\bar{M}}\, d\xi =R\partial_{x_1}^k\left[\bar{\theta}\int_{\mathbb{R}^{3}}\bar{B}_{1i}(\frac{\xi-\bar{u}}{\sqrt{R\bar{\theta}}})\frac{\bar{\Pi}_{1}}{\bar{M}} \,d\xi\right]\le C\bar{\delta}\partial_{x_1}^k D_{-1}, \end{align}\] and \[\begin{align} &\partial_{x_1}^k \int_{\mathbb{R}^{3}} (\frac{1}{2}\xi_{1}|\xi|^{2}-\xi_{1}\xi\cdot \bar{u})L^{-1}_{\bar{M}}\bar{\Pi}_{1} \,d\xi= \partial_{x_1}^k \int_{\mathbb{R}^{3}} L^{-1}_{\bar{M}}\{\bar{P}_{1}(\frac{1}{2}\xi_{1}|\xi|^{2}-\xi_{1}\xi\cdot \bar{u})\bar{M}\}\frac{\bar{\Pi}_{1}}{\bar{M}}\, d\xi \nonumber\\ =&\partial_{x_1}^k \int_{\mathbb{R}^{3}} L^{-1}_{\bar{M}}\{(R\bar{\theta})^{\frac{3}{2}}\hat{A}_{1}(\frac{\xi-\bar{u}}{\sqrt{R\bar{\theta}}})\bar{M}\}\frac{\bar{\Pi}_{1}}{\bar{M}}\, d\xi =\partial_{x_1}^k \left[(R\bar{\theta})^{\frac{3}{2}}\int_{\mathbb{R}^{3}}\bar{A}_{1}(\frac{\xi-\bar{u}}{\sqrt{R\tilde{\theta}}})\frac{\bar{\Pi}_{1}}{\bar{M}}\, d\xi\right] \le C \bar{\delta}\partial_{x_1}^k D_{-1}. \end{align}\] Combining 43 and ?? , the function \(\tilde{\mathbf{U}}=(\tilde{\rho},\tilde{m}_1,\tilde{m}_2,\tilde{m}_3,\tilde{\mathbb{E}})^{t}:=\bar{\mathbf{U}}+\Xi\) defined as the superposition of the background solution \(\bar{\mathbf{U}}\) and the coupled diffusion wave \(\Xi\) satisfies \[\begin{align} \label{New-Ansatz} \partial_t\tilde{\mathbf{U}}+\partial_{x_1}F(\tilde{\mathbf{U}})&=\partial_{x_1}\Big(\mathcal{B}(\tilde{\mathbf{U}})\partial_{x_1}\tilde{\mathbf{U}}\Big) +\partial_{x_1}\tilde{\mathcal{R}} -\partial_{x_1} \mathcal{R}_{G}, \end{align}\tag{50}\] where \[\begin{align} \label{new-error} \tilde{\mathcal{R}}=\Big(F(\tilde{\mathbf{U}})-F(\bar{\mathbf{U}})-F'(\bar{\mathbf{U}})\Xi-\frac{1}{2}\Xi^{t}F''(\bar{\mathbf{U}})\Xi\Big)+\Big[\left( \mathcal{B}(\bar{\mathbf{U}})+\Xi^{t}\mathcal{B}'(\bar{\mathbf{U}})-\mathcal{B}(\tilde{\mathbf{U}}) \right) \partial_{x_1}\tilde{\mathbf{U}} - \Xi^{t}\mathcal{B}'(\bar{\mathbf{U}})\partial_{x_1}\Xi \Big]. \end{align}\tag{51}\] It can be seen from 36 and ?? that \(\mathbf{U}-\tilde{\mathbf{U}}\) still satisfies the zero mass condition, i.e. \[\begin{align} \label{initial-zero-mass-2} \int_{\Omega} \mathbf{U}(0,x)-\tilde{\mathbf{U}}(0,x)dx=\int_{\Omega} \mathbf{U}(0,x)-\bar{\mathbf{U}}(0,x)dx-\int_{\Omega} \Xi (0,x)dx=0. \end{align}\tag{52}\] Moreover, as can be seen from Corollary 15, \(\tilde{\mathcal{R}}\approx \bar{\delta}\tilde{D}_{-\frac{3}{2}}\) turns out to be satisfied. For \(i=1,2,3\), we also define the background non-conserved quantities: velocity \(\tilde{u}_i\), temperature \(\tilde{\theta}\) and pressure \(\tilde{p}\) as follows \[\begin{align} \label{non-conserved32quantities} \tilde{u}_i:=\frac{\tilde{m}_i}{\tilde{\rho}},\qquad \qquad\tilde{\theta}:=\frac{\tilde{\mathbb{E}}}{\tilde{\rho}}-\frac{\left\lvert\tilde{m}\right\rvert^2}{2\tilde{\rho}^2},\qquad\qquad\tilde{p}=\frac{2}{3}\tilde{\mathbb{E}}-\frac{\left\lvert\tilde{m}\right\rvert^2}{3\tilde{\rho}}. \end{align}\tag{53}\] The \(L^p\) norm decay rate of the coupled diffusion wave \(\Xi\) ?? aligns with that of the diffusion wave \(\Theta\) 34 . For details, we refer to Theorem 11 and Corollary 15 later one.
We define the perturbation for the new ansatz 47 , 50 and 53 \[\begin{align} \label{ori-perb-1} (\phi,\varphi,h,\psi,\zeta,\sqrt{\mu}g):=(\rho-\tilde{\rho},m-\tilde{m},\mathbb{E}-\tilde{\mathbb{E}},u-\tilde{u},\theta-\tilde{\theta},G-\bar{G}_0). \end{align}\tag{54}\] For convenience, we use the following notation \[\begin{align} \label{2025-11-5-1} {\mathbf{V}}:=(\Phi,\Psi,H)^{t},\qquad{\check{\mathbf{V}}}:=(\check{\Phi},\check{\Psi},\check{H})^{t},\qquad \check{\mathbf{V}}^{\ast}:=(0,\check{\Psi},\check{H}),\qquad \mathbf{v}:=(\phi,\psi,\zeta)^{t},\qquad {\mathbf{v}}^{\ast}:=(0,\psi,\zeta)^{t}, \end{align}\tag{55}\] where \((\Phi,\Psi,H)\) and \((\check{\Phi},\check{\Psi},\check{H})\) are defined in 128 and 143 respectively. Next, for the solution \(f\) of landau equation 1 and the definition of \(\mu\) ?? , we introduce the instant energy functionals \(\mathcal{E}_{1,\omega} (t),\mathcal{E}_{2,\omega}(t)\) with weight \(\omega\) 31 and the unweighted instant energy functionals \(\mathcal{E}_{1} (t),\mathcal{E}_{2} (t),\mathcal{E}_{3} (t)\), \(\tilde{\mathcal{E}}_{2} (t)\) by \[\begin{align} \mathcal{E}_{1,\omega}(t)=&\left\lVert\check{\mathbf{V}}\right\rVert_{H^1}^2+\left\lVert\partial_{x_1}^2\left(\check{\Phi},\check{H},\sum_{i=2}^3 \check{\Psi}_i\right)\right\rVert_{L^2}^2 + \left\lVert\partial_{x_1}\left( \tilde{\theta}\partial_{x_1} \check{\Psi}_1\right)\right\rVert_{L^2}^2 +\left\lVert\mathbf{v}_{\neq}\right\rVert_{H^1}^2+\left\lVert\nabla_x^2 \mathbf{v}\right\rVert_{L^2}^2+\tilde{C}\sum_{\left\lvert\alpha\right\rvert=3}\left\lVert\frac{\partial^{\alpha}f}{\sqrt{\mu}}\right\rVert_{2,\omega}^2 \notag\\ &+\sum_{l=1}^4\mathcal{X}^l+\tilde{c}\left(\sum_{k=0}^1\int_{\mathbb{R}} \partial_{x_1}^{k+1}\check{\Phi}\partial_{x_1}^k\check{\Psi}_1 dx_1 +\sum_{k=1}^2\int_{\Omega} \nabla_x^k \psi \cdot\nabla_x^{k+1} \phi dx\right)+\tilde{C}\sum_{0\le \left\lvert\alpha\right\rvert\le 2\atop 0\leq\left\lvert\alpha\right\rvert+\left\lvert\beta\right\rvert\leq3}\|\partial^\alpha_\beta g\|^2_{2,\omega},\tag{56}\\ \mathcal{E}_{2,\omega}(t)=&\left\lVert\partial_{x_1}\check{\mathbf{V}}\right\rVert_{L^2}^2+\left\lVert\partial_{x_1}^2\left(\check{\Phi},\check{H},\sum_{i=2}^3 \check{\Psi}_i\right)\right\rVert_{L^2}^2 + \left\lVert\partial_{x_1}\left( \tilde{\theta}\partial_{x_1} \check{\Psi}_1\right)\right\rVert_{L^2}^2 +\left\lVert\mathbf{v}_{\neq}\right\rVert_{H^1}^2+\left\lVert\nabla_x^2 \mathbf{v}\right\rVert_{L^2}^2 \notag\\ &+c\left(\int_{\mathbb{R}} \partial_{x_1}^{2}\check{\Phi}\partial_{x_1}\check{\Psi}_1 dx_1 +\sum_{k=1}^2\int_{\Omega} \nabla_x^k \psi \cdot\nabla_x^{k+1} \phi dx\right)+\tilde{C}\sum_{0\le \left\lvert\alpha\right\rvert\le 2\atop 0\leq\left\lvert\alpha\right\rvert+\left\lvert\beta\right\rvert\leq3}\|\partial^\alpha_\beta g\|^2_{2,\omega}+\tilde{C}\sum_{\left\lvert\alpha\right\rvert=3}\left\lVert\frac{\partial^{\alpha}f}{\sqrt{\mu}}\right\rVert_{2,\omega}^2,\tag{57}\\ \mathcal{E}_{1}(t)=&\left\lVert\check{\mathbf{V}}\right\rVert_{H^1}^2+\left\lVert\partial_{x_1}^2\left(\check{\Phi},\check{H},\sum_{i=2}^3 \check{\Psi}_i\right)\right\rVert_{L^2}^2 + \left\lVert\partial_{x_1}\left( \tilde{\theta}\partial_{x_1} \check{\Psi}_1\right)\right\rVert_{L^2}^2 +\left\lVert\mathbf{v}_{\neq}\right\rVert_{H^1}^2+\left\lVert\nabla_x^2 \mathbf{v}\right\rVert_{L^2}^2 +\tilde{C}\sum_{\left\lvert\alpha\right\rvert=3}\left\lVert\frac{\partial^{\alpha}f}{\sqrt{\mu}}\right\rVert^2_{2}\notag\\ &+\sum_{l=1}^4\mathcal{X}^l+\tilde{c}\left(\sum_{k=0}^1\int_{\mathbb{R}} \partial_{x_1}^{k+1}\check{\Phi}\partial_{x_1}^k\check{\Psi}_1 dx_1 +\sum_{k=1}^2\int_{\Omega} \nabla_x^k \psi \cdot\nabla_x^{k+1} \phi dx\right)+\tilde{C}\sum_{0\le \left\lvert\alpha\right\rvert\leq 2}\|\partial^\alpha g\|^2_{2},\tag{58}\\ \mathcal{E}_{2}(t)=&\left\lVert\partial_{x_1}\check{\mathbf{V}}\right\rVert_{L^2}^2+\left\lVert\partial_{x_1}^2\left(\check{\Phi},\check{H},\sum_{i=2}^3 \check{\Psi}_i\right)\right\rVert_{L^2}^2 + \left\lVert\partial_{x_1}\left( \tilde{\theta}\partial_{x_1} \check{\Psi}_1\right)\right\rVert_{L^2}^2+\left\lVert\mathbf{v}_{\neq}\right\rVert_{H^1}^2+\left\lVert\nabla_x^2 \mathbf{v}\right\rVert_{L^2}^2 \notag\\ &+\tilde{c}\left(\int_{\mathbb{R}} \partial_{x_1}^{2}\check{\Phi}\partial_{x_1}\check{\Psi}_1 dx_1 +\sum_{k=1}^2\int_{\Omega} \nabla_x^k \psi \cdot\nabla_x^{k+1} \phi dx\right)+\tilde{C}\sum_{0\le \left\lvert\alpha\right\rvert\leq 2}\|\partial^\alpha g\|^2_{2}+\tilde{C}\sum_{\left\lvert\alpha\right\rvert=3}\left\lVert\frac{\partial^{\alpha}f}{\sqrt{\mu}}\right\rVert^2_{2},\tag{59}\\ \tilde{\mathcal{E}}_{2}(t)=&\left\lVert\partial_{x_1}\check{\mathbf{V}}\right\rVert_{L^2}^2+\left\lVert\partial_{x_1}^2\left(\check{\Phi},\check{H},\sum_{i=2}^3 \check{\Psi}_i\right)\right\rVert_{L^2}^2 + \left\lVert\partial_{x_1}\left( \tilde{\theta}\partial_{x_1} \check{\Psi}_1\right)\right\rVert_{L^2}^2+\left\lVert\mathbf{v}_{\neq}\right\rVert_{H^1}^2+\left\lVert\nabla_x^2 \mathbf{v}\right\rVert_{L^2}^2 \notag\\ &+\tilde{c}\left(\int_{\mathbb{R}} \partial_{x_1}^{2}\check{\Phi}\partial_{x_1}\check{\Psi}_1 dx_1 +\sum_{k=1}^2\int_{\Omega} \nabla_x^k \psi \cdot\nabla_x^{k+1} \phi dx\right)+\tilde{C}\sum_{1\le \left\lvert\alpha\right\rvert\leq 2}\|\partial^\alpha g\|^2_{2}+\tilde{C}\sum_{\left\lvert\alpha\right\rvert=3}\left\lVert\frac{\partial^{\alpha}f}{\sqrt{\mu}}\right\rVert^2_{2},\tag{60}\\ \mathcal{E}_{3}(t)=&\left\lVert\partial_{x_1}^2\left(\check{\Phi},\check{H},\sum_{i=2}^3 \check{\Psi}_i\right)\right\rVert_{L^2}^2 + \left\lVert\partial_{x_1}\left( \tilde{\theta}\partial_{x_1} \check{\Psi}_1\right)\right\rVert_{L^2}^2 +\left\lVert\nabla_x\mathbf{v}_{\neq}\right\rVert_{L^2}^2+\left\lVert\nabla_x^2 \mathbf{v}\right\rVert_{L^2}^2 \notag\\ &+\tilde{c}\sum_{k=1}^2\int_{\Omega} \nabla_x^k \psi \cdot\nabla_x^{k+1} \phi dx+\tilde{C}\sum_{1\le \left\lvert\alpha\right\rvert\leq 2}\|\partial^\alpha g\|^2_{2}+\tilde{C}\sum_{\left\lvert\alpha\right\rvert=3}\left\lVert\frac{\partial^{\alpha}f}{\sqrt{\mu}}\right\rVert^2_{2},\tag{61} \end{align}\] where \(\mathcal{X}^l\) is defined in ?? and \(\tilde{c}\) and \(\tilde{C}\) are sufficiently small and sufficiently large constants, respectively, used to ensure that all the transient energy functionals defined above are positive.
Remark 7. For \(N=1,2\), a direct calculation shows that the transient energy functionals defined above have the following equivalent relationship \[\begin{align} & \mathcal{E}_{N,\omega}\approx\left\lvert N-2\right\rvert\left\lVert\check{\mathbf{V}}\right\rVert_{L^2}^2+\left\lVert\mathbf{v}\right\rVert_{H^2}^2+\sum_{0\le \left\lvert\alpha\right\rvert\le 2\atop 0\leq\left\lvert\alpha\right\rvert+\left\lvert\beta\right\rvert\leq3}\|\partial^\alpha_\beta g\|^2_{2,\omega}+\sum_{\left\lvert\alpha\right\rvert=3}\left\lVert\frac{\partial^{\alpha}f}{\sqrt{\mu}}\right\rVert_{2,\omega}^2,\\ &\mathcal{E}_{N}\approx\left\lvert N-2\right\rvert\left\lVert\check{\mathbf{V}}\right\rVert_{L^2}^2+\left\lVert\mathbf{v}\right\rVert_{H^2}^2+\sum_{0\le \left\lvert\alpha\right\rvert\le 2}\|\partial^\alpha g\|^2_{2}+\sum_{\left\lvert\alpha\right\rvert=3}\left\lVert\frac{\partial^{\alpha}f}{\sqrt{\mu}}\right\rVert_{2}^2,\\ &\tilde{\mathcal{E}}_{2}\approx\left\lVert\mathbf{v}\right\rVert_{H^2}^2+\sum_{1\le \left\lvert\alpha\right\rvert\le 2}\|\partial^\alpha g\|^2_{2}+\sum_{\left\lvert\alpha\right\rvert=3}\left\lVert\frac{\partial^{\alpha}f}{\sqrt{\mu}}\right\rVert_{2}^2,\qquad\mathcal{E}_{3}\approx\left\lVert\nabla_x\mathbf{v}\right\rVert_{H^1}^2+\sum_{1\le \left\lvert\alpha\right\rvert\le 2}\|\partial^\alpha g\|^2_{2}+\sum_{\left\lvert\alpha\right\rvert=3}\left\lVert\frac{\partial^{\alpha}f}{\sqrt{\mu}}\right\rVert_{2}^2. \end{align}\]
And the corresponding dissipation energy functionals \(\mathcal{D}_{1,\omega} (t),\mathcal{D}_{2,\omega} (t),\mathcal{D}_{3,\omega} (t)\) with weight \(\omega\) and the unweighted dissipation energy functionals \(\mathcal{D}_{1} (t),\mathcal{D}_{2} (t),\mathcal{D}_{3} (t),\tilde{\mathcal{D}}_{2} (t)\) by \[\begin{align} &\mathcal{D}_{1,\omega}(t)=\left\lVert\partial_{x_1}\check{\mathbf{V}}\right\rVert_{H^2}^2+\left\lVert\mathbf{v}_{\neq}\right\rVert_{H^2}^2+\left\lVert\nabla_x^3 \mathbf{v}\right\rVert_{L^2}^2+\sum_{0\leq|\alpha|+|\beta|\leq 3}\|\partial^\alpha_\beta g\|^2_{\sigma,\omega}, \tag{62}\\ &\mathcal{D}_{2,\omega}(t)=\left\lVert\partial_{x_1}^2\check{\mathbf{V}}\right\rVert_{H^1}^2+\left\lVert\nabla_x \mathbf{v}_{\neq}\right\rVert_{H^1}^2+\left\lVert\nabla_x^3 \mathbf{v}\right\rVert_{L^2}^2+\sum_{0\leq|\alpha|+|\beta|\leq 3}\|\partial^\alpha_\beta g\|^2_{\sigma,\omega},\tag{63}\\ &\mathcal{D}_{3,\omega}(t)=\left\lVert\partial_{x_1}^3\check{\mathbf{V}}\right\rVert_{L^2}^2+\left\lVert\nabla_x^2 \mathbf{v}_{\neq}\right\rVert_{L^2}^2+\left\lVert\nabla_x^3 \mathbf{v}\right\rVert_{L^2}^2+\sum_{1\leq|\alpha|+|\beta|\leq 3}\|\partial^\alpha_\beta g\|^2_{\sigma,\omega},\tag{64}\\ &\mathcal{D}_{1}(t)=\left\lVert\partial_{x_1}\check{\mathbf{V}}\right\rVert_{H^2}^2+\left\lVert\mathbf{v}_{\neq}\right\rVert_{H^2}^2+\left\lVert\nabla_x^3 \mathbf{v}\right\rVert_{L^2}^2+\sum_{0\leq|\alpha|\leq 3}\|\partial^\alpha g\|^2_{\sigma}, \tag{65}\\ &\mathcal{D}_{2}(t)=\left\lVert\partial_{x_1}^2\check{\mathbf{V}}\right\rVert_{H^1}^2+\left\lVert\nabla_x \mathbf{v}_{\neq}\right\rVert_{H^1}^2+\left\lVert\nabla_x^3 \mathbf{v}\right\rVert_{L^2}^2+\sum_{0\leq|\alpha|\leq 3}\|\partial^\alpha g\|^2_{\sigma},\tag{66}\\ &\tilde{\mathcal{D}}_{2}(t)=\left\lVert\partial_{x_1}^2\check{\mathbf{V}}\right\rVert_{H^1}^2+\left\lVert\nabla_x \mathbf{v}_{\neq}\right\rVert_{H^1}^2+\left\lVert\nabla_x^3 \mathbf{v}\right\rVert_{L^2}^2+\sum_{1\leq|\alpha|\leq 3}\|\partial^\alpha g\|^2_{\sigma},\tag{67}\\ &\mathcal{D}_{3}(t)=\left\lVert\partial_{x_1}^3\check{\mathbf{V}}\right\rVert_{L^2}^2+\left\lVert\nabla_x^2 \mathbf{v}_{\neq}\right\rVert_{L^2}^2+\left\lVert\nabla_x^3 \mathbf{v}\right\rVert_{L^2}^2+\sum_{1\leq|\alpha|\leq 3}\|\partial^\alpha g\|^2_{\sigma}.\tag{68} \end{align}\]
Now we are ready to state the main results of this paper.
Theorem 8. Let \(M_{[\check{\rho},\check{u},\check{\theta}](t,x_1)}(\xi)\) be the viscous entropy wave defined in 27 with the small wave strength \(\delta=|\theta_{+}-\theta_{-}|>0\). Then, there is a sufficiently small constant \(\varepsilon_{0}>0\) such that if the initial data \(f_{0}(x,\xi)\geq 0\) satisfies \[\begin{align} \notag \left\lVert(\rho-\check{\rho},m-\check{m},\mathbb{E}-\check{\mathbb{E}})|_{t=0}\right\rVert_{L^1}^2+\mathcal{E}_{1,\omega}(0)\leq C\varepsilon^{2}_{0}, \end{align}\] where \(\mathcal{E}_{1,\omega}\) is defined in 56 with \(l\geq 2\) being arbitrarily given in 31 , then the Cauchy problem 1 and 3 on the Landau equation with Coulomb interaction admits a unique global solution \(f(t,x,\xi)\geq 0\) satisfying \[\begin{align} \notag \Big\|\frac{f-M_{[\check{\rho},\check{u},\check{\theta}]}}{\sqrt{\mu}}\Big\|_{L_{x}^{\infty}L_{\xi}^{2}}\leq C(\varepsilon_0^{\frac{1}{2}}+{\delta}^{\frac{1}{2}})(1+t)^{-\frac{1}{2}}, \quad \left\lVert\mathbf{D}_{\neq}\left(\frac{f-M_{[\check{\rho},\check{u},\check{\theta}](t,x_1)}}{\sqrt{\mu}}\right)\right\rVert_{L_x^\infty L_{\xi}^2}^2\leq C (\delta+\varepsilon_0) e^{-ct^{\frac{2}{3}}}. \end{align}\]
In order to prove Theorem 8, we first show the following result.
Theorem 9. Under the same assumptions of Theorem 8, it holds that \[\begin{align} \notag \mathcal{E}_{2,\omega}\le C(\varepsilon_0+\delta),\qquad\quad\mathcal{E}_{2}(t)\leq C(\varepsilon_0+\delta)(1+t)^{-\frac{1}{2}},\qquad\quad\mathcal{E}_{3}(t)\leq C(\varepsilon_0+\delta)(1+t)^{-\frac{3}{2}}, \end{align}\] where \(\mathcal{E}_{2,\omega},\mathcal{E}_{2}(t),\mathcal{E}_{3}(t)\) are defined in 57 , 59 and 61 . Moreover, for the non-zero modes, it holds that \[\begin{align} \left\lVert\mathbf{D}_{\neq}\left(\frac{f-M_{[\check{\rho},\check{u},\check{\theta}](t,x_1)}}{\sqrt{\mu}}\right)\right\rVert_{L_x^\infty L_{\xi}^2}^2\leq C (\delta+\varepsilon_0) e^{-ct^{\frac{2}{3}}}, \end{align}\] where \(C\) is a positive constant independent of \(\varepsilon_0\) and \(\delta\), and \(\mathbf{D}_{\neq}\) is defined in 33 .
Once we have Theorem 9, we are ready to prove Theorem 8.
In terms of 5 , 34 , 35 , 50 , 59 and 61 , one has \[\begin{align} &\left\lVert\frac{f(t,x,\xi)-M_{[\check{\rho},\tilde{u},\check{\theta}]}}{\sqrt{\mu}}\right\rVert_{L_{x}^{\infty}L_{\xi}^{2}}\le \left\lVert\frac{M_{[\rho,u,\theta]}-M_{[\tilde{\rho},\tilde{u},\tilde{\theta}]}+M_{[\tilde{\rho},\tilde{u},\tilde{\theta}]}-M_{[\check{\rho},\check{u},\check{\theta}]}+\bar{G}_0+\sqrt{\mu}g}{\sqrt{\mu}}\right\rVert_{L_{x}^{\infty}L_{\xi}^{2}}\\ &\qquad\qquad\quad\le C\left(\left\lVert\mathbf{v}\right\rVert_{L_x^{\infty}}+\sum_{i=\{1,3\}}\bar{\Theta}_i\left\lVert\Theta_i\right\rVert_{L^\infty} +\sum_{i=2}^3\left\lVert\bar{m}_i\right\rVert_{L^\infty}+\left\lVert\Xi\right\rVert_{L^\infty}+\left\lVert\left\lvert\bar{G}_0\right\rvert_2\right\rVert_{L_x^\infty}+\left\lVert\left\lvert g\right\rvert_2\right\rVert_{L_x^\infty}\right)\\ &\qquad\qquad\quad\le C\left( \mathcal{E}_2^{\frac{1}{4}}\mathcal{E}_3^{\frac{1}{4}}+\mathcal{E}_3^{\frac{1}{2}}+\sum_{i=\{1,3\}}\bar{\Theta}_i\left\lVert\Theta_i\right\rVert_{L^\infty} +\sum_{i=2}^3\left\lVert\bar{m}_i\right\rVert_{L^\infty}+\left\lVert\Xi\right\rVert_{L^\infty}+\left\lVert\left\lvert\bar{G}_0\right\rvert_2\right\rVert_{L_x^\infty} \right). \end{align}\] According to 25 , 34 , 35 , 47 , Theorem 9 and the estimate of coupled diffusion wave \(\Xi_i\) (see Corollary 15 below), we have completed the proof of Theorem 8. 0◻
The local existence of the solutions to the Landau system 1 near a global Maxwellian was proved in [16]. By a straightforward modification of the argument there, we can obtain the local existence of the solutions to the Landau system 1 and 3 with \(f(t,x,\xi)\ge0\) under the assumptions in Theorem 8. As for the local existence of the anti-derivative variables \((\Phi,\Psi,W)\), since \(f\) already exists, the corresponding macroscopic system has a similar structure to the Navier-Stokes equations, and thus its local existence can be established analogously. For brevity, we omit the proof.
Thus, by the continuity method, it suffices to provide a uniform a priori estimate. The a priori assumptions are given by \[\begin{align} \label{apa} \sup_{0\le t \le T} \bigg\{ &\left\lVert\mathbf{V}\right\rVert_{L_x^\infty}^2+\sum_{i=0}^1(1+t)^{\frac{1}{2}+i}\left\lVert\nabla_x^i \mathbf{v}\right\rVert_{L^2}^2 + \sum_{k=2}^3(1+t)^{\frac{3}{2}}\left\lVert\nabla_x^k \mathbf{v}\right\rVert_{L^2}^2+(1+t)^{\frac{1}{2}}\left\lVert g\right\rVert_2^2+\sum_{\left\lvert\alpha\right\rvert=1}^2(1+t)^{\frac{3}{2}}\left\lVert\partial^{\alpha}g\right\rVert_2^2 \notag\\ &\qquad+\sum_{\left\lvert\alpha\right\rvert=3}(1+t)^{\frac{3}{2}}\left\lVert\frac{\partial^{\alpha}f}{\sqrt{\mu}}\right\rVert_2^2+\sum_{\left\lvert\alpha\right\rvert+\left\lvert\beta\right\rvert=0}^2 \left\lVert\partial_{\beta}^{\alpha}g\right\rVert_{2,\omega}^2+\sum_{\left\lvert\beta\right\rvert=3} \left\lVert\partial_{\beta}g\right\rVert_{2,\omega}^2+\sum_{\left\lvert\alpha\right\rvert=3}\left\lVert\frac{\partial^{\alpha}f}{\sqrt{\mu}}\right\rVert_{2,\omega}^2\bigg\} \leq \chi^2. \end{align}\tag{69}\] Then we need to close the a priori assumptions. Indeed, we are able to prove the following a priori estimates.
Proposition 10 (a priori estimates). Assume that \((\mathbf{V},\mathbf{v},g,f)\) 55 is the unique solution given in the local existence and satisfies the a priori* assumptions 69 , then the following estimates hold \[\begin{align} &\left\lVert\mathbf{V}\right\rVert_{L_x^\infty}^2+\sum_{i=0}^1(1+t)^{\frac{1}{2}+i}\left\lVert\nabla_x^i \mathbf{v}\right\rVert_{L^2}^2 + \sum_{k=2}^3(1+t)^{\frac{3}{2}}\left\lVert\nabla_x^k \mathbf{v}\right\rVert_{L^2}^2+(1+t)^{\frac{1}{2}}\left\lVert g\right\rVert_2^2+\sum_{\left\lvert\alpha\right\rvert=1}^2(1+t)^{\frac{3}{2}}\left\lVert\partial^{\alpha}g\right\rVert_2^2 \notag\\ &\qquad\qquad+\sum_{\left\lvert\alpha\right\rvert=3}(1+t)^{\frac{3}{2}}\left\lVert\frac{\partial^{\alpha}f}{\sqrt{\mu}}\right\rVert_2^2+\sum_{\left\lvert\alpha\right\rvert+\left\lvert\beta\right\rvert=0}^2 \left\lVert\partial_{\beta}^{\alpha}g\right\rVert_{2,\omega}^2+\sum_{\left\lvert\beta\right\rvert=3} \left\lVert\partial_{\beta}g\right\rVert_{2,\omega}^2+\sum_{\left\lvert\alpha\right\rvert=3}\left\lVert\frac{\partial^{\alpha}f}{\sqrt{\mu}}\right\rVert_{2,\omega}^2 \le C(\delta+\varepsilon_0), \end{align}\] where \(C>0\) is a universal constant independent of any small parameters in this paper.*
By the local existence, Proposition 10 and Theorem 31, using a continuity argument, we obtain Theorem 9. We will prove Proposition 10 and the estimate of the coupled diffusion wave \(\Xi_i\) (Corollary 15) in the subsequent sections.
Combining 41 , 42 , 43 , 44 and 45 , the coupled diffusion wave system ?? can be written as \[\begin{align} \label{omega} \left\{\begin{aligned} &\partial_t\Xi+\partial_{x_1}\big(\mathcal{A}_1(\bar{\mathbf{U}})\Xi\big)=\partial_{x_1}\big(\mathcal{B}_1(\bar{\mathbf{U}})\partial_{x_1}\Xi\big)-\partial_{x_1}\big( \mathcal{A}_1^{*}(\bar{\mathbf{U}})\Xi \big)+\partial_{x_1}\big(\mathcal{B}_1^{*}(\bar{\mathbf{U}}){\partial_{x_1}\Xi}\big) \\ &\qquad\qquad\qquad\qquad\quad-\frac{1}{2}\partial_{x_1}(\Xi^{t}F''(\bar{\mathbf{U}})\Xi)+\partial_{x_1}\big(\Xi^{t}\mathcal{B}{'}(\bar{\mathbf{U}}){\partial_{x_1}\bar{\mathbf{U}}}\big) -\partial_{x_1}\bar{\mathcal{R}}-\partial_{x_1}\mathcal{R}_{G}, \\ &\Xi(0,x_1)=0. \end{aligned}\right. \end{align}\tag{70}\] Since \(\Xi|_{t=0} = 0\), we can define the anti-derivative of \(\Xi_i\) as \(W_i=\int_{-\infty}^{x_1}\Xi_i(\cdot,t)dy\), and thus system 70 becomes \[\begin{align} \label{W} \partial_t\mathbf{W}+\mathcal{A}_1(\bar{\mathbf{U}})\partial_{x_1}\mathbf{W}=&\mathcal{B}_1(\bar{\mathbf{U}})\partial_{x_1}^2\mathbf{W}- \mathcal{A}_1^{*}(\bar{\mathbf{U}})\partial_{x_1}\mathbf{W}+\mathcal{B}_{1}^{*}(\bar{\mathbf{U}})\partial_{x_1}^2\mathbf{W}\notag\\ &-\frac{1}{2}\mathbf{W}_{x_1}^{t}F''(\bar{\mathbf{U}})\partial_{x_1}\mathbf{W}+\partial_{x_1}\mathbf{W}^{t}\mathcal{B}{'}(\bar{\mathbf{U}}){\partial_{x_1}\bar{\mathbf{U}}} -\bar{\mathcal{R}}-\mathcal{R}_{G}, \end{align}\tag{71}\] for \(\mathbf{W}=(W_1,\cdots,W_5)^{t}\). To ensure that 71 satisfies the left-right structural conditions, we make the following transformations \[\begin{align} \label{trans-W-aa} \bar{W}_1=\bar{\theta}W_1,\quad \bar{W}_2=W_2,\quad \bar{W}_3=W_3,\quad \bar{W}_4=W_4,\quad \bar{W}_5=W_5-\bar{\theta}W_1, \end{align}\tag{72}\] then 71 becomes \[\begin{align} \label{Wt} \partial_t\bar{\mathbf{W}}+\bar{\mathcal{A}}_1(\bar{\mathbf{U}})\partial_{x_1}\bar{\mathbf{W}}=\bar{\mathcal{B}}_1(\bar{\mathbf{U}})\partial_{x_1}^2\bar{\mathbf{W}}+\mathcal{S}_{\bar \mathbf{W}}, \end{align}\tag{73}\] where \(\bar{\mathbf{W}}=(\bar{W}_1,\cdots,\bar{W}_5)^{t}\), \[\begin{align} \bar{\mathcal{A}}_1(\bar{\mathbf{U}})=\left(\begin{array}{ccccc} 0 &\bar{\theta}&0 &0&0 \\ \frac{2}{3}& 0&0&0 &\frac{2}{3}\\ 0 &0&0&0&0\\ 0&0&0&0&0\\ 0&\frac{2}{3}\bar{\theta}&0&0&0 \end{array}\right),\quad \bar{\mathcal{B}}_{1}(\bar{\mathbf{U}})=\left(\begin{array}{ccccc} 0 &0 &0&0&0 \\ -\frac{4}{3}\frac{\mu(\bar{\theta})\bar{m}_1}{\bar{\rho}^2\bar{\theta}} & \frac{4}{3}\frac{\mu(\bar{\theta})}{\bar{\rho}} &0&0&0\\ 0&0&\frac{\mu(\bar{\theta})}{\bar{\rho}}&0&0\\ 0&0&0&\frac{\mu(\bar{\theta})}{\bar{\rho}}&0\\ 0&0&0&0&\frac{\kappa(\bar{\theta})}{\bar{\rho}} \end{array}\right), \end{align}\] and \[\begin{align} &\mathcal{S}_{\bar \mathbf{W}}=\left( (\mathcal{S}_{\bar \mathbf{W}})_1, (\mathcal{S}_{\bar \mathbf{W}})_2, (\mathcal{S}_{\bar \mathbf{W}})_3,(\mathcal{S}_{\bar \mathbf{W}})_4,(\mathcal{S}_{\bar \mathbf{W}})_5 \right)^{t}, \notag \\ &(\mathcal{S}_{\bar \mathbf{W}})_1= \bar{\theta}(\mathcal{S}_{\mathbf{W}})_1-\partial_t\bar{\theta}W_1,\quad (\mathcal{S}_{\bar \mathbf{W}})_2=(\mathcal{S}_{\mathbf{W}})_2-\frac{4 \mu(\bar{\theta}) \bar{m}_1}{3 \bar{\rho}^2} \left[ \partial_{x_1}^2 \left( \frac{\bar{W}_1}{\bar{\theta}} \right)-\frac{\partial_{x_1}^2\bar{W}_1}{\bar{\theta}} \right] ,\quad (\mathcal{S}_{\bar \mathbf{W}})_3=(\mathcal{S}_{\mathbf{W}})_3, \notag\\ &(\mathcal{S}_{\bar \mathbf{W}})_4=(\mathcal{S}_{\mathbf{W}})_4,\quad (\mathcal{S}_{\bar \mathbf{W}})_5= \frac{\kappa(\bar{\theta})}{\bar{\rho}}\partial_{x_1}^2 \bar{W}_1-\frac{\bar{\theta}\kappa(\bar{\theta})}{\bar{\rho}}\partial_{x_1}^2 \left( \frac{\bar{W}_1}{\bar{\theta}} \right)+(\mathcal{S}_{\mathbf{W}})_5-(\mathcal{S}_{\bar{\mathbf{W}}})_1, \notag\\ &\left((\mathcal{S}_{\mathbf{W}})_1,\cdots,(\mathcal{S}_{\mathbf{W}})_5 \right)^{t}=- \mathcal{A}_1^{*}(\bar{\mathbf{U}})\partial_{x_1}\mathbf{W}+\mathcal{B}_{1}^{*}(\bar{\mathbf{U}})\partial_{x_1}^2\mathbf{W}-\frac{1}{2}\mathbf{W}_{x_1}^{t}F''(\bar{\mathbf{U}})\partial_{x_1}\mathbf{W}+\partial_{x_1}\mathbf{W}^{t}\mathcal{B}{'}(\bar{\mathbf{U}}){\partial_{x_1}\bar{\mathbf{U}}} -\bar{\mathcal{R}}-\mathcal{R}_{G}. \notag \end{align}\] By the definition of \(\mathcal{A}_1^{*},\mathcal{B}_{1}^{*},\Xi^{t}F''(\bar{\mathbf{U}}),\Xi^{t}\mathcal{B}{'}(\bar{\mathbf{U}})\) 41 45 and \(\mathcal{R}_{G}\) 49 , one has \[\begin{align} & \left\lvert(\mathcal{S}_{\bar \mathbf{W}})_1\right\rvert\lesssim \bar{\delta}D_{-1} + \bar{\delta}D_{-1} \left\lvert\bar{W}_1\right\rvert,\quad \left\lvert(\mathcal{S}_{\bar \mathbf{W}})_2\right\rvert\lesssim \bar{\delta}D_{-1}+\left\lvert\partial_{x_1}\bar{\mathbf{W}}\right\rvert^2+\bar{\delta}D_{-\frac{1}{2}}\left\lvert\partial_{x_1}\bar{\mathbf{W}}\right\rvert+\bar{\delta}D_{-1}\left\lvert\bar{\mathbf{W}}\right\rvert,\tag{74}\\ &\left\lvert(\mathcal{S}_{\bar \mathbf{W}})_3\right\rvert\thickapprox\left\lvert(\mathcal{S}_{\bar \mathbf{W}})_4\right\rvert\lesssim \bar{\delta}D_{-1}+\left\lvert\partial_{x_1}\bar{\mathbf{W}}\right\rvert^2+\bar{\delta}D_{-\frac{1}{2}}\left\lvert\partial_{x_1}^2\bar{\mathbf{W}}\right\rvert+\bar{\delta}D_{-\frac{1}{2}}\left\lvert\partial_{x_1}\bar{\mathbf{W}}\right\rvert+\bar{\delta}D_{-1}\left\lvert\bar{\mathbf{W}}\right\rvert,\tag{75}\\ &\left\lvert(\mathcal{S}_{\bar \mathbf{W}})_5\right\rvert\lesssim \bar{\delta}D_{-1}+\left\lvert\partial_{x_1}\bar{\mathbf{W}}\right\rvert^2+\bar{\delta}D_{-\frac{1}{2}}\sum_{i=1}^4\left\lvert\partial_{x_1}^2\bar{W}_i\right\rvert+\bar{\delta}D_{-\frac{1}{2}}\left\lvert\partial_{x_1}\bar{\mathbf{W}}\right\rvert+\bar{\delta}D_{-1}\left\lvert\bar{\mathbf{W}}\right\rvert\tag{76}, \end{align}\] where \(\bar{\delta}:=\delta+\varepsilon_0\). Now, let us state the results on the coupled diffusion wave \(\bar{\mathbf{W}}\) 73 .
Theorem 11. Under the same assumptions of Theorem 8, one has \[\begin{align} \left\lVert\bar{\mathbf{W}}\right\rVert_{L^\infty}\le C(\varepsilon_0^{\frac{1}{2}}+\delta^{\frac{1}{2}}),\qquad \left\lVert\partial_{x_1}^k \bar{\mathbf{W}}\right\rVert_{L^2}^2 \leq C(\varepsilon_0 +\delta)(1+t)^{\frac{1}{2}-k}, \quad k \geq 1 \;\text{and}\; k\in \mathbb{N}. \end{align}\]
The proof of Theorem 11 for small \(t_0\) is standard and the details are omitted for simplicity of presentation. Next, the a priori estimates will be carried out under the following a priori assumptions \[\begin{align} \label{aps-Wt} \sup_{T_0 \leq t \leq T_0+ t_0}\left\{\left\lVert\bar{\mathbf{W}}\right\rVert_{L^\infty}^2+(1+t)^{k-\frac{1}{2}}\left\lVert\partial_{x_1}^k \bar{\mathbf{W}}\right\rVert_{L^2}^2\right\}\leq \chi^2. \end{align}\tag{77}\] Under the a priori assumptions 77 , we obtain the following a priori estimate.
Proposition 12 (a priori estimates for \(\bar{\mathbf{W}}\)). Under the same assumptions of Theorem 8, assuming further that \(\bar{\mathbf{W}}\) is the unique solution for 73 in the interval \([T_0,T_0+t_0]\), satisfying the a priori* assumptions 77 , it holds that \[\begin{align} &\left\lVert\bar{\mathbf{W}}\right\rVert_{L^\infty}\leq C(\varepsilon_0^{\frac{1}{2}}+\delta^{\frac{1}{2}}),\qquad \left\lVert\partial_{x_1}^k \bar{\mathbf{W}}\right\rVert_{L^2}^2 \leq C(\varepsilon_0+\delta) (1+t)^{\frac{1}{2}-k}. \end{align}\]*
In the rest of this section, we are devoted to the proof of Proposition 12 above. It is noted that in the linear part of 73 , \(\bar{\mathbf{W}}^{\#}:=(\bar{W}_1,\bar{W}_2,\bar{W}_5)\) and \((\bar{W}_3,\bar{W}_4)\) are decoupled. Therefore, we first estimate \(\bar{\mathbf{W}}^{\#}\). Indeed, \(\bar{\mathbf{W}}^{\#}\) satisfies the following equation \[\begin{align} \label{W-t-1} \partial_t\bar{\mathbf{W}}^{\#}+\bar{\mathcal{A}}^{\#}_1(\bar{\mathbf{U}})\partial_{x_1}\bar{\mathbf{W}}^{\#}=\bar{\mathcal{B}}^{\#}_1(\bar{\mathbf{U}})\partial_{x_1}^2\bar{\mathbf{W}}^{\#}+(\mathcal{S}_{\bar \mathbf{W}})^{\#}, \end{align}\tag{78}\] where \[\begin{align} &\bar{\mathcal{A}}^{\#}_1(\bar{\mathbf{U}})=\left(\begin{array}{ccc} 0 &\bar{\theta}&0 \\ \frac{2}{3}& 0 &\frac{2}{3}\\ 0&\frac{2}{3}\bar{\theta}&0 \end{array}\right),\qquad \bar{\mathcal{B}}^{\#}_{1}(\bar{\mathbf{U}})=\left(\begin{array}{ccc} 0 &0 &0 \\ 0 & \frac{4}{3}\frac{\mu(\bar{\theta})}{\bar{\rho}} &0\\ 0&0&\frac{\kappa(\bar{\theta})}{\bar{\rho}} \end{array}\right), \\ &(\mathcal{S}_{\bar \mathbf{W}})^{\#}=((\mathcal{S}_{\bar \mathbf{W}})_1,(\mathcal{S}_{\bar \mathbf{W}})_2-\frac{4\mu(\bar{\theta})\bar{m}_1}{3\bar{\rho}^2 \bar{\theta}}\partial_{x_1}^2\bar{W}_1,(\mathcal{S}_{\bar \mathbf{W}})_5)^{t}. \end{align}\] Direct computation yields the eigenvalues of \(\bar{\mathcal{A}}^{\#}_1(\bar{\mathbf{U}})\) as \[\begin{align} \notag \bar{\lambda}_1=-\sqrt{\frac{10}{9}\bar{\theta}},\quad \lambda_2=0,\quad\bar{\lambda}_3=\sqrt{\frac{10}{9}\bar{\theta}},\qquad \Lambda:=\text{diag}\{\bar{\lambda}_1,0,\bar{\lambda}_3\}, \end{align}\] and the corresponding eigenvectors as \[\begin{align} &\bar{L}:=\left(\begin{array}{ccc} \sqrt{\frac{3}{10}} & \sqrt{\frac{3}{10}}\frac{3\bar{\lambda}_1}{2} &\sqrt{\frac{3}{10}} \\ \sqrt{\frac{2}{5}} & 0 & -\sqrt{\frac{9}{10}}\\ \sqrt{\frac{3}{10}} & -\sqrt{\frac{3}{10}}\frac{3\bar{\lambda}_1}{2} &\sqrt{\frac{3}{10}} \end{array}\right),\quad &\bar{R}:=\left(\begin{array}{ccc} \sqrt{\frac{3}{10}} & \sqrt{\frac{2}{5}} &\sqrt{\frac{3}{10}} \\ \sqrt{\frac{3}{10}}\frac{\bar{\lambda}_1}{\bar{\theta}} & 0 & -\sqrt{\frac{3}{10}}\frac{\bar{\lambda}_1}{\bar{\theta}}\\ \frac{2}{3}\sqrt{\frac{3}{10}} & -\sqrt{\frac{2}{5}} &\frac{2}{3}\sqrt{\frac{3}{10}} \end{array}\right). \end{align}\]
Remark 13. Direct calculations yield that the two sides of structural conditions 22 are satisfied for the system of \(\bar{\mathbf{W}}^{\#}\) 78 .
Set \(\mathbf{B}=\bar{L}\bar{\mathbf{W}}^{\#}=(b_1,b_2,b_3)^t\), then \(\bar{\mathbf{W}}^{\#}=\bar{R}\mathbf{B}\) and 78 can be written in a diagonalized form \[\begin{align} \label{eq-diaB} \partial_t\mathbf{B}+\Lambda \partial_{x_1} \mathbf{B}=\bar{L}\bar{\mathcal{B}}^{\#}_1\bar{R}\partial_{x_1} ^2\mathbf{B}+2\bar{L}\bar{\mathcal{B}}^{\#}_1\partial_{x_1} \bar{R}\partial_{x_1} \mathbf{B}+\left[\left(\partial_t\bar{L}+\Lambda \partial_{x_1} \bar{L}\right)\bar{R}+\bar{L}\bar{\mathcal{B}}^{\#}_1 \partial_{x_1} ^2\bar{R}\right]\mathbf{B}+\bar{L} (\mathcal{S}_{\bar \mathbf{W}})^{\#}. \end{align}\tag{79}\] Set \(v_{1}=\frac{\check{\rho}}{\rho_{+}}\) , one has \(\left|v_{1}-1\right| \leq C \delta .\) For the sake of convenience, we denote \(\bar{\mathcal{A}}_4=\bar{L}\bar{\mathcal{B}}^{\#}_1\bar{R}\), \[\begin{align} \bar{\mathcal{A}}_4=\left(\begin{array}{ccc} \frac{2\bar{\mu}}{3}+\frac{1}{5}\bar{\kappa} & -\frac{\sqrt{3}}{5}\bar{\kappa}& -\frac{2\bar{\mu}}{3}+\frac{1}{5}\bar{\kappa}\\ -\frac{\sqrt{3}}{5}\bar{\kappa}& \frac{3}{5}\bar{\kappa} &-\frac{\sqrt{3}}{5}\bar{\kappa}\\ -\frac{2\bar{\mu}}{3}+\frac{1}{5}\bar{\kappa}& -\frac{\sqrt{3}}{5}\bar{\kappa} &\frac{2\bar{\mu}}{3}+\frac{1}{5}\bar{\kappa} \end{array}\right),\label{A953} \text{\quad where\quad} \bar{\mu}=\frac{\mu(\bar{\theta})}{\bar{\rho}} ,\quad\bar{\kappa}=\frac{\kappa(\bar{\theta})}{\bar{\rho}} . \end{align}\tag{80}\] And we also denote \[\begin{align} \notag E^{\#}_k:=&\int_{\mathbb{R}}\frac{v_1^N}{2} \left\lvert\partial_{x_1}^kb_1\right\rvert^2+\frac{1}{2} \left\lvert\partial_{x_1}^kb_2\right\rvert^2+\frac{v_1^{-N}}{2} \left\lvert\partial_{x_1}^kb_3\right\rvert^2dx_1, \qquad K^{\#}_k:=\int_{\mathbb{R}}\partial_{x_1}^{k+1}B^t\bar{\mathcal{A}}_4\partial_{x_1}^{k+1}Bdx_1, \end{align}\] where \(N=4[\delta^{-\frac{1}{2}}]+1\) is a large positive integer. For \(k=0\), \(E^{\#}_0=C\left\lVert\bar{\mathbf{W}}^{\#}\right\rVert_{L^2}^2\), and for \(k\ge 1\), \[\begin{align} \label{sec-n-1} \left\lVert\partial_{x_1}^{k}\bar{\mathbf{W}}^{\#}\right\rVert_{L^2}^2-\bar{\delta}\sum_{j=0}^{k-1}(1+t)^{-(k-j)}\left\lVert\partial_{x_1}^{j}\bar{\mathbf{W}}^{\#}\right\rVert_{L^2}^2\lesssim E^{\#}_{k}\lesssim \left\lVert\partial_{x_1}^{k}\bar{\mathbf{W}}^{\#}\right\rVert_{L^2}^2+\bar{\delta}\sum_{j=0}^{k-1}(1+t)^{-(k-j)}\left\lVert\partial_{x_1}^{j}\bar{\mathbf{W}}^{\#}\right\rVert_{L^2}^2. \end{align}\tag{81}\] We further denote \[\label{sec-n-2} \left\{\begin{align} K_i:=&\int_{\mathbb{R}} \left\lvert\partial_{x_1}^{i+1}\bar{\mathbf{W}}\right\rvert^2dx_1,\\ {E}_i:=&E^{\#}_i+\int_{\mathbb{R}}\left\lvert\partial_{x_1}^{i}\bar{W}_3\right\rvert^2+\left\lvert\partial_{x_1}^{i}\bar{W}_4\right\rvert^2 dx_1 +\bar{c} \int_{\mathbb{R}} \partial_{x_1}^i\bar{W}_2\partial_{x_1}^{i+1}\bar{W}_1 +\frac{2\mu(\bar{\theta})}{3\bar{\theta}\bar{\rho}} \left\lvert\partial_{x_1}^{i+1} \bar{W}_1\right\rvert^2dx_1, \end{align} \right.\tag{82}\] where \(\bar{c}\) is a sufficiently small constant chosen to ensure \(E_i\) is positive. Then we have the following result.
Lemma 14. Under the same assumptions of Proposition 12, it holds that \[\begin{align} &\frac{d}{dt}\big(\sum_{i=0}^n E_i\big)+\sum_{i=0}^n(K_i+G_{i})\leq C\check{\delta}(1+t)^{-1}\big(\sum_{i=0}^n {E}_i\big)+C\bar{\delta}(1+t)^{-\frac{1}{2}},\label{11}\\ &\frac{d}{dt}\big(\sum_{i=1}^n {E}_i\big)+\sum_{i=1}^n (K_i+G_{i})\leq C\check{\delta}\left[(1+t)^{-1}\big(\sum_{i=1}^n{E}_i+G_0\big)+(1+t)^{-2}{E}_0\right]+C\bar{\delta}(1+t)^{-\frac{3}{2}},\label{22}\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad......., \notag\\ &\frac{d}{dt}\big(\sum_{i=n-1}^n {E}_i\big)+\sum_{i=n-1}^n (K_i+G_{i})\leq C\check{\delta}\bigg[(1+t)^{-1}\sum_{i=n-1}^n E_i +\sum_{i=0}^{n-2}(1+t)^{i-n}E_i\notag\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\;+\sum_{i=0}^{n-2}(1+t)^{i+1-n}G_i\bigg]+C\bar{\delta}(1+t)^{\frac{1}{2}-n},\notag\\ &\frac{d}{dt}{E}_n+{K}_n+G_{n}\leq C\check{\delta}\left[\sum_{i=0}^{n}(1+t)^{i-n-1}E_i+\sum_{i=0}^{n-1}(1+t)^{i-n}G_i \right]+C\bar{\delta}(1+t)^{-\frac{1}{2}-n},\label{kk} \end{align}\] {#eq: sublabel=eq:11,eq:22,eq:kk} where \(G_i\), \(i=1,...,n\) is defined in 85 , \(\bar{\delta}:=\delta+\varepsilon_0\), and \(\check{\delta}:=\chi+\bar{\delta}^{\frac{1}{2}}\).
Proof.
Step 1. The diagonalized system.
Applying \(\partial_{x_1}^k\) with \(k=0,1,2,\dots,n\) to 79 , one has \[\begin{align} \label{equ-Bk} \begin{aligned} \partial_t\partial_{x_1}^k\mathbf{B}+\Lambda \partial_{x_1}^{k+1}\mathbf{B}=\bar{\mathcal{A}}_4 \partial_{x_1}^{k+2}\mathbf{B}+\mathcal{S}_{\mathbf{B}k}, \end{aligned} \end{align}\tag{83}\] where \[\begin{align} \mathcal{S}_{\mathbf{B}k}:=&\sum_{j=1}^{k}\bigg[-\partial_{x_1}^j\Lambda\partial_{x_1}^{k-j+1}\mathbf{B}+\partial_{x_1}^j\bar{\mathcal{A}}_4\partial_{x_1}^{k-j+2}\mathbf{B}\bigg]+2\sum_{i=0}^{k}\partial_{x_1}^i\big(\bar{L} \bar{\mathcal{B}}^{\#}_1 \partial_{x_1}\bar{R}\big) \partial_{x_1}^{k-i+1}\mathbf{B}\notag\\ &+\sum_{i=0}^{k}\bigg\{\partial_{x_1}^i\left[\left(\partial_t\bar{L}+\Lambda \partial_{x_1}\bar{L}\right) \bar{R}+\bar{L} \bar{\mathcal{B}}^{\#}_1 \partial_{x_1}^2 \bar{R}\right] \partial_{x_1}^{k-i}\mathbf{B}\bigg\}+\partial_{x_1}^{k}\left(\bar{L} (\mathcal{S}_{\bar \mathbf{W}})^{\#}\right) \notag\\ \leq&C\sum_{j=1}^{k+1}\left(\left\lvert\partial_{x_1}^j \check{\rho}\right\rvert \left\lvert\partial_{x_1}^{k-j+1}b_1\right\rvert,0,\left\lvert\partial_{x_1}^j\check{\rho}\right\rvert \left\lvert\partial_{x_1}^{k-j+1}b_3\right\rvert\right)^{t} +C\bar{\delta} \sum_{j=1}^{k+2}\left\lvert D_{-\frac{j}{2}}\partial_{x_1}^{k-j+2}\mathbf{B}\right\rvert+\partial_{x_1}^{k}\left(\bar{L} (\mathcal{S}_{\bar \mathbf{W}})^{\#}\right):=\sum_{i=1}^{3}\mathcal{S}_{\mathbf{B}k}^{(i)}.\notag \end{align}\] We shall use a weighted energy method to derive the intrinsic dissipation. Without loss of generality, we assume that \(\partial_{x_1}{\check{\rho}}>0\) since the proof in the case \(\partial_{x_1}{\check{\rho}}<0\) is similar.
Applying \(\partial_{x_1}^k\) to 79 and then multiplying the resulting equations by \(\bar{\mathbf{B}}^{(k)}=\left(v_{1}^{N} \partial_{x_1}^kb_{1}, \partial_{x_1}^kb_{2}, v_{1}^{-N} \partial_{x_1}^kb_{3}\right)\), one has \[\begin{align} \label{d-k-b} &\int_\mathbb{R}\partial_t\left(\frac{v_1^N}{2} \left\lvert\partial_{x_1}^kb_1\right\rvert^2+\frac{1}{2} \left\lvert\partial_{x_1}^kb_2\right\rvert^2+\frac{v_1^{-N}}{2} \left\lvert\partial_{x_1}^kb_3\right\rvert^2\right)+\partial_{x_1}\bar{\mathbf{B}}^{(k)} \bar{\mathcal{A}}_4 \partial_{x_1}^{k+1}\mathbf{B}dx_1 +\int_{\mathbb{R}} a_1\left\lvert\partial_{x_1}^kb_1\right\rvert^2+ a_3\left\lvert\partial_{x_1}^kb_3\right\rvert^2dx_1\nonumber\\ =&\int_{\mathbb{R}}\bar{\mathbf{B}}^{(k)}\partial_{x_1} \bar{\mathcal{A}}_{4}\partial_{x_1}^{k+1} \mathbf{B}+\bigg[\left(\frac{v_1^N}{2}\right)_t \left\lvert\partial_{x_1}^kb_1\right\rvert^2+\left(\frac{v_1^{-N}}{2}\right)_t \left\lvert\partial_{x_1}^kb_3\right\rvert^2\bigg]+\bar{\mathbf{B}}^{(k)}\mathcal{S}_{\mathbf{B}k}dx_1:=I_1+I_2+I_3, \end{align}\tag{84}\] where \[\begin{align} &a_1:=-\frac{v_1^{N-1}}{2}\left(N \bar\lambda_1 \partial_{x_1} v_{1}+v_1\partial_{x_1} \bar\lambda_{1 }\right)\ge C\delta ^{-\frac{1}{2}}\partial_{x_1} \check{\rho}-C\bar{\delta}D_{-1},\\ &a_3:=\frac{v_1^{-N-1}}{2}\left(N \bar\lambda_3 \partial_{x_1} v_{1 }-v_1 \partial_{x_1} \bar\lambda_{3 }\right)\ge C\delta ^{-\frac{1}{2}}\partial_{x_1} \check{\rho}-C\bar{\delta}D_{-1}. \end{align}\] Thus, we obtain an additional dissipative structure through diagonalization: \[\begin{align} \label{20254664607-1} G_k:=C\delta^{-\frac{1}{2}}\int_{\mathbb{R}} \partial_{x_1}\check{\rho}\left\lvert\partial_{x_1}^kb_1\right\rvert^2+ \partial_{x_1}\check{\rho}\left\lvert\partial_{x_1}^kb_3\right\rvert^2dx_1. \end{align}\tag{85}\] Recalling the definition of \(\bar{\kappa}\) and \(\bar{\mu}\) 80 , we find that the dissipation matrix \(\bar{\mathcal{A}}_4\) 80 is non-negative definite. Furthermore, it holds \[\begin{align} \label{q-z-x} \partial_{x_1}^{k+1}{\mathbf{B}}^t\bar{\mathcal{A}}_4\partial_{x_1}^{k+1}\mathbf{B}=&\frac{\bar{\kappa}}{5}\left[(\partial_{x_1}^{k+1} b_{1}+\partial_{x_1}^{k+1} b_{3})-\sqrt{3}\partial_{x_1}^{k+1} b_{2}\right]^2+\frac{2\bar{\mu}}{3}\left[(\partial_{x_1}^{k+1} b_{3}-\partial_{x_1}^{k+1} b_{1})\right]^2 \notag\\ \geq&C\big(|{\partial_{x_1}^{k+1}}\bar{W}_2|^2+|{\partial_{x_1}^{k+1}}\bar{W}_5|^2\big)-C\bar{\delta}\sum_{j=1}^{k+1}D_{-j}\left\lvert\partial_{x_1}^{k+1-j}\bar{W}_2\right\rvert^2. \end{align}\tag{86}\] And we also have \[\begin{align} \label{2025-11-11-1} \int_{\mathbb{R}} \partial_{x_1} \left(\bar{\mathbf{B}}^{(k)}-\partial_{x_1}^{k}\mathbf{B}^t\right)\bar{\mathcal{A}}_4\partial_{x_1}^{k+1}\mathbf{B}dx_1 \leq C(\chi+\bar{\delta}^{\frac{1}{2}})(1+t)^{-1}E_k+C(\chi+\bar{\delta}^{\frac{1}{2}})K_k. \end{align}\tag{87}\] By 81 , 82 and the a priori assumptions 77 , we obtain \[\begin{align} \label{I12} I_1+I_2\leq C \left(\bar{\delta}^{\frac{1}{2}}+\chi\right)(1+t)^{-1}E_k+\left(\bar{\delta}^{\frac{1}{2}}+\chi\right) K_k. \end{align}\tag{88}\] The estimate of \(I_3\) is more intricate. We first present the estimate of \(\int_{\mathbb{R}}\bar{\mathbf{B}}^{(k)}\mathcal{S}_{\mathbf{B}k}^{(2)}dx_1\) as \[\begin{align} \int_{\mathbb{R}}\bar{\mathbf{B}}^{(k)}\mathcal{S}_{\mathbf{B}k}^{(2)}dx_1\leq C\bar{\delta}\sum_{i=0}^{k+1}(1+t)^{-i}\left\lVert\partial_{x_1}^{k-i+1}\mathbf{B}\right\rVert_{L^2}^2, \quad \text{for}\quad k\ge 0.\label{I14} \end{align}\tag{89}\] Then we consider the estimate of \(\int_{\mathbb{R}}\bar{\mathbf{B}}^{(k)}\mathcal{S}_{\mathbf{B}k}^{(1)}dx_1\). For the case of \(k=0\), it is easy to see \[\begin{align} \label{IL12} \int_{\mathbb{R}}\bar{\mathbf{B}}^{(0)}\mathcal{S}_{\mathbf{B}0}^{(1)}dx_1\le C\int_{\mathbb{R}} \partial_{x_1} \check{\rho}\left\lvert b_1\right\rvert^2+\partial_{x_1} \check{\rho}\left\lvert b_3\right\rvert^2 dx_1. \end{align}\tag{90}\] For \(k \ge 1\), we have \[\begin{align} \label{2025-11-3-1} &\int_{\mathbb{R}}\bar{\mathbf{B}}^{(k)}\mathcal{S}_{\mathbf{B}k}^{(1)}dx_1\lesssim \sum_{l=1,3}\sum_{j=0}^k \int_{\mathbb{R}} \left\lvert\partial_{x_1}^{j+1}\check{\rho}\right\rvert\left\lvert\partial_{x_1}^{k-j} b_{l}\right\rvert \left\lvert\partial_{x_1}^k b_l\right\rvert dx_1. \end{align}\tag{91}\] Noticing \[\begin{align} &\left\lvert\partial_{x_1}^{2m} \check{\rho}\right\rvert\lesssim \sum_{i=0}^{m-1}(1+t)^{i-m+1}\left\lvert\frac{x_1}{1+t}\right\rvert^{2i+1}\partial_{x_1}\check{\rho},\quad \left\lvert\partial_{x_1}^{2m+1} \check{\rho}\right\rvert\lesssim \sum_{i=0}^{m}(1+t)^{i-m}\left\lvert\frac{x_1}{1+t}\right\rvert^{2i}\partial_{x_1}\check{\rho}, \end{align}\] we have \[\begin{align} &\int_{\mathbb{R}} \left\lvert\partial_{x_1}^{j+1}\check{\rho}\right\rvert\left\lvert\partial_{x_1}^{k-j} b_{l}\right\rvert \left\lvert\partial_{x_1}^k b_l\right\rvert dx_1\notag\\ &\lesssim \sum_{i=0}^{\frac{j}{2}}(1+t)^{i-\frac{j}{2}}\int_{\mathbb{R}} \left\lvert\frac{x_1}{1+t}\right\rvert^{2i}\partial_{x_1}\check{\rho}\left\lvert\partial_{x_1}^{k-j}b_l\right\rvert\left\lvert\partial_{x_1}^k b_l\right\rvert dx_1 \notag\\ &\lesssim (1+t)^{-j} \int_{\mathbb{R}} \partial_{x_1}\check{\rho}\left\lvert\partial_{x_1}^{k-j}b_l\right\rvert^2 dx_1 + \sum_{i=0}^{\frac{j}{2}} \int_{\mathbb{R}} \left\lvert\frac{x_1^2}{1+t}\right\rvert^{2i} \partial_{x_1}\check{\rho}\left\lvert\partial_{x_1}^k b_l\right\rvert^2 dx_1\notag\\ &\lesssim (1+t)^{-j} \int_{\mathbb{R}} \partial_{x_1}\check{\rho}\left\lvert\partial_{x_1}^{k-j}b_l\right\rvert^2 dx_1 + \bar{\delta}\int_{\mathbb{R}} \Upsilon_{-\frac{1}{2}} \left\lvert\partial_{x_1}^k b_l\right\rvert^2 dx_1, \quad \text{for} \; j \; \text{is} \; \text{even}, \label{2025-11-3-2} \end{align}\tag{92}\] and \[\begin{align} &\int_{\mathbb{R}} \left\lvert\partial_{x_1}^{j+1}\check{\rho}\right\rvert\left\lvert\partial_{x_1}^{k-j} b_{l}\right\rvert \left\lvert\partial_{x_1}^k b_l\right\rvert dx_1\notag\\ &\lesssim \sum_{i=0}^{\frac{j}{2}-\frac{1}{2}}(1+t)^{i-\frac{j}{2}+\frac{1}{2}}\int_{\mathbb{R}} \left\lvert\frac{x_1}{1+t}\right\rvert^{2i+1}\partial_{x_1}\check{\rho}\left\lvert\partial_{x_1}^{k-j}b_l\right\rvert\left\lvert\partial_{x_1}^k b_l\right\rvert dx_1 \notag\\ &\lesssim (1+t)^{-j} \int_{\mathbb{R}} \partial_{x_1}\check{\rho}\left\lvert\partial_{x_1}^{k-j}b_l\right\rvert^2 dx_1 + \sum_{i=0}^{\frac{j}{2}-\frac{1}{2}} \int_{\mathbb{R}} \left\lvert\frac{x_1^2}{1+t}\right\rvert^{2i+1} \partial_{x_1}\check{\rho}\left\lvert\partial_{x_1}^k b_l\right\rvert^2 dx_1\notag\\ &\lesssim (1+t)^{-j} \int_{\mathbb{R}} \partial_{x_1}\check{\rho}\left\lvert\partial_{x_1}^{k-j}b_l\right\rvert^2 dx_1 + \bar{\delta}\int_{\mathbb{R}} \Upsilon_{-\frac{1}{2}} \left\lvert\partial_{x_1}^k b_l\right\rvert^2 dx_1, \quad \text{for} \; j \; \text{is} \; \text{odd},\label{2025-11-3-3} \end{align}\tag{93}\] where we have used the definition of \(\Upsilon_{-\frac{1}{2}}\) in 39 . Combining 91 , 92 and 93 , we then obtain \[\begin{align} \label{2025-11-3-4} \int_{\mathbb{R}}\bar{\mathbf{B}}^{(k)}\mathcal{S}_{\mathbf{B}k}^{(1)}dx_1\lesssim \sum_{l=1,3}\sum_{j=0}^k (1+t)^{-j} \int_{\mathbb{R}} \partial_{x_1}\check{\rho}\left\lvert\partial_{x_1}^{k-j}b_l\right\rvert^2 dx_1 +\bar{\delta}\sum_{l=1,3} \int_{\mathbb{R}} \Upsilon_{-\frac{1}{2}} \left\lvert\partial_{x_1}^k b_l\right\rvert^2 dx_1. \end{align}\tag{94}\] Next, we estimate terms involving \(\mathcal{S}_{\mathbf{B}k}^{(3)}\). For \(k=0\) \[\begin{align} \label{d-k-M-00} \left\lVert\bar{\mathbf{B}}^{(0)}\mathcal{S}_{\mathbf{B}0}^{(3)}\right\rVert_{L^1(\mathbb{R})}\leq&C\int_{\mathbb{R}} \left\{\bar{\delta}\left[D_{-1}+D_{-1}\left\lvert\bar{\mathbf{W}}\right\rvert+D_{-\frac{1}{2}}\left(\left\lvert\partial_{x_1}\bar{\mathbf{W}}\right\rvert+\left\lvert\partial_{x_1}^2 \bar{\mathbf{W}}\right\rvert\right)\right]+\left\lvert\partial_{x_1} \bar{\mathbf{W}}\right\rvert^2\right\}\left\lvert\bar{\mathbf{B}}^{(0)}\right\rvert dx_1 :=\sum_{j=1}^4 I_{4}^{(j)}. \end{align}\tag{95}\] Then we estimate the above \(I_4\) term by term in the way that \[\begin{align} \notag &I_{4}^{(1)}\leq C \bar{\delta}(1+t)^{-\frac{1}{2}}+C\check{\delta}(1+t)^{-1}\left\lVert\mathbf{B}\right\rVert^2_{L^2},\notag\\ &I_{4}^{(2)}+I_{4}^{(3)}+I_{4}^{(4)}\leq C\check{\delta}(1+t)^{-1}\left\lVert\bar{\mathbf{W}}\right\rVert^2_{L^2}+C\check{\delta}\left\lVert\partial_{x_1} \bar{\mathbf{W}}\right\rVert_{H^1}^2,\notag \end{align}\] where we have used the a priori assumptions 77 . For \(k \geq 1, \; k \in \mathbb{N}\), we have \[\begin{align} \label{d-k-M-k11} &\left\lVert\bar{\mathbf{B}}^{(k)}\mathcal{S}_{\mathbf{B}k}^{(3)}\right\rVert_{L^1(\mathbb{R})}\le\left\lvert\int_{\mathbb{R}} \partial_{x_1}^{k+1}b_2 \partial_{x_1}^{k-1}\left( \sqrt{\frac{2}{5}}(\mathcal{S}_{\bar \mathbf{W}})_1-\sqrt{\frac{9}{10}}(\mathcal{S}_{\bar \mathbf{W}})_5 \right) dx_1\right\rvert \notag\\ &\quad\;+\left\lvert\int_{\mathbb{R}} \partial_{x_1}\left( v_1^N \partial_{x_1}^k b_1 \right)\partial_{x_1}^{k-1} \left[\sqrt{\frac{3}{10}} (\mathcal{S}_{\bar \mathbf{W}})_1+ \sqrt{\frac{3}{10}}\frac{3\bar{\lambda}_1}{2}\left( (\mathcal{S}_{\bar \mathbf{W}})_2-\frac{4\mu(\bar{\theta})\bar{m}_1}{3\bar{\rho}^2\bar{\theta}}\partial_{x_1}^2\bar{W}_1\right)+\sqrt{\frac{3}{10}} (\mathcal{S}_{\bar \mathbf{W}})_5 \right]dx_1 \right\rvert\notag\\ &\quad\;+\left\lvert\int_{\mathbb{R}} \partial_{x_1}\left( v_1^{-N} \partial_{x_1}^k b_3 \right)\partial_{x_1}^{k-1} \left[\sqrt{\frac{3}{10}} (\mathcal{S}_{\bar \mathbf{W}})_1-\sqrt{\frac{3}{10}}\frac{3\bar{\lambda}_1}{2}\left( (\mathcal{S}_{\bar \mathbf{W}})_2-\frac{4\mu(\bar{\theta})\bar{m}_1}{3\bar{\rho}^2\bar{\theta}}\partial_{x_1}^2\bar{W}_1\right)+\sqrt{\frac{3}{10}} (\mathcal{S}_{\bar \mathbf{W}})_5 \right]dx_1 \right\rvert. \end{align}\tag{96}\] By 74 , 75 and 76 , we present the calculations of the error term \(\partial_{x_1}^{k-1}D_{-1}\) and the nonlinear terms \(\partial_{x_1}^{k-1} |\partial_{x_1} \bar{\mathbf{W}}|^2\) among \(\mathcal{S}_{\mathbf{B}k}^{(3)}\). Applying the a priori assumptions 77 , we have \[\begin{align} &\bar{\delta}\int_{\mathbb{R}} \partial_{x_1}^{k-1}D_{-1} \left[\partial_{x_1}^{k+1}b_2+\partial_{x_1} \left( v_1^N \partial_{x_1}^kb_1\right)+\partial_{x_1} \left( v_1^{-N} \partial_{x_1}^kb_3\right) \right] dx_1 \notag\\ \leq& C\bar{\delta}(1+t)^{-\frac{1}{2}-k}+C\check{\delta}(1+t)^{-1}\left\lVert\partial_{x_1}^k \mathbf{B}\right\rVert_{L^2}^2 +C\bar{\delta}\left\lVert\partial_{x_1}^{k+1}\mathbf{B}\right\rVert_{L^2}^2 ,\\ & \int_{\mathbb{R}}\partial_{x_1}^{k-1}\left\lvert\partial_{x_1}\bar{\mathbf{W}}\right\rvert^2\left( \left\lvert\partial_{x_1}^{k+1}b_2\right\rvert+\left\lvert\partial_{x_1}(v_1^N\partial_{x_1}^kb_1)\right\rvert+\left\lvert\partial_{x_1}(v_1^{-N}\partial_{x_1}^k b_3)\right\rvert\right)dx_1 \notag\\ \le& C\sum_{i=0}^{[\frac{k-1}{2}]} \left\lVert\partial_{x_1}^{i+1} \bar{\mathbf{W}}\right\rVert_{L^\infty}\left\lVert\partial_{x_1}^{k-i} \bar{\mathbf{W}}\right\rVert_{L^2}\left\lVert\partial_{x_1}^{k+1} \bar{\mathbf{W}}\right\rVert_{L^2}+C\sum_{i=[\frac{k-1}{2}]+1}^{k-1}\left\lVert\partial_{x_1}^{k-i} \bar{\mathbf{W}}\right\rVert_{L^\infty}\left\lVert\partial_{x_1}^{i+1} \bar{\mathbf{W}}\right\rVert_{L^2}\left\lVert\partial_{x_1}^{k+1} \bar{\mathbf{W}}\right\rVert_{L^2} \notag\\ &+C \bar{\delta}(1+t)^{-\frac{1}{2}} \left\lVert\partial_{x_1}^k\bar{\mathbf{W}}\right\rVert_{L^2}\left(\sum_{i=0}^{[\frac{k-1}{2}]} \left\lVert\partial_{x_1}^{i+1} \bar{\mathbf{W}}\right\rVert_{L^\infty}\left\lVert\partial_{x_1}^{k-i} \bar{\mathbf{W}}\right\rVert_{L^2}+ \sum_{i=[\frac{k-1}{2}]+1}^{k-1}\left\lVert\partial_{x_1}^{k-i} \bar{\mathbf{W}}\right\rVert_{L^\infty}\left\lVert\partial_{x_1}^{i+1} \bar{\mathbf{W}}\right\rVert_{L^2}\right)\notag\\ \leq& C\check{\delta}\sum_{i=0}^{k} (1+t)^{i-k-1}\left\lVert\partial_{x_1}^i \bar{\mathbf{W}}\right\rVert_{L^2}^2 +C\check{\delta}\left\lVert\partial_{x_1}^{k+1}\bar{\mathbf{W}}\right\rVert_{L^2}^2 . \end{align}\] The calculations of the remaining terms in 96 are similar. Then we have \[\begin{align} \label{d-k-M-k2} \left\lVert\bar{\mathbf{B}}^{(k)}\mathcal{S}_{\mathbf{B}k}^{(3)}\right\rVert_{L^1(\mathbb{R})}\le C\check{\delta}\sum_{i=0}^{k} (1+t)^{i-k-1}\left\lVert\partial_{x_1}^i \bar{\mathbf{W}}\right\rVert_{L^2}^2 +C\check{\delta}\left\lVert\partial_{x_1}^{k+1}\bar{\mathbf{W}}\right\rVert_{L^2}^2 +C\bar{\delta}(1+t)^{-\frac{1}{2}-k} . \end{align}\tag{97}\]
Step 2. Estimates on \(\int_{\mathbb{R}} \Upsilon_{-\frac{1}{2}} \left\lvert\partial_{x_1}^k b_i\right\rvert^2 dx_1\).
Notice that we still need to control \(\int_{\mathbb{R}} \Upsilon_{-\frac{1}{2}} \left\lvert\partial_{x_1}^k b_i\right\rvert^2 dx_1\) in 94 . From 25 and 39 , we have \[\bar{\delta}\Upsilon_{-1/2}\approx\bar{\delta}(1+t)^{-\frac{1}{2}}e^{-\frac{\tilde{d} x_1^2}{1+t}},\qquad \partial_{x_1}\check{\rho}\approx\bar{\delta}(1+t)^{-\frac{1}{2}}e^{-\frac{d x_1^2}{1+t}},\] with \(\tilde{d}<d\). Therefore \(\int_{\mathbb{R}} \Upsilon_{-\frac{1}{2}} \left\lvert\partial_{x_1}^k b_i\right\rvert^2 dx_1\) cannot be directly controlled by \(G_k\) 85 . We need to further develop weighted heat kernel inequalities 99 and 100 in order to obtain the control of \(\int_{\mathbb{R}} \Upsilon_{-\frac{1}{2}} \left\lvert\partial_{x_1}^k b_i\right\rvert^2 dx_1\) for \(k \ge 1\). To derive these estimates, let \[\tilde{h}(t,x_1)=\int_{-\infty}^{x_1} \Upsilon_{-\frac{1}{2}}(t,y)dy,\] then it follows that \(\|\tilde{h}\|_{L^\infty}\leq C\) and \(4\tilde{d} \partial_t \tilde{h}= \partial_{x_1}\Upsilon_{-\frac{1}{2}}\). Multiplying 83 \(_1\) by \(\tilde{h}\partial_{x_1}^k{b}_1\), we thus have \[\begin{align} \label{2025-10-30-3} & \frac{1}{2} \frac{d}{dt}\left[ \tilde{h} (\partial_{x_1}^kb_1)^2 \right]-\frac{1}{2}\partial_t \tilde{h}\left(\partial_{x_1}^k b_1 \right)^2 + \frac{1}{2}\partial_{x_1} \left[ \tilde{h} \bar{\lambda}_1 \left(\partial_{x_1}^k b_1 \right)^2 \right] - \frac{1}{2} \partial_{x_1} \left( \tilde{h} \bar{\lambda}_1\right)\left( \partial_{x_1}^k b_1\right)^2\notag\\ =&\left(\mathcal{S}_{\mathbf{B}k}^{(1)} \right)_1\tilde{h}\partial_{x_1}^k b_1+ \left[ \left( \bar{\mathcal{A}}_4 \partial_{x_1}^{k+2} \mathbf{B}\right)_1 + \sum_{i=2}^3 \left(\mathcal{S}_{\mathbf{B}k}^{(i)} \right)_1 \right] \tilde{h} \partial_{x_1}^k b_1. \end{align}\tag{98}\] Using the same method as for obtaining 91 , 92 , 93 and 94 , one has \[\begin{align} \int_{\mathbb{R}}\left(\mathcal{S}_{\mathbf{B}k}^{(1)} \right)_1\tilde{h}\partial_{x_1}^k b_1 dx_1 \lesssim \bar{\delta}\int_{\mathbb{R}} \Upsilon_{-\frac{1}{2}} \left\lvert\partial_{x_1}^k b_1\right\rvert^2 dx_1 + \sum_{i=0}^k(1+t)^{-i} \int_{\mathbb{R}} \partial_{x_1}\check{\rho}\left\lvert\partial_{x_1}^{k-i}b_1\right\rvert^2 dx_1. \end{align}\] Applying an argument similar to 97 , it holds that \[\begin{align} \int_{\mathbb{R}}\Big[ \left(\bar{\mathcal{A}}_4 \partial_{x_1}^{k+2} \mathbf{B}\right)_1 + \sum_{i=2}^3 \left(\mathcal{S}_{\mathbf{B}k}^{(i)} \right)_1 \Big] \tilde{h} \partial_{x_1}^k b_1 dx_1 \lesssim \sum_{i=0}^{k+1}(1+t)^{j-k-1}\left\lVert\partial_{x_1}^j \bar{\mathbf{W}}\right\rVert_{L^2}^2+(1+t)^{-\frac{1}{2}-k}. \end{align}\] Thus, integrating 98 with respect to \(x_1\), we obtain \[\begin{align} \label{2025-10-4} \int_{\mathbb{R}} \Upsilon_{-\frac{1}{2}} \left\lvert\partial_{x_1}^k b_1\right\rvert^2 dx_1 +\partial_t \int_{\mathbb{R}} \tilde{h} \left(\partial_{x_1}^k b_1 \right)^2 dx_1 \lesssim & \sum_{j=0}^{k+1} (1+t)^{j-k-1}\left\lVert\partial_{x_1}^j \bar{\mathbf{W}}\right\rVert_{L^2}^2 + (1+t)^{-\frac{1}{2}-k} \notag \\ &+ \sum_{j=0}^k (1+t)^{-j} \int_{\mathbb{R}} \partial_{x_1}\check{\rho}\left\lvert\partial_{x_1}^{k-j}b_1\right\rvert^2 dx_1 . \end{align}\tag{99}\] Using the same argument as for obtaining 99 , we have \[\begin{align} \label{2025-10-5} \int_{\mathbb{R}} \Upsilon_{-\frac{1}{2}} \left\lvert\partial_{x_1}^k b_3\right\rvert^2 dx_1- \partial_t \int_{\mathbb{R}} \tilde{h} \left(\partial_{x_1}^k b_3 \right)^2 dx_1 \lesssim &\sum_{j=0}^{k+1} (1+t)^{j-k-1}\left\lVert\partial_{x_1}^j \bar{\mathbf{W}}\right\rVert_{L^2}^2 + (1+t)^{-\frac{1}{2}-k} \notag \\ &+ \sum_{j=0}^k (1+t)^{-j} \int_{\mathbb{R}} \partial_{x_1}\check{\rho}\left\lvert\partial_{x_1}^{k-j}b_3\right\rvert^2 dx_1 . \end{align}\tag{100}\] Combining 84 90 , 94 , 97 , 99 and 100 , one has \[\begin{align} &\frac{d}{dt}\big(\sum_{i=0}^n E_i^{\#}\big)+\sum_{i=0}^n(K^{\#}_i+G_{i})\notag\\ \leq& C\bar{\delta}(1+t)^{-\frac{1}{2}}+ C\check{\delta}\left[(1+t)^{-1}\big(\sum_{i=0}^n {E}_i\big)+\left\lVert\partial_{x_1}\bar{W}_1\right\rVert_{H^n}^2+\left\lVert\partial_{x_1}\bar{W}_3\right\rVert_{H^n}^2+\left\lVert\partial_{x_1}\bar{W}_4\right\rVert_{H^n}^2\right],\tag{101}\\ &\frac{d}{dt}\big(\sum_{i=1}^n {E}^{\#}_i\big)+\sum_{i=1}^n (K^{\#}_i+G_{i})\notag \leq C\bar{\delta}(1+t)^{-\frac{3}{2}}+C\check{\delta}\bigg[(1+t)^{-1}\big(\sum_{i=1}^n{E}_i+\sum_{i=0}^{n-1}G_i\big)+(1+t)^{-2}{E}_0\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\;+\left\lVert\partial_{x_1}^2\bar{W}_1\right\rVert_{H^{n-1}}^2+\left\lVert\partial_{x_1}^2\bar{W}_3\right\rVert_{H^{n-1}}^2+\left\lVert\partial_{x_1}^2\bar{W}_4\right\rVert_{H^{n-1}}^2\bigg],\tag{102}\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\dots\dots, \notag\\ &\frac{d}{dt}\big(\sum_{i=n-1}^n {E}_i^{\#}\big)+\sum_{i=n-1}^n (K^{\#}_i+G_{i})\leq C\bar{\delta}(1+t)^{\frac{1}{2}-n}+C\check{\delta}\bigg[(1+t)^{-1}\sum_{i=n-1}^n E_i +\sum_{i=0}^{n-2}(1+t)^{i-n}E_i\notag\\ &\qquad\qquad\qquad\qquad\qquad\qquad+\sum_{i=0}^{n-2}(1+t)^{i+1-n}G_i+\left\lVert\partial_{x_1}^{n}\bar{W}_1\right\rVert_{H^1}^2+\left\lVert\partial_{x_1}^{n}\bar{W}_3\right\rVert_{H^1}^2+\left\lVert\partial_{x_1}^{n}\bar{W}_4\right\rVert_{H^1}^2\bigg],\tag{103}\\ &\frac{d}{dt}{E}^{\#}_n+K^{\#}_n+G_{n}\le C\bar{\delta}(1+t)^{-\frac{1}{2}-n}+ C\check{\delta}\bigg[\sum_{i=0}^{n}(1+t)^{i-n-1}E_i\notag\\ & \qquad\qquad\qquad\qquad\qquad\qquad+\sum_{i=0}^{n-1}(1+t)^{i-n}G_i+\left\lVert\partial_{x_1}^{n+1}\bar{W}_1\right\rVert_{L^2}^2+\left\lVert\partial_{x_1}^{n+1}\bar{W}_3\right\rVert_{L^2}^2+\left\lVert\partial_{x_1}^{n+1}\bar{W}_4\right\rVert_{L^2}^2 \bigg].\tag{104} \end{align}\]
Step 3. Estimates on \(\left\lVert\partial_{x_1}^{k+1}\bar{W}_1\right\rVert_{L^2}\).
We still need to estimate \(\left\lVert\partial_{x_1}^{k+1}\bar{W}_1\right\rVert_{L^2}\). Applying \(\partial_{x_1}^k\) to 73 \(_2\) and then multiplying the resulting equation by \(\partial_{x_1}^{k+1}\bar{W}_1\), one has \[\begin{align} \label{eq-phi-x} &\frac{d}{dt}\int_{\mathbb{R}} \partial_{x_1}^k\bar{W}_2\partial_{x_1}^{k+1}\bar{W}_1 dx_1 + \frac{2}{3}\left\lVert\partial_{x_1}^{k+1}\bar{W}_1\right\rVert_{L^2}^2-\frac{4}{3}\int_{\mathbb{R}}\frac{\mu(\bar{\theta})}{\bar{\rho}} \partial_{x_1}^{k+2}\bar{W}_2\partial_{x_1}^{k+1}\bar{W}_1 dx_1 + \int_{\mathbb{R}} \partial_t \partial_{x_1}^k \bar{W}_1 \partial_{x_1}^{k+1}\bar{W}_2 dx_1\notag\\ =&\frac{4}{3}\sum_{i=0}^{k-1}\int_{\mathbb{R}}\partial_{x_1}^{k-i}\left( \frac{\mu(\bar{\theta})}{\bar{\rho}} \right)\partial_{x_1}^{i+2}\bar{W}_2 \partial_{x_1}^{k+1}\bar{W}_1 dx_1 -\frac{2}{3}\int_{\mathbb{R}}\partial_{x_1}^{k+1}\bar{W}_5 \partial_{x_1}^{k+1}\bar{W}_1 dx_1 \notag\\ &+\int_{\mathbb{R}} \partial_{x_1}^k (\mathcal{S}_{\bar \mathbf{W}})_2\partial_{x_1}^{k+1}\bar{W}_1 dx_1-\frac{4}{3}\int_{\mathbb{R}} \partial_{x_1}^{k}\left[ \frac{\mu(\bar{\theta})\bar{m}_1}{\bar{\rho}^2 \bar{\theta}} \partial_{x_1}^2\bar{W}_{1} \right] \partial_{x_1}^{k+1} \bar{W}_1. \end{align}\tag{105}\] Then we use 73 \(_1\) to deal with \(-\frac{4}{3}\int_{\mathbb{R}}\frac{\mu(\bar{\theta})}{\bar{\rho}} \partial_{x_1}^{k+2}\bar{W}_2\partial_{x_1}^{k+1}\bar{W}_1 dx_1\) as \[\begin{align} \label{px-k431-W1} &-\frac{4}{3}\int_{\mathbb{R}}\frac{\mu(\bar{\theta})}{\bar{\rho}} \partial_{x_1}^{k+2}\bar{W}_2\partial_{x_1}^{k+1}\bar{W}_1 dx_1 \notag\\ =&\frac{4}{3}\int_{\mathbb{R}}\frac{\mu(\bar{\theta})}{\bar{\rho}} \partial_{x_1}^{k+1} \left( \frac{\partial_t\bar{W}_1}{\bar{\theta}} \right) \partial_{x_1}^{k+1}\bar{W}_1 dx_1 -\frac{4}{3}\int_{\mathbb{R}}\frac{\mu(\bar{\theta})}{\bar{\rho}} \partial_{x_1}^{k+1} \left( \frac{(\mathcal{S}_{\bar \mathbf{W}})_1}{\bar{\theta}} \right) \partial_{x_1}^{k+1}\bar{W}_1 dx_1 \notag\\ =&\frac{2}{3}\frac{d}{dt}\left(\int_{\mathbb{R}} \frac{\mu(\bar{\theta})}{\bar{\theta}\bar{\rho}} \left\lvert\partial_{x_1}^{k+1} \bar{W}_1\right\rvert ^2 dx_1\right)-\frac{2}{3}\int_{\mathbb{R}} \partial_t\left(\frac{\mu(\bar{\theta})}{\bar{\theta}\bar{\rho}} \right)\left\lvert\partial_{x_1}^{k+1} \bar{W}_1\right\rvert ^2 dx_1 -\frac{4}{3}\int_{\mathbb{R}}\frac{\mu(\bar{\theta})}{\bar{\rho}} \partial_{x_1}^{k+1} \left( \frac{(\mathcal{S}_{\bar \mathbf{W}})_1}{\bar{\theta}} \right) \partial_{x_1}^{k+1}\bar{W}_1 dx_1\notag\\ & - \frac{4}{3}\sum_{i=0}^k \int_{\mathbb{R}}\frac{\mu(\bar{\theta})}{\bar{\rho}} \partial_{x_1}^{k+1-i}\left( \frac{1}{\bar{\theta}} \right) \left[\partial_{x_1}^i \left( \bar{\theta}\bar{W}_{2x_1}\right) -\partial_{x_1}^i (\mathcal{S}_{\bar \mathbf{W}})_1 \right]\partial_{x_1}^{k+1} \bar{W}_1 dx_1. \end{align}\tag{106}\] And using 73 \(_1\) again, one has \[\begin{align} \label{px-k431-W1-1} \int_{\mathbb{R}}\partial_t \partial_{x_1}^k \bar{W}_1 \partial_{x_1}^{k+1}\bar{W}_2 dx_1 = -\int_{\mathbb{R}} \partial_{x_1}^k\left(\tilde{\theta}\bar{W}_{2x_1} \right) \partial_{x_1}^{k+1}\bar{W}_2 dx_1 + \int_{\mathbb{R}}\partial_{x_1}^k(\mathcal{S}_{\bar \mathbf{W}})_1 \partial_{x_1}^{k+1}\bar{W}_2 dx_1. \end{align}\tag{107}\] Combining 105 , 106 , and 107 , we obtain \[\begin{align} \label{px-k431-W1-2} &\frac{2}{3}\frac{d}{dt}\left\lVert \sqrt{\frac{\mu(\bar{\theta})}{\bar{\theta}\bar{\rho}}} \partial_{x_1}^{k+1} \bar{W}_1\right\rVert_{L^2}^2 + \frac{d}{dt}\int_{\mathbb{R}} \partial_{x_1}^k\bar{W}_2\partial_{x_1}^{k+1}\bar{W}_1 dx_1 + \frac{2}{3}\left\lVert\partial_{x_1}^{k+1}\bar{W}_1\right\rVert_{L^2}^2=I_5+I_6, \end{align}\tag{108}\] where \[\begin{align} I_5=&\frac{4}{3}\sum_{i=0}^{k-1}\int_{\mathbb{R}}\partial_{x_1}^{k-i}\left( \frac{\mu(\bar{\theta})}{\bar{\rho}} \right)\partial_{x_1}^{i+2}\bar{W}_2 \partial_{x_1}^{k+1}\bar{W}_1 dx_1 -\frac{2}{3}\int_{\mathbb{R}}\partial_{x_1}^{k+1}\bar{W}_5\partial_{x_1}^{k+1}\bar{W}_1 dx_1 +\int_{\mathbb{R}} \partial_{x_1}^k\left(\bar{\theta}\bar{W}_{2x_1} \right) \partial_{x_1}^{k+1}\bar{W}_2 dx_1 \\ & +\frac{2}{3} \int_{\mathbb{R}}\partial_t \left( \frac{\mu(\bar{\theta})}{\bar{\theta}\bar{\rho}} \right) \left\lvert\partial_{x_1}^{k+1}\bar{W}_1\right\rvert^2 dx_1 + \frac{4}{3}\sum_{i=0}^k \int_{\mathbb{R}}\frac{\mu(\bar{\theta})}{\bar{\rho}} \partial_{x_1}^{k+1-i}\left( \frac{1}{\bar{\theta}} \right) \partial_{x_1}^i \left( \bar{\theta}\bar{W}_{2x_1}\right) \partial_{x_1}^{k+1} \bar{W}_1 dx_1,\\ I_6=& \int_{\mathbb{R}} \partial_{x_1}^k (\mathcal{S}_{\bar \mathbf{W}})_2\partial_{x_1}^{k+1}\bar{W}_1 dx_1 +\frac{4}{3}\int_{\mathbb{R}}\frac{\mu(\bar{\theta})}{\bar{\rho}} \partial_{x_1}^{k+1} \left( \frac{(\mathcal{S}_{\bar \mathbf{W}})_1}{\bar{\theta}} \right) \partial_{x_1}^{k+1}\bar{W}_1 dx_1 -\frac{4}{3}\int_{\mathbb{R}} \partial_{x_1}^{k}\left[ \frac{\mu(\bar{\theta})\bar{m}_1}{\bar{\rho}^2 \bar{\theta}} \partial_{x_1}^2\bar{W}_{1} \right] \partial_{x_1}^{k+1} \bar{W}_1 dx_1 \notag\\ &+ \frac{4}{3}\sum_{i=0}^k \int_{\mathbb{R}}\frac{\mu(\bar{\theta})}{\bar{\rho}} \partial_{x_1}^{k+1-i}\left( \frac{1}{\bar{\theta}} \right)\partial_{x_1}^i (\mathcal{S}_{\bar \mathbf{W}})_1\partial_{x_1}^{k+1} \bar{W}_1 dx_1 - \int_{\mathbb{R}}\partial_{x_1}^k(\mathcal{S}_{\bar \mathbf{W}})_1 \partial_{x_1}^{k+1}\bar{W}_2 dx_1. \end{align}\] By Hölder’s inequality, we have \[\begin{align} \label{I-7} I_5 \le&C\check{\delta}\sum_{i=1}^k (1+t)^{i-k-1}\left\lVert\partial_{x_1}^i \bar{\mathbf{W}}\right\rVert_{L^2}^2 + \frac{1}{16000} \left\lVert\partial_{x_1}^{k+1}\bar{W}_1\right\rVert_{L^2}^2 +C\sum_{i=2}^5 \left\lVert\partial_{x_1}^{k+1} \bar{W}_i\right\rVert_{L^2}^2. \end{align}\tag{109}\] For \(I_6\), we apply integration by parts to control \(-\frac{4}{3}\int_{\mathbb{R}} \partial_{x_1}^{k}\left[ \frac{\mu(\bar{\theta})\bar{m}_1}{\bar{\rho}^2 \bar{\theta}} \partial_{x_1}^2\bar{W}_{1} \right] \partial_{x_1}^{k+1} \bar{W}_1 dx_1\) as \[\begin{align} & -\frac{4}{3}\int_{\mathbb{R}} \partial_{x_1}^{k}\left[ \frac{\mu(\bar{\theta})\bar{m}_1}{\bar{\rho}^2 \bar{\theta}} \partial_{x_1}^2\bar{W}_{1} \right] \partial_{x_1}^{k+1} \bar{W}_1 dx_1 \\ =& \frac{2}{3}\int_{\mathbb{R}} \partial_{x_1} \left( \frac{\mu(\bar{\theta})\bar{m}_1}{\bar{\rho}^2 \bar{\theta}} \right) \left\lvert\partial_{x_1}^{k+1} \bar{W}_1\right\rvert^2 d{x_1} -\frac{4}{3} \sum_{i=0}^{k-1} \int_{\mathbb{R}} \partial_{x_1}^{k-i} \left( \frac{\mu(\bar{\theta})\bar{m}_1}{\bar{\rho}^2 \bar{\theta}} \right) \partial_{x_1}^{i+2} \bar{W}_1 \partial_{x_1}^{k+1} \bar{W}_1 dx_1 \\ \le& C\check{\delta}\sum_{i=0}^k (1+t)^{i-k-1}\left\lVert\partial_{x_1}^i \bar{W}_1\right\rVert_{L^2}^2 + C\check{\delta}\left\lVert\partial_{x_1}^{k+1}\bar{W}_1\right\rVert_{L^2}^2. \end{align}\] Subsequently, we address the control of \(\bar{\delta}D_{-1}\) in terms of \(I_6\) that involves \((\mathcal{S}_{\bar \mathbf{W}})_1\) and \((\mathcal{S}_{\bar \mathbf{W}})_2\) 74 . It holds that \[\begin{align} \bar{\delta}\int_{\mathbb{R}} \partial_{x_1}^k D_{-1} \left( \partial_{x_1}^{k+1} \bar{W}_1 + \partial_{x_1}^{k+1} \bar{W}_2\right) dx_1 \le C \bar{\delta}(1+t)^{-k-\frac{3}{2}}+C\bar{\delta}\left\lVert\partial_{x_1}^{k+1}\bar{\mathbf{W}}\right\rVert_{L^2}^2. \end{align}\] Then, by the a priori assumptions 77 , the nonlinear terms of \((\mathcal{S}_{\bar \mathbf{W}})_2\) 74 in \(I_6\) can be bounded as \[\begin{align} &\sum_{i=1}^5 \int_{\mathbb{R}} \partial_{x_1}^k \left(\partial_{x_1}\bar{W}_i \right)^2 \partial_{x_1}^{k+1} \bar{W}_1 dx_1 \\ =& \sum_{i=1}^5 \left( \sum_{j=1}^{[\frac{k-1}{2}]} + \sum_{j=[\frac{k-1}{2}]+1}^{k-1} \right) \int_{\mathbb{R}} \partial_{x_1}^{j+1}\bar{W}_i \partial_{x_1}^{k+1-j}\bar{W}_i \partial_{x_1}^{k+1}\bar{W}_1dx_1+2\sum_{i=1}^5 \int_{\mathbb{R}} \partial_{x_1} \bar{W}_i \partial_{x_1}^{k+1}\bar{W}_i \partial_{x_1}^{k+1} \bar{W}_1 dx_1\\ \le& C\left\lVert\partial_{x_1}^{k+1}\bar{\mathbf{W}}\right\rVert_{L^2}\left( \sum_{j=1}^{[\frac{k-1}{2}]} \left\lVert\partial_{x_1}^{j+1}\bar{\mathbf{W}}\right\rVert_{L^\infty} \left\lVert\partial_{x_1}^{k+1-j}\bar{\mathbf{W}}\right\rVert_{L^2}+\sum_{j=[\frac{k-1}{2}]+1}^{k-1}\left\lVert\partial_{x_1}^{k+1-j}\bar{W}\right\rVert_{L^\infty} \left\lVert\partial_{x_1}^{j+1}\bar{\mathbf{W}}\right\rVert_{L^2}\right)\\ \le& C\check{\delta}\sum_{i=0}^k (1+t)^{i-k-1}\left\lVert\partial_{x_1}^i \bar{\mathbf{W}}\right\rVert_{L^2}^2+C\check{\delta}\left\lVert\partial_{x_1}^{k+1}\bar{\mathbf{W}}\right\rVert_{L^2}^2. \end{align}\] The other terms in \(I_6\) can be controlled by the Hölder’s inequality and the a priori assumptions 77 . Thus we obtain \[\begin{align} \label{I-8} I_6 \le& \sum_{i=0}^k C\check{\delta}(1+t)^{i-k-1}\left\lVert\partial_{x_1}^i \bar{\mathbf{W}}\right\rVert_{L^2}^2+C\check{\delta}\left\lVert\partial_{x_1}^{k+1}\bar{\mathbf{W}}\right\rVert_{L^2}^2+C\bar{\delta}(1+t)^{-k-\frac{3}{2}}. \end{align}\tag{110}\] Combining 108 , 109 and 110 , one has \[\begin{align} \label{px-k431-W1-estimate} & \frac{d}{dt}\left\lVert \sqrt{\frac{\mu(\bar{\theta})}{\bar{\theta}\bar{\rho}}} \partial_{x_1}^{k+1} \bar{W}_1\right\rVert_{L^2}^2 + \frac{d}{dt}\int_{\mathbb{R}} \partial_{x_1}^k\bar{W}_2\partial_{x_1}^{k+1}\bar{W}_1 dx_1 + c\left\lVert\partial_{x_1}^{k+1}\bar{W}_1\right\rVert_{L^2}^2 \notag\\ \le &C \check{\delta}\sum_{i=0}^k(1+t)^{i-k-1}\left\lVert\partial_{x_1}^i \bar{\mathbf{W}}\right\rVert_{L^2}^2+C\bar{\delta}(1+t)^{-k-\frac{3}{2}} +C\sum_{i=2}^5\left\lVert\partial_{x_1}^{k+1}W_i\right\rVert_{L^2}^2. \end{align}\tag{111}\]
Step 4. Estimates on \(\bar{W}_i\), \(i=3,4\).
Applying \(\partial_{x_1}^k\) to 73 \(_{3,4}\), multiplying them by \(\partial^k\bar{W}_i\), \(i=3,4\) and then integrating the resulting equation over \(\mathbb{R}\) with respect to \(x_1\), one has \[\begin{align} &\frac{d}{dt}\left\lVert\partial_{x_1}^k\bar{W}_i\right\rVert_{L^2}^2+\int_{\mathbb{R}}\frac{\mu(\bar{\theta})}{\bar{\rho}}\left\lvert\partial_{x_1}^{k+1}\bar{W}_i\right\rvert^2dx_1\\ \lesssim& \left\lvert\int_{\mathbb{R}}\partial_{x_1}^k\left( \frac{\mu(\bar{\theta})}{\bar{\rho}} \partial_{x_1}^2 \bar{W}_i\right)\partial_{x_1}^k \bar{W}_i+ \frac{\mu(\bar{\theta})}{\bar{\rho}}\left\lvert\partial_{x_1}^{k+1}\bar{W}_i\right\rvert^2dx_1\right\rvert+\int_{\mathbb{R}}\left\lvert\partial_{x_1}^{k+1}\bar{W}_i \partial_{x_1}^{k_1}(\mathcal{S}_{\bar \mathbf{W}})_i\right\rvert dx_1\nonumber:=I_{W}^{k1}+I_{W}^{k2}. \end{align}\] The direct calculation yields \[\begin{align} \notag I_{W}^{k1}\leq C\check{\delta}\sum_{j=0}^{k}(1+t)^{-j}\left\lVert\partial_{x_1}^{k+1-j}\bar{\mathbf{W}}\right\rVert_{L^2}^2, \end{align}\] and the estimate of \(I_{W}^{k2}\) is similar to 95 97 . Finally, we have \[\begin{align} & \sum_{i=3}^4\left[\frac{d}{dt}\left\lVert\partial_{x_1}^k\bar{W}_i\right\rVert_{L^2}^2+\int_{\mathbb{R}}\frac{\mu(\bar{\theta})}{\bar{\rho}}\left\lvert\partial_{x_1}^{k+1}\bar{W}_i\right\rvert^2dx_1\right]\label{psi2}\\ \leq& C\bar{\delta} (1+t)^{-\frac{1}{2}-k}+C\check{\delta}\left[\left\lVert\partial_{x_1}^{k+1}\bar{W}_1\right\rVert_{L^2}^2+\left\lVert\partial_{x_1}^{k+1}\bar{W}_2\right\rVert_{L^2}^2+\left\lVert\partial_{x_1}^{k+1}\bar{W}_5\right\rVert_{L^2}^2+\sum_{j=1}^{k}(1+t)^{-j}\left\lVert\partial_{x_1}^{k+1-j}\bar{\mathbf{W}}\right\rVert_{L^2}^2\right].\notag \end{align}\tag{112}\] Combining 101 104 , 111 and 112 , we complete the proof of Lemma 14. ◻
Proof of Proposition 12:
In the sequel, we use the following fact which comes from 81 and 82 : \[\begin{align} \notag E_n \le C K_{n-1}+C\bar{\delta}\sum_{j=0}^{n-2}(1+t)^{j+1-n}K_j+C\bar{\delta}(1+t)^{-n}E_0. \end{align}\] By Grönwall’s inequality and ?? , one has \[\begin{align} \label{basic-W-E} \sum_{i=0}^n E_i\leq C(\varepsilon^2_0+\bar{\delta})(1+t)^{\frac{1}{2}},\qquad\quad\sum_{i=0}^n\int_{\mathbb{R}}(K_i+G_{i})dx\leq C(\varepsilon^2_0+\bar{\delta})(1+t)^{\frac{1}{2}}. \end{align}\tag{113}\] Multiplying ?? by \((1+t)\) and then integrating on \([0,t]\), one has \[\begin{align} \notag (1+t)\sum_{i=1}^nE_i+\sum_{i=1}^n\int_{0}^t(1+\tau)(K_i+G_{i})d\tau\leq C(\varepsilon^2_0+\bar{\delta})(1+t)^{\frac{1}{2}}. \end{align}\] Then it follows that \[\begin{align} \label{E1} \sum_{i=1}^nE_i\leq C(\varepsilon^2_0+\bar{\delta})(1+t)^{-\frac{1}{2}},\qquad\quad\sum_{i=1}^n\int_{0}^t(1+\tau)(K_i+G_{i})d\tau\leq C(\varepsilon^2_0+\bar{\delta})(1+t)^{\frac{1}{2}}. \end{align}\tag{114}\] By the same argument, we obtain \[\begin{align} \label{Ei} \sum_{i=k}^nE_i\leq C(\varepsilon^2_0+\bar{\delta})(1+t)^{\frac{1}{2}-k},\qquad\sum_{i=k}^n\int_{0}^t(1+\tau)^k(K_i+G_{i})d\tau\leq C(\varepsilon^2_0+\bar{\delta})(1+t)^{\frac{1}{2}}. \end{align}\tag{115}\] Finally, multiplying ?? by \((1+t)^n\), one has \[\begin{align} &(1+t)^nE_n+\int_{0}^t(1+\tau)^n(K_n+G_{n})d\tau\\ \leq&\sum_{i=0}^{n-1}\int_0^t (1+\tau)^{i}\left(K_{i}+G_i \right)d\tau + \int_0^t (1+\tau)^{-1} E_0 d\tau+ C\bar{\delta}(1+t)^{\frac{1}{2}},\\ \leq& C(\bar{\delta}+\varepsilon^2_0)(1+t)^{\frac{1}{2}}. \end{align}\] Then it holds that \[\begin{align} \label{E2} E_n\leq C(\varepsilon^2_0+\bar{\delta})(1+t)^{\frac{1}{2}-n}. \end{align}\tag{116}\] By 81 , 113 and 114 , we have \[\begin{align} \label{W-final-1} \left\lVert\bar{\mathbf{W}}\right\rVert_{L^\infty} \le C \left\lVert\bar{\mathbf{W}}\right\rVert_{L^2}^{\frac{1}{2}}\left\lVert\partial_{x_1} \bar{\mathbf{W}}\right\rVert_{L^2}^{\frac{1}{2}} \le C E_0^{\frac{1}{2}}E_1^{\frac{1}{2}} \leq C\bar{\delta}^{\frac{1}{2}}. \end{align}\tag{117}\] Combining 81 and 114 117 , we complete the proof of Proposition 12.0◻
Corollary 15. Under the same assumptions of Theorem 8, for \(k\geq0\), one has \[\begin{align} \left\lVert\mathbf{W}\right\rVert_{L^{\infty}}\leq C \bar{\delta}^{\frac{1}{2}}, \qquad\quad \left\lVert\partial_{x_1}^k \Xi\right\rVert_{L^2}^2\leq C \bar{\delta}(1+t)^{-\frac{1+2k}{2}}, \qquad\quad \left\lVert\partial_{x_1}^k\tilde{\mathcal{R}}\right\rVert_{L^2}^2\leq C \bar{\delta}^{3} (1+t)^{-\frac{5+2k}{2}}. \end{align}\]
Proof. Applying Theorem 11 and the relationship between \(\Xi=\partial_{x_1}\mathbf{W}\) and \(\bar{\mathbf{W}}\) 72 , one has \(\left\lVert\mathbf{W}\right\rVert_{L^{\infty}}\leq C \bar{\delta}^{\frac{1}{2}}\) and \(\left\lVert\partial_{x_1}^k \Xi\right\rVert_{L^2}^2\leq C \bar{\delta}(1+t)^{-\frac{1+2k}{2}}\). From the expression of \(\tilde{\mathcal{R}}\) 51 , for \(k=0\), we have \[\begin{align} \left\lVert\tilde{\mathcal{R}}\right\rVert_{L^2}^2\le C\left\lVert\left\lvert\Xi\right\rvert^3\right\rVert_{L^2}^2+C\bar{\delta}\left\lVert D_{-\frac{1}{2}} \left\lvert\Xi\right\rvert^2\right\rVert_{L^2}^2+C\left\lVert\left\lvert\Xi\right\rvert\left\lvert\Xi_{x_1}\right\rvert\right\rVert_{L^2}^2\leq C \bar{\delta}^{3}(1+t)^{-\frac{5}{2}}. \end{align}\] Similarly, for \(k\geq1\), we have \(\left\lVert\partial_{x_1}^k\tilde{\mathcal{R}}\right\rVert_{L^2}^2\leq C \bar{\delta}^{3}(1+t)^{-\frac{5+2k}{2}}\). Then we have completed the proof of Corollary 15. ◻
In this section, we study the estimates for the perturbation 54 near the local Maxwellian \(M_{[\tilde{\rho},\tilde{u},\tilde{\theta}]}\) with its macroscopic quantities constructed in 50 for the Landau equation 1 .
Recall the definition of the perturbation \((\phi,\varphi,h,\psi,\zeta,\sqrt{\mu}g)\) 54 , by equation 17 and profile 53 , we have \[\begin{align} \label{perturbation-1} \begin{cases} \partial_t\phi+\tilde{\rho}\operatorname{div}_x \psi=\mathcal{S}_{\phi}^{f}, \\ \partial_t\psi+\frac{2}{3}\frac{\tilde{\theta}}{\tilde{\rho}} \nabla_x \phi+ \frac{2}{3} \nabla_x \zeta-\frac{\mu(\tilde{\theta})}{\tilde{\rho}}\left(\Delta_x \psi+\frac{1}{3} \nabla_x \operatorname{div}_x \psi\right)=\mathcal{S}_{\psi}^{f1}+\mathcal{S}_{\psi}^{f2}+\mathcal{S}_{\psi}^{m},\\ \partial_t\zeta+\frac{2}{3} \tilde{\theta}\operatorname{div}_x \psi-\frac{\kappa(\tilde{\theta})}{\tilde{\rho}}\Delta_x \zeta=\mathcal{S}_{\zeta}^{f1}+\mathcal{S}_{\zeta}^{f2}+\mathcal{S}_{\zeta}^{m}, \end{cases} \end{align}\tag{118}\] where \[\begin{align} &\mathcal{S}_{\phi}^{f}:=-u \cdot \nabla_x \phi-\phi \operatorname{div}_x \psi-\psi \cdot \nabla_x \tilde{\rho}-\phi \operatorname{div}_x \tilde{u}-\partial_{x_1}\tilde{\mathcal{R}}_{0}, \tag{119} \\ &\mathcal{S}_{\psi}^{f1}:=-u \cdot \nabla_x \psi-\frac{\nabla_x(p-\tilde{p})}{\tilde{\rho}}+\frac{2}{3}\frac{\tilde{\theta}}{\tilde{\rho}}\nabla_x\phi + \frac{2}{3}\nabla_x\zeta- \psi \cdot \nabla_x \tilde{u}+\frac{\phi}{\tilde{\rho}\rho}\nabla_x\tilde{p}- \left(\frac{1}{\rho}-\frac{1}{\tilde{\rho}} \right) \nabla_x\left(p-\tilde{p}\right),\tag{120} \\ &\mathcal{S}_{\psi}^{f2}:=\frac{\partial_{x_1}\tilde{\mathcal{R}}_0 \tilde{u}-\partial_{x_1} (\tilde{\mathcal{R}}_1,\tilde{\mathcal{R}}_2,\tilde{\mathcal{R}}_3)^t}{\tilde{\rho}}+\frac{\mu(\theta)-\mu(\tilde{\theta})}{\rho}\left(\Delta_x u+\frac{1}{3} \nabla_x \operatorname{div}_x u\right)\notag\\ &\quad+\frac{\mu^{\prime}(\tilde{\theta})}{\rho} \nabla_x \tilde{\theta}\cdot\left(\nabla_x \psi+(\nabla_x \psi)^{t}-\frac{2}{3}\mathbb{I} \operatorname{div}_x \psi\right)-\mu'(\tilde{\theta})\frac{\nabla_x\tilde{\theta}\phi}{\tilde{\rho}\rho}\cdot\left(\nabla_x \tilde{u}+(\nabla_x \tilde{u})^{t}-\frac{2}{3}\mathbb{I} \operatorname{div}_x \tilde{u}\right)\nonumber\\ &\quad+\frac{\left({\mu^{\prime}(\theta)} \nabla_x \theta-\mu'(\tilde{\theta})\nabla_x\tilde{\theta}\right)}{\rho} \cdot\left(\nabla_x {u}+(\nabla_x {u})^{t}-\frac{2}{3}\mathbb{I} \operatorname{div}_x {u}\right) +\mu(\tilde{\theta})\left(\frac{1}{\rho}-\frac{1}{\tilde{\rho}}\right)\left(\Delta_x u+\frac{1}{3} \nabla_x \operatorname{div}_x u\right), \tag{121}\\ &\mathcal{S}_{\psi}^{m} :=-\frac{1}{\rho} \int_{\mathbb{R}^3} \xi \otimes \xi \cdot \nabla_{x} \left(L_{M}^{-1}\Pi-L_{\bar{M}}^{-1}\bar{\Pi}_1\right) d \xi-\frac{\phi}{\tilde{\rho}\rho} \int_{\mathbb{R}^3} \xi \otimes \xi \cdot \nabla_{x} L_{\bar{M}}^{-1}\bar{\Pi}_1 d \xi,\tag{122}\\ &\mathcal{S}_{\zeta}^{f1}:= -u \cdot \nabla_x \zeta - \frac{2}{3} \zeta \operatorname{div}_x \psi - \psi \cdot \nabla_x \tilde{\theta} - \frac{2}{3} \zeta \operatorname{div}_x \tilde{u}, \tag{123} \\ &\mathcal{S}_{\zeta}^{f2}\nonumber:=-\frac{1}{\tilde{\rho}}\left[\partial_{x_1} \tilde{\mathcal{R}}_4-\partial_{x_1} \tilde{\mathcal{R}}\cdot \tilde{u} -\left(\tilde{\theta}+\frac{\left\lvert\tilde{u}\right\rvert^2}{2}\right) \partial_{x_1} \tilde{\mathcal{R}}_0\right]+\left(\kappa(\theta)-\kappa(\tilde{\theta})\right) \frac{\Delta_x \theta}{\rho}+\frac{{\kappa^{\prime}(\theta)}|\nabla_x \theta|^{2}-\kappa^{\prime}(\tilde{\theta})|\nabla_x \tilde{\theta}|^{2}}{\rho} \notag \\ &\quad+\frac{\mu(\theta)}{\rho}\left[\frac{\left(\nabla_x u+(\nabla_x u)^{t}\right)^{2}}{2}-\frac{2}{3}(\operatorname{div}_x u)^{2}\right]-\frac{\mu(\tilde{\theta})}{\rho}\left[\frac{\left(\nabla_x \tilde{u}+(\nabla_x \tilde{u})^{t}\right)^{2}}{2}-\frac{2}{3}(\operatorname{div}_x \tilde{u})^{2}\right] \nonumber\\ &\quad+\mu(\tilde{\theta})\left(\frac{1}{\rho}-\frac{1}{\tilde{\rho}}\right)\left[\frac{\left(\nabla_x \tilde{u}+(\nabla_x \tilde{u})^{t}\right)^{2}}{2}-\frac{2}{3}(\operatorname{div}_x \tilde{u})^{2}\right] + \kappa(\tilde{\theta}) \left(\frac{1}{\rho}-\frac{1}{\tilde{\rho}}\right) \Delta_x \theta+\kappa^{\prime}(\tilde{\theta})\left(\frac{1}{\rho}-\frac{1}{\tilde{\rho}}\right)|\nabla_x \tilde{\theta}|^{2}, \tag{124}\\ &\mathcal{S}_{\zeta}^{m}:=\frac{1}{\rho}\left(-\int_{\mathbb{R}^3} \frac{1}{2}|\xi|^{2} \xi \cdot (\nabla_{x} L_{M}^{-1} \Pi-\nabla_{x} L_{\bar{M}}^{-1}\bar{\Pi}_1 )d \xi+\tilde{u} \cdot \int_{\mathbb{R}^3} \xi \otimes \xi \cdot(\nabla_{x} L_{M}^{-1}\Pi-\nabla_{x} L_{\bar{M}}^{-1}\bar{\Pi}_1) d \xi \right) \notag\\ &\quad+\frac{\psi}{\rho}\cdot \int_{\mathbb{R}^3} \xi \otimes \xi \cdot\nabla_{x} L_{M}^{-1}\Pi d \xi+\frac{\phi}{\tilde{\rho}\rho}\left(\int_{\mathbb{R}^3} \frac{1}{2}|\xi|^{2} \xi \cdot \nabla_{x} L_{\bar{M}}^{-1}\bar{\Pi}_1 d \xi-\tilde{u} \cdot \int_{\mathbb{R}^3} \xi \otimes \xi \cdot \nabla_{x} L_{\bar{M}}^{-1}\bar{\Pi}_1d \xi\right). \tag{125} \end{align}\] Since \(G=\bar{G}_0+\sqrt{\mu}g\), by 10 and 47 , we derive the equation of the microscopic component \(g\) as \[\begin{align} \label{mic-perturbation} &\partial_t g + \xi \cdot \nabla_x g -\mathcal{L} g \notag\\[1.4mm] =&-\frac{1}{\sqrt{\mu}}P_1 \left[\xi_1 \left(\frac{\left\lvert\xi-u\right\rvert^2\partial_{x_1} \zeta}{2 R \theta^2} +\frac{\left( \xi -u\right)\cdot \partial_{x_1} \psi}{R \theta} \right)M \right] -\sum_{i=2}^3\frac{1}{\sqrt{\mu}}P_1 \left[\xi_i \left( \frac{\left\lvert\xi-u\right\rvert^2\partial_{x_i} \zeta}{2 R \theta^2} +\frac{\left( \xi -u\right)\cdot \partial_{x_i} \psi}{R \theta} \right) M\right] \notag\\ &+ \Gamma\left(g,\frac{M-\mu}{\sqrt{\mu}} \right)+\Gamma\left(\frac{M-\mu}{\sqrt{\mu}},g \right) -\frac{1}{\sqrt{\mu}}P_1 \left[\xi_1 \left(\frac{\left\lvert\xi-u\right\rvert^2\partial_{x_1} (\tilde{\theta}-\bar{\theta})}{2 R \theta^2} +\frac{\left( \xi -u\right)\cdot \partial_{x_1} (\tilde{u}-\bar u)}{R \theta} \right)M \right] \notag\\ & +\Gamma(\frac{G}{\sqrt{\mu}},\frac{G}{\sqrt{\mu}}) +\frac{P_0 \left( \xi \cdot \sqrt{\mu}\nabla_x g \right)}{\sqrt{\mu}} -\frac{P_1 \left( \xi_1 \partial_{x_1} \bar{G}_0 \right)}{\sqrt{\mu}}-\frac{\partial_t \bar{G}_0}{\sqrt{\mu}}, \end{align}\tag{126}\] where \(\Gamma\) and \(\mathcal{L}\) are defined by \[\begin{align} \label{2465} \Gamma(f,g):=\frac{1}{\sqrt{\mu}}Q(\sqrt{\mu}f,\sqrt{\mu}g), \quad \mathcal{L}f:=\Gamma(\sqrt{\mu},f)+\Gamma(f,\sqrt{\mu}). \end{align}\tag{127}\] Here we have used the fact that \[\frac{1}{\sqrt{\mu}} L_{M}(\sqrt{\mu}f)= \frac{1}{\sqrt{\mu}}\{Q(M,\sqrt{\mu}f)+Q(\sqrt{\mu}f,M)\} =\mathcal{L}f+\Gamma(f,\frac{M-\mu}{\sqrt{\mu}})+ \Gamma(\frac{M-\mu}{\sqrt{\mu}},f).\] Note that the linearized Landau operator \(\mathcal{L}\) is self-adjoint and non-positive definite, and its null space \(\ker\mathcal{L}\) is spanned by the five functions \(\{\sqrt{\mu},\xi\sqrt{\mu},|\xi|^{2}\sqrt{\mu}\}\), cf. [16].
In order to use the anti-derivative technique, it is convenient to study the perturbation system for \((\phi,\varphi,h)\) 54 since these quantities are conserved. According to zero mass 52 , we define the anti-derivative by \[\begin{align} \label{anti-11-5} (\Phi,\Psi,H)(t,x_1):=\int_{-\infty}^{x_1}\int_{\mathbb{T}^2}(\rho-\tilde{\rho},m-\tilde{m},\mathbb{E}-\tilde{\mathbb{E}})(t,y)dy'dy_1. \end{align}\tag{128}\] From \(\int_{\mathbb{T}^2}\)14 \(dx_2dx_3\) and 50 , we then derive the system for \((\Phi,\Psi,H)\) as \[\begin{align} \label{conserv-1-system} \left\{ \begin{aligned} &\partial_t \Phi + \partial_{x_1} \Psi_{1}=-\tilde{\mathcal{R}}_{0},\\ &\partial_t \Psi_1 + \frac{2}{3} \partial_{x_1} H -\frac{4}{3}\mu(\tilde{\theta}) \partial_{x_1}\mathring{\psi}_{1} + \mathbf{D}_0\int_{{\mathbb{R}}^{3}} \xi_1^2\left( L_M^{-1} \Pi -L_{\bar{M}}^{-1}\bar{\Pi}_1\right)d \xi= \mathcal{S}^{f}_{\Psi1}, \\ &\partial_t \Psi_i - \mu(\tilde{\theta}) \partial_{x_1}\mathring{\psi}_{i} + \mathbf{D}_0\int_{\mathbb{R}^3} \xi_1 \xi_i\left( L_M^{-1} \Pi -L_{\bar{M}}^{-1}\bar{\Pi}_1\right) d\xi = \mathcal{S}^{f}_{\Psi i}, \quad \text{for} \quad i=2,3,\\ &\partial_t H + \frac{5}{3}\tilde{\theta}\partial_{x_1} \Psi_{1} - \kappa(\tilde{\theta}) \partial_{x_1}\mathring{\zeta}+ \frac{1}{2}\mathbf{D}_0\int_{\mathbb{R}^3} \xi_1 \left\lvert\xi\right\rvert^2\left( L_M^{-1} \Pi-L_{\bar{M}}^{-1}\bar{\Pi}_1\right) d\xi= \mathcal{S}^{f}_{H} , \end{aligned}\right. \end{align}\tag{129}\] where \[\begin{align} &\mathcal{S}^{f}_{\Psi1}= \mathbf{D}_0\left[\left( \frac{\tilde{m}_1^2}{\tilde{\rho}}-\frac{1}{3}\frac{\left\lvert\tilde{m}\right\rvert^2}{\tilde{\rho}} \right) - \left( \frac{m_1^2}{\rho} - \frac{1}{3}\frac{\left\lvert m\right\rvert^2}{\rho} \right)\right] + \frac{4}{3} \mathbf{D}_0\left[\left( \mu(\theta)-\mu(\tilde{\theta}) \right) \partial_{x_1} u_{1} \right]-\tilde{\mathcal{R}}_1,\\ &\mathcal{S}^{f}_{\Psi i}= \mathbf{D}_0\left[\frac{\tilde{m}_1\tilde{m}_i}{\tilde{\rho}} - \frac{m_1 m_i}{\rho} \right]+ \mathbf{D}_0\left[ \left( \mu(\theta)-\mu(\tilde{\theta}) \right)\partial_{x_1} u_{i } \right] -\tilde{\mathcal{R}}_i,\\ &\mathcal{S}^{f}_{H}= \mathbf{D}_0\left[ \frac{5}{3}\tilde{\theta}\partial_{x_1} \Psi_{1} -\left( \frac{m_1 \mathbb{E}}{\rho}+\frac{m_1 p}{\rho}-\frac{\tilde{m}_1 \tilde{\mathbb{E}}}{\tilde{\rho}} - \frac{\tilde{m}_1 \tilde{p}}{\tilde{\rho}} \right)\right] + \mathbf{D}_0\left[\left( \kappa(\theta) - \kappa(\tilde{\theta}) \right)\partial_{x_1}\theta\right]\notag\\ & \qquad\;\; +\mathbf{D}_0\left( \frac{4 }{3} \mu(\theta) u_1 \partial_{x_1} u_{1} - \frac{4 }{3} \mu(\tilde{\theta}) \tilde{u}_1 \partial_{x_1}\tilde{u}_{1} + \sum_{i=2}^3 \mu(\theta) u_i \partial_{x_1}u_{i}-\sum_{i=2}^3 \mu(\tilde{\theta}) \tilde{u}_i \partial_{x_1}\tilde{u}_{i} \right) -\tilde{\mathcal{R}}_4. \end{align}\] For the non-fluid part, it holds that \[\begin{align} & \mathbf{D}_0\left(\int_{\mathbb{R}^3} \xi_1 \xi_i L_{M}^{-1} \Pi d\xi- \int_{\mathbb{R}^3} \xi_1 \xi_i L_{\bar{M}}^{-1} \bar{\Pi}_1 d\xi\right) :=(\mathcal{S}^{m}_{\Psi})^{i}=\sum_{j=1}^3\sum_{l=a,b}(\mathcal{S}^{m}_{\Psi})_{jl}^i+(\mathcal{S}^{m}_{\Psi})_{4b}^i,\tag{130}\\ &\mathbf{D}_0\left(\frac{1}{2}\int_{\mathbb{R}^3}\xi_1\left\lvert\xi\right\rvert^{2}L_M^{-1} \Pi d\xi-\frac{1}{2} \int_{\mathbb{R}^3}\xi_1\left\lvert\xi\right\rvert^{2} L_{\bar{M}}^{-1} \bar{\Pi}_1 d\xi \right) :=(\mathcal{S}^{m}_{H})=\sum_{j=1}^3\sum_{l=a,b}(\mathcal{S}^{m}_{H})_{jl}+(\mathcal{S}^{m}_{H})_{4b}+(\mathcal{S}^{m}_{H})_{4}.\tag{131} \end{align}\] Using 258 , 259 and the self-adjoint property of \(L^{-1}_{M}\), the terms in 130 and 131 are respectively given as \[\begin{align} &(\mathcal{S}^{m}_{\Psi})_{1a}^i= R\mathbf{D}_0\int_{\mathbb{R}^3} \theta B_{1i}\left(\frac{\xi-u}{\sqrt{R\theta}} \right)\frac{\sqrt{\mu}\partial_t g}{M} d\xi ,\quad(\mathcal{S}^{m}_{\Psi})_{2a}^i=R\mathbf{D}_0\int_{\mathbb{R}^3}\theta B_{1i}\left(\frac{\xi-u}{\sqrt{R\theta}} \right)\frac{ P_1(\xi \sqrt{\mu} \cdot \nabla_x g)}{M} d\xi, \tag{132} \\ &(\mathcal{S}^{m}_{\Psi})_{3a}^i= R\mathbf{D}_0\int_{\mathbb{R}^3} \theta B_{1i}\left(\frac{\xi-u}{\sqrt{R\theta}} \right) \frac{\sqrt{\mu}}{M} \left[ \Gamma\left(\frac{\bar{G}_0}{\sqrt{\mu}},g \right)+\Gamma\left(g,\frac{\bar{G}_0}{\sqrt{\mu}} \right)+\Gamma\left(g,g \right) \right]d\xi ,\tag{133}\\ &(\mathcal{S}^{m}_{\Psi})_{1b}^i= R\mathbf{D}_0\int_{\mathbb{R}^3}\theta B_{1i}\left(\frac{\xi-u}{\sqrt{R\theta}} \right)\frac{\partial_t \bar{G}_0}{M} d\xi,\;\; (\mathcal{S}^{m}_{\Psi})_{2b}^i= \mathbf{D}_0\int_{\mathbb{R}^3} \xi_1 \xi_i \left( L_M^{-1}Q(\bar{G}_0,\bar{G}_0)- L_{\bar{M}}^{-1} Q(\bar{G},\bar{G}) \right)d\xi, \tag{134}\\ & (\mathcal{S}^{m}_{\Psi})_{3b}^i=\mathbf{D}_0\int_{\mathbb{R}^3} \xi_1 \xi_iL_M^{-1} (P_1 \xi_1 \partial_{x_1} \bar G_0)-\xi_1\xi_iL_{\bar{M}}^{-1} (\bar{P}_1 \xi_1 \partial_{x_1}\bar G) d\xi, \tag{135}\\ & (\mathcal{S}^{m}_{\Psi})_{4b}^i= \sum_{l=2}^3 R\mathbf{D}_0\int_{\mathbb{R}^3} \theta B_{1i}\left(\frac{\xi-u}{\sqrt{R\theta}} \right)\frac{P_1 \left( \xi_l \partial_{x_l} \bar G_0 \right)}{M} d\xi,\quad (\mathcal{S}^{m}_{H})_{1a}=\mathbf{D}_0\int_{\mathbb{R}^3} \left( R \theta \right)^{\frac{3}{2}}A_1 \left( \frac{\xi-u}{\sqrt{R\theta}} \right)\frac{\sqrt{\mu}\partial_t g}{M} d\xi, \tag{136}\\ & (\mathcal{S}^{m}_{H})_{2a}=\mathbf{D}_0\int_{\mathbb{R}^3}\left( R \theta \right)^{\frac{3}{2}} A_1 \left( \frac{\xi-u}{\sqrt{R\theta}} \right) \frac{P_1\left( \xi \cdot \sqrt{\mu}\nabla_x g \right)}{M}d\xi,\tag{137}\\ &(\mathcal{S}^{m}_{H})_{3a}=\mathbf{D}_0\int_{\mathbb{R}^3}\left( R \theta \right)^{\frac{3}{2}}A_1 \left( \frac{\xi-u}{\sqrt{R\theta}} \right)\frac{\sqrt{\mu}}{M} \left[ \Gamma\left(\frac{\bar{G}_0}{\sqrt{\mu}},g \right)+\Gamma\left(g,\frac{\bar{G}_0}{\sqrt{\mu}} \right)+\Gamma\left(g,g \right) \right]d\xi ,\tag{138}\\ & (\mathcal{S}^{m}_{H})_{4}=\mathbf{D}_0\left( \int_{\mathbb{R}^3}u \cdot \xi \xi_1 L_M^{-1}\Pi- \bar u \cdot \xi \xi_1 L_{\bar{M}}^{-1} \bar{\Pi}_1 d\xi \right) ,\;\;(\mathcal{S}^{m}_{H})_{1b}=\mathbf{D}_0\left[\left( R \theta \right)^{\frac{3}{2}} \int_{\mathbb{R}^3} A_1 \left( \frac{\xi-u}{\sqrt{R\theta}} \right)\frac{\partial_t \bar{G}_0}{M} d\xi \right],\tag{139}\\ &(\mathcal{S}^{m}_{H})_{2b}=\mathbf{D}_0\int_{\mathbb{R}^3}\left( \frac{1}{2}\xi_1\left\lvert\xi\right\rvert^{2}-\xi_1\xi\cdot u \right) L_M^{-1} (P_1 \xi_1 \partial_{x_1}\bar{G}_0)-\left( \frac{1}{2}\xi_1\left\lvert\xi\right\rvert^{2}-\xi_1\xi\cdot \bar u \right)L_{\bar{M}}^{-1} (\bar{P}_1 \partial_{x_1} \bar{G}) d\xi,\tag{140}\\ &(\mathcal{S}^{m}_{H})_{3b}=\mathbf{D}_0\int_{\mathbb{R}^3}\left( \frac{1}{2}\xi_1\left\lvert\xi\right\rvert^{2}-\xi_1\xi\cdot u \right) L_M^{-1} Q\left( \bar{G}_0, \bar{G}_0\right) -\left( \frac{1}{2}\xi_1\left\lvert\xi\right\rvert^{2}-\xi_1\xi\cdot \bar u \right)L_{\bar{M}}^{-1} Q\left( \bar{G}, \bar{G}\right)d\xi,\tag{141}\\ & (\mathcal{S}^{m}_{H})_{4b}=\sum_{l=2}^3\mathbf{D}_0\int_{\mathbb{R}^3}\left( R \theta \right)^{\frac{3}{2}} A_1 \left( \frac{\xi-u}{\sqrt{R\theta}} \right) \frac{P_1\left( \xi_l \partial_{x_l} \bar{G}_0 \right)}{M}d\xi.\tag{142} \end{align}\] For the same reason as for introducing the transformation 72 , we define the following transformation \[\begin{align} \label{anti-trans-qwe} \check{\Phi}=\tilde{\theta}\Phi, \quad \check{\Psi}=\Psi,\quad\check{H}=H-\check{\Phi}. \end{align}\tag{143}\] By 129 and 143 , we have \[\begin{align} \label{T-A-D-1} \left\{ \begin{aligned} &\partial_t \check{\Phi}+ \tilde{\theta}\partial_{x_1}\check{\Psi}_{1}=\mathcal{S}_{\check{\Phi}}^{f},\\ &\partial_t \check{\Psi}_1 + \frac{2}{3} \partial_{x_1}\check{H}+\frac{2}{3}\partial_{x_1}\check{\Phi}-\frac{4\mu(\tilde{\theta})}{3\tilde{\rho}} \partial_{x_1}^2 \check{\Psi}_1 +\sum_{j=1}^3 (\mathcal{S}^{m}_{\Psi})_{ja}^1= \mathcal{S}_{\check{\Psi}1}^{f}+\mathcal{S}^{m}_{\check{\Psi}1}, \\ &\partial_t \check{\Psi}_i - \frac{\mu(\tilde{\theta})}{\tilde{\rho}}\partial_{x_1}^2 \check{\Psi}_i +\sum_{j=1}^3 (\mathcal{S}^{m}_{\Psi})_{ja}^i = \mathcal{S}_{\check{\Psi}i}^{f}+\mathcal{S}^{m}_{\check{\Psi}i}, \quad \text{for} \quad i=2,3,\\ &\partial_t \check{H}+ \frac{2}{3}\tilde{\theta}\partial_{x_1}\check{\Psi}_{1} - \frac{\kappa(\tilde{\theta})}{\tilde{\rho}}\partial_{x_1}^2 \check{H}+ \sum_{i=1}^3(\mathcal{S}^{m}_{H})_{ia}= \mathcal{S}_{\check{H}}^{f}+\mathcal{S}^{m}_{\check{H}}-(\mathcal{S}^{m}_{H})_{4}, \end{aligned}\right. \end{align}\tag{144}\] where \[\begin{align} &\mathcal{S}_{\check{\Phi}}^{f}=-\tilde{\theta}\tilde{\mathcal{R}}_0+\frac{\partial_t\tilde{\theta}}{\tilde{\theta}}\check{\Phi}, \quad \mathcal{S}_{\check{\Psi}1}^{f}=\frac{4\mu(\tilde{\theta})}{3}\left( \partial_{x_1}\mathring{\psi}_{1}-\frac{\partial_{x_1}^2\check{\Psi}_1}{\tilde{\rho}} \right)+ \mathcal{S}^{f}_{\Psi1},\quad \mathcal{S}^{m}_{\check{\Psi}1}=-\sum_{j=1}^4(\mathcal{S}^{m}_{\Psi})_{jb}^1, \\ &\mathcal{S}_{\check{\Psi}i}^{f}=\mu(\tilde{\theta})\left( \partial_{x_1}\mathring{\psi}_{i}-\frac{\partial_{x_1}^2\check{\Psi}_i}{\tilde{\rho}} \right)+ \mathcal{S}^{f}_{\Psi i}, \quad \mathcal{S}^{m}_{\check{\Psi}i}=-\sum_{j=1}^4(\mathcal{S}^{m}_{\Psi})_{jb}^i, \quad \text{for}\quad i=2,3, \\ &\mathcal{S}_{\check{H}}^{f}=\kappa(\tilde{\theta})\left( \partial_{x_1}\mathring{\zeta}-\frac{\partial_{x_1}^2 \check{H}}{\tilde{\rho}} \right)+ \mathcal{S}^{f}_{H}+\tilde{\theta}\tilde{\mathcal{R}}_0-\frac{\partial_t\tilde{\theta}}{\tilde{\theta}}\check{\Phi},\qquad \mathcal{S}^{m}_{\check{H}}=-\sum_{j=1}^4 (\mathcal{S}^{m}_{H})_{jb}. \end{align}\]
Recall the definition of \(\check{\mathbf{V}}\) 55 , instant energy functionals \(\mathcal{E}_i\) 56 61 and dissipation energy functionals \(\mathcal{D}_i\) 62 68 . Then, we can present the \(H^2\) energy estimate for the zero mode 144 as follows.
Theorem 16. Under the same assumptions of Proposition 10, it holds that \[\begin{align} & {\frac{d}{dt}}\left[\left\lVert\check{\mathbf{V}}\right\rVert_{L^2}^2+\sum_{l=1}^4\mathcal{X}^l+ \tilde{c}\left(\left\lVert\partial_{x_1}\check{\Phi}\right\rVert_{L^2}^2+\int_{\mathbb{R}} \partial_{x_1}\check{\Phi}\check{\Psi}_1 dx_1 \right)\right] +\tilde{c}\left\lVert\partial_{x_1}\check{\mathbf{V}}\right\rVert_{L^2}\\ &\leq C\bar{\delta}(1+t)^{-\frac{3}{2}}+C\check{\delta}\left[(1+t)^{-1}\mathcal{E}_1 + \mathcal{D}_1 +\left\lVert\partial_t\nabla_x \mathbf{v}^{\ast}\right\rVert_{L^2}^2\right]+C_\eta\sum_{\left\lvert\alpha\right\rvert\le 2}\left\lVert\partial^{\alpha} g\right\rVert_{\sigma}^2, \end{align}\] and \[\begin{align} &{\frac{d}{dt}}\left[\left\lVert\partial_{x_1}\check{\mathbf{V}}\right\rVert_{L^2}^2+ \tilde{c}\left(\left\lVert\partial_{x_1}^2\check{\Phi}\right\rVert_{L^2}^2+\int_{\mathbb{R}} \partial_{x_1}^2\check{\Phi}\partial_{x_1}\check{\Psi}_1 dx_1 \right) \right]+\tilde{c}\left\lVert\partial_{x_1}^2\check{\mathbf{V}}\right\rVert_{L^2}\\ &\leq C\bar{\delta}(1+t)^{-\frac{5}{2}}+C\check{\delta}\left[(1+t)^{-2}\mathcal{E}_1 +(1+t)^{-1} \mathcal{D}_1+\mathcal{D}_2 +\left\lVert\partial_t\nabla_x \mathbf{v}^{\ast}\right\rVert_{L^2}^2\right]+C_\eta\sum_{1\le\left\lvert\alpha\right\rvert\le 2}\left\lVert\partial^{\alpha} g\right\rVert_{\sigma}^2, \end{align}\] and \[\begin{align} &{\frac{d}{dt}}\left(\left\lVert\partial_{x_1}^2\left(\check{\Phi},\sum_{i=2,3}\check{\Psi}_i,\check{H}\right)\right\rVert_{L^2}^2 + \left\lVert\partial_{x_1}\left( \tilde{\theta}\partial_{x_1} \check{\Psi}_1\right)\right\rVert_{L^2}^2 \right) + \left\lVert\partial_{x_1}^3\left(\check{\Psi},\check{H}\right)\right\rVert_{L^2}^2\\ &\leq C\bar{\delta}(1+t)^{-\frac{5}{2}}+C\check{\delta}\left[(1+t)^{-3}\mathcal{E}_1 +\sum_{j=1}^3(1+t)^{-3+j} \mathcal{D}_j+\sum_{\left\lvert\alpha\right\rvert= 1}\left\lVert\partial^{\alpha} g\right\rVert_{\sigma}^2+\left\lVert\partial_t\nabla_x \mathbf{v}^{\ast}\right\rVert_{L^2}^2\right]+C_\eta\sum_{\left\lvert\alpha\right\rvert= 2}\left\lVert\partial^{\alpha} g\right\rVert_{\sigma}^2, \end{align}\] where \(\bar{\delta}:=\delta+\varepsilon_0\), \(\check{\delta}:=\chi+\bar{\delta}^{\frac{1}{2}}\) and for \(i=1,2,3\), \[\begin{align} \label{definition-of-Xm} & \mathcal{X}^i= R\int_{\mathbb{R}} \mathbf{D}_0\int_{\mathbb{R}^3} \theta B_{1i}\left( \frac{\xi-u}{\sqrt{R\theta}}\right) \frac{\sqrt{\mu}}{M} g d\xi \check{\Psi}_i dx_1 , \quad \mathcal{X}^4= R^{\frac{3}{2}}\int_{\mathbb{R}} \mathbf{D}_0\int_{\mathbb{R}^3} \theta^{\frac{3}{2}} A_{1}\left( \frac{\xi-u}{\sqrt{R\theta}}\right) \frac{\sqrt{\mu}}{M} g d\xi \frac{\check{H}}{\tilde{\theta}} dx_1. \end{align}\qquad{(4)}\]
The proof of Theorem 16 will be decomposed into three parts including Lemma 18, Lemma 20 and Lemma 21. Before proving these three lemmas, we should study the source terms in 144 . Recall the definition of \(\tilde{D}_{-\alpha}\) 40 and \((\mathbf{V},\check{\mathbf{V}},\mathbf{v})\) 55 . We use the following notations for the sake of convenience: \[\begin{align} \label{useful-notation} \begin{aligned} &\mathbf{D}^{(k)}:=\bar{\delta}^{\frac{1}{2}} \sum_{j=1}^{k+2} \tilde{D}_{-\frac{j}{2}} \left\lvert\partial_{x_1}^{k-j+2} \check{\mathbf{V}}\right\rvert,\qquad \mathbf{T}^{(0)}:=\left\lvert\partial_{x_1} \check{\mathbf{V}}\right\rvert^2+ \bar{\delta}^{\frac{1}{2}} \tilde{D}_{-\frac{1}{2}}\left\lvert\partial_{x_1} \check{\mathbf{V}}\right\rvert,\\ &\mathbf{T}^{(1)}:=\left\lvert\partial_{x_1}^2 \check{\mathbf{V}}\right\rvert\left\lvert\partial_{x_1} \check{\mathbf{V}}\right\rvert+\bar{\delta}^{\frac{1}{2}} \left(\tilde{D}_{-\frac{1}{2}}\left\lvert\partial_{x_1} \check{\mathbf{V}}\right\rvert^2+\tilde{D}_{-1}\left\lvert\partial_{x_1} \check{\mathbf{V}}\right\rvert+\tilde{D}_{-\frac{1}{2}}\left\lvert\partial_{x_1}^2 \check{\mathbf{V}}\right\rvert\right),\\ &\mathbf{T}^{(2)}:=\left\lvert\partial_{x_1}^3 \check{\mathbf{V}}\right\rvert\left\lvert\partial_{x_1} \check{\mathbf{V}}\right\rvert+\left\lvert\partial_{x_1}^2 \check{\mathbf{V}}\right\rvert^2+ \bar{\delta}^{\frac{1}{2}} \tilde{D}_{-\frac{3}{2}}\left\lvert\partial_{x_1} \check{\mathbf{V}}\right\rvert\\ &\qquad\qquad +\bar{\delta}^{\frac{1}{2}} \tilde{D}_{-\frac{1}{2}}\big(\left\lvert\partial_{x_1} \check{\mathbf{V}}\right\rvert\left\lvert\partial_{x_1}^2 \check{\mathbf{V}}\right\rvert+\left\lvert\partial_{x_1}^3 \check{\mathbf{V}}\right\rvert\big)+ \bar{\delta}^{\frac{1}{2}} \tilde{D}_{-1}\big(\left\lvert\partial_{x_1}^2 \check{\mathbf{V}}\right\rvert+\left\lvert\partial_{x_1} \check{\mathbf{V}}\right\rvert^2\big),\\ &\mathbf{Z}^{(0)}:=\left\lvert\mathbf{v}_{\neq}\right\rvert^2,\qquad\qquad\mathbf{Z}^{(1)}:=\left\lvert\mathbf{v}_{\neq}\right\rvert\left\lvert\nabla_x \mathbf{v}_{\neq}\right\rvert,\qquad\qquad\mathbf{Z}^{(2)}:=\left\lvert\nabla_x \mathbf{v}_{\neq}\right\rvert^2+\left\lvert\nabla_x^2 \mathbf{v}_{\neq}\right\rvert\left\lvert\mathbf{v}_{\neq}\right\rvert. \end{aligned} \end{align}\tag{145}\]
Lemma 17. Under the same assumptions of Proposition 10, for \(k=0,1\), one has \[\begin{align} &\sum_{i=1}^3\left\lvert\mathcal{S}_{\check \Psi i}^{m}\right\rvert+\left\lvert\mathcal{S}_{\check H}^m\right\rvert\le C \bar{\delta}\tilde{D}_{-\frac{3}{2}}+C \bar{\delta}\tilde{D}_{-1}\left\lvert\mathbf{v}\right\rvert+C\bar{\delta}\tilde{D}_{-\frac{1}{2}}(\left\lvert\nabla_x\mathbf{v}^{\ast}\right\rvert+\left\lvert\partial_t\mathbf{v}^{\ast}\right\rvert),\label{lem-2-1-n-3-1-1-1}\\ &\left\lvert(\mathcal{S}^{m}_{\Psi})^i\right\rvert+\left\lvert(\mathcal{S}^{m}_{H})\right\rvert\le C\bar{\delta}\tilde{D}_{-1}+C\sum_{\left\lvert\alpha\right\rvert=1}\left\lvert\partial^{\alpha}g\right\rvert_{\sigma}+C\left( \left\lvert g\right\rvert_{2}+\bar{\delta}\tilde{D}_{-\frac{1}{2}}\right)\left\lvert g\right\rvert_{\sigma}+C\bar{\delta}\tilde{D}_{-\frac{1}{2}}(\left\lvert\nabla_x\mathbf{v}^{\ast}\right\rvert+\left\lvert\partial_t\mathbf{v}^{\ast}\right\rvert),\label{lem-2-1-n-3}\\ &\left\lvert\partial_{x_1}(\mathcal{S}^{m}_{\Psi})^i\right\rvert+\left\lvert\partial_{x_1}(\mathcal{S}^{m}_{H})\right\rvert\le C\sum_{\left\lvert\alpha\right\rvert=2}\left\lvert\partial^{\alpha}g\right\rvert_{\sigma}+C\sum_{\left\lvert\gamma\right\rvert=1}\bigg[\left\lvert g\right\rvert_{\sigma}\left\lvert\partial^{\gamma}g\right\rvert_{\sigma}+\bar{\delta}\tilde{D}_{-\frac{1}{2}}(\left\lvert\nabla_x^2\mathbf{v}^{\ast}\right\rvert+\left\lvert\partial_t\nabla_x\mathbf{v}^{\ast}\right\rvert)\notag\\ &\qquad\qquad\qquad\qquad\quad+\left(\bar{\delta}^{\frac{1}{2}} \tilde{D}_{-\frac{1}{2}}+\left\lvert\partial_t \mathbf{v}^{\ast}\right\rvert+\left\lvert\nabla_x \mathbf{v}\right\rvert\right)\left( \bar{\delta}\tilde{D}_{-1}+\left\lvert\partial^{\gamma}g\right\rvert_{\sigma}+\left\lvert g\right\rvert_2\left\lvert g\right\rvert_{\sigma}+\bar{\delta}\tilde{D}_{-\frac{1}{2}}\left\lvert g\right\rvert_\sigma \right)\bigg],\label{lem-2-1-n-4}\\ &\left\lvert(\mathcal{S}^{m}_{H})_{4}\right\rvert\le C (\bar{\delta}^{\frac{1}{2}}+\chi) (1+t)^{-\frac{1}{2}}\left\lvert(\mathcal{S}^{m}_{\Psi})^i\right\rvert+C\bar{\delta}\left(\tilde{D}_{-1} \left\lvert\mathbf{v}^{\ast}\right\rvert+\tilde{D}_{-\frac{3}{2}}\right),\label{lem-2-1-n-5-1-1}\\ &\left\lvert\partial_{x_1} (\mathcal{S}^{m}_{H})_{4}\right\rvert\lesssim (\bar{\delta}^{\frac{1}{2}}+\chi) \left[(1+t)^{-\frac{3}{4}}\left\lvert(\mathcal{S}^{m}_{\Psi})^i\right\rvert + (1+t)^{-\frac{1}{2}}\left\lvert\partial_{x_1}(\mathcal{S}^{m}_{\Psi})^i\right\rvert\right]+\bar{\delta}\left(\tilde{D}_{-\frac{3}{2}} \left\lvert\mathbf{v}^{\ast}\right\rvert+\tilde{D}_{-1}\left\lvert\nabla_x\mathbf{v}^{\ast}\right\rvert+\tilde{D}_{-\frac{5}{2}}\right),\label{lem-2-1-n-5}\\ &\left\lvert\partial_{x_1}^k\mathcal{S}_{\check{\Phi}}^{f}\right\rvert\le C\bar{\delta}\tilde{D}_{-\frac{3+k}{2}}+\mathbf{D}^{(k)}, \label{2026-6-8}\\ &\sum_{i=1}^3\left\lvert\partial_{x_1}^k\mathcal{S}_{\check{\Psi}i}^{f}\right\rvert+\left\lvert\partial_{x_1}^k\mathcal{S}_{\check H}^f\right\rvert\leq C\bar{\delta}\tilde{D}_{-\frac{3+k}{2}} +\mathbf{D}^{(k)}+\mathbf{T}^{(k)}+\mathbf{T}^{(k+1)}+\mathbf{Z}^{(k)}+\mathbf{Z}^{(k+1)}\label{lem-2-1-n-2}. \end{align}\] {#eq: sublabel=eq:lem-2-1-n-3-1-1-1,eq:lem-2-1-n-3,eq:lem-2-1-n-4,eq:lem-2-1-n-5-1-1,eq:lem-2-1-n-5,eq:2026-6-8,eq:lem-2-1-n-2}
Proof. We first present the calculation of the microscopic parts, that is, to prove ?? ?? . For \(\mathcal{S}_{\check{\Psi}_i}^m\) and \(\mathcal{S}_{\check{H}}^m\) in 144 , by the definition of \(\bar{G}_0\) 47 , estimate of \(\bar{G}_0\) Lemma 40 and the definition of \((\mathcal{S}^{m}_{\Psi})_{1b}^i\) 134 , \((\mathcal{S}^{m}_{\Psi})_{4b}^i\) 136 , \((\mathcal{S}^{m}_{H})_{1b}\) 139 and \((\mathcal{S}^{m}_{H})_{4b}\) 142 , it holds that \[\begin{align} \notag \left\lvert(\mathcal{S}^{m}_{\Psi})_{1b}^i\right\rvert+\left\lvert(\mathcal{S}^{m}_{\Psi})_{4b}^i\right\rvert+\left\lvert(\mathcal{S}^{m}_{H})_{1b}\right\rvert+\left\lvert(\mathcal{S}^{m}_{H})_{4b}\right\rvert\le C \bar{\delta}\tilde{D}_{-\frac{3}{2}}+C\bar{\delta}\tilde{D}_{-\frac{1}{2}} \left\lvert\partial_t\mathbf{v}^{\ast}\right\rvert+C\bar{\delta}\tilde{D}_{-\frac{1}{2}} \left\lvert\nabla_x\mathbf{v}^{\ast}\right\rvert. \end{align}\] Moreover, \((\mathcal{S}^{m}_{\Psi})_{2b}^{i}\) 134 , \((\mathcal{S}^{m}_{\Psi})_{3b}^{i}\) 135 , \((\mathcal{S}^{m}_{H})_{2b}\) 140 and \((\mathcal{S}^{m}_{H})_{3b}\) 141 can be treated in the same way, so we only calculate \((\mathcal{S}^{m}_{\Psi})_{3b}^{i}\) 135 . By the self-adjoint properties of \(L_M^{-1}\) and expansion of \(\partial_{x_1}\bar G_0\) 260 , we have \[\begin{align} \label{2026-4-24-2} &\mathbf{D}_0\int_{\mathbb{R}^3} \xi_1\xi_i L_M^{-1} (P_1 \xi_1 \partial_{x_1}\bar G_0) d\xi \notag\\ =&\mathbf{D}_0\int_{\mathbb{R}^3} R\theta B_{1i}\left(\frac{\xi-u}{\sqrt{R\theta}} \right) \frac{ \xi_1 \partial_{x_1}\bar G_0}{M} d\xi-\sum_{j=0}^4\mathbf{D}_0\int_{\mathbb{R}^3} R\theta B_{1i}\left(\frac{\xi-u}{\sqrt{R\theta}} \right) \frac{ \langle\xi_1 \partial_{x_1}\bar G_0,\chi_j\rangle\chi_j}{M} d\xi\notag\\ =&\sum_{i,j=1}^3 \mathbf{D}_0\left( \mathcal{A}_1 \partial_{x_1}^2\bar{\theta}+\mathcal{A}_2\partial_{x_1}\theta\partial_{x_1}\bar{\theta}+\mathcal{A}_3\partial_{x_1}u_i\partial_{x_1}\bar{\theta}+\mathcal{A}_4\partial_{x_1}^2\bar u_i+\mathcal{A}_5 \partial_{x_1}\bar u_i\partial_{x_1} u_j+\mathcal{A}_6\partial_{x_1}\bar u_i \partial_{x_1}\theta\right). \end{align}\tag{146}\] Using the same method, we also have \[\begin{align} \label{2026-4-24-3} &\mathbf{D}_0\int_{\mathbb{R}^3} \xi_1\xi_i L_{\bar M}^{-1} (\bar P_1 \xi_1 \partial_{x_1}\bar G) d\xi \notag\\ =&\sum_{i,j=1}^3 \mathbf{D}_0\left( \bar{\mathcal{A}}_1 \partial_{x_1}^2\bar{\theta}+\bar{\mathcal{A}}_2\partial_{x_1}\bar{\theta}\partial_{x_1}\bar{\theta}+\bar{\mathcal{A}}_3\partial_{x_1}\bar{u}_i\partial_{x_1}\bar{\theta}+\bar{\mathcal{A}}_4\partial_{x_1}^2\bar u_i+\bar{\mathcal{A}}_5 \partial_{x_1}\bar u_i\partial_{x_1} \bar{u}_j+\bar{\mathcal{A}}_6\partial_{x_1}\bar u_i \partial_{x_1}\bar{\theta}\right), \end{align}\tag{147}\] where \(\mathcal{A}_{1,\dots,6}\) and \(\bar{\mathcal{A}}_{1,\dots,6}\) are the smooth function of \((\rho,u,\theta)\) and \((\bar\rho,\bar{u},\bar{\theta})\), respectively. By 146 , 147 and \(\left\lvert(\rho,u,\theta)-(\bar{\rho},\bar{u},\bar{\theta})\right\rvert\lesssim \bar{\delta}\tilde{D}_{-\frac{1}{2}}+\left\lvert\mathbf{v}\right\rvert\), we have \[\begin{align} \left\lvert(\mathcal{S}^{m}_{\Psi})_{3b}^{i}\right\rvert\le C\bar{\delta}\tilde{D}_{-\frac{3}{2}}+C\bar{\delta}\tilde{D}_{-1} \left\lvert\mathbf{v}\right\rvert+C\bar{\delta}\tilde{D}_{-\frac{1}{2}} \left\lvert\nabla_x \mathbf{v}^{\ast}\right\rvert. \end{align}\] Based on the above estimates, we then obtain \[\begin{align} \notag \sum_{i=1}^3\left\lvert\mathcal{S}_{\check{\Psi}i}^{m}\right\rvert+\left\lvert\mathcal{S}_{\check H}^m\right\rvert \le C \bar{\delta}\tilde{D}_{-\frac{3}{2}}+C \bar{\delta}\tilde{D}_{-1}\left\lvert\mathbf{v}\right\rvert+C\bar{\delta}\tilde{D}_{-\frac{1}{2}}(\left\lvert\nabla_x\mathbf{v}^{\ast}\right\rvert+\left\lvert\partial_t\mathbf{v}^{\ast}\right\rvert). \end{align}\] Then we have finished the proof of ?? .
From the definition of \((\mathcal{S}^{m}_{\Psi})^i,\;(\mathcal{S}^{m}_{H})\) 130 and 131 , and by noting \(\Pi=\partial_t G + P_1(\xi \cdot \nabla_x G)-Q(G,G)\), \(G=\bar{G}_0+\sqrt{\mu}g\), and \((\rho,u,\theta)=(\tilde{\rho},\tilde{u},\tilde{\theta})+(\phi,\psi,\zeta)\), it can be directly calculated that \[\begin{align} \notag &\left\lvert(\mathcal{S}^{m}_{\Psi})^i\right\rvert+\left\lvert(\mathcal{S}^{m}_{H})\right\rvert\le C\bar{\delta}\tilde{D}_{-1}+C\sum_{\left\lvert\alpha\right\rvert=1}\left\lvert\partial^{\alpha}g\right\rvert_{\sigma}+C\left( \left\lvert g\right\rvert_{2}+\bar{\delta}\tilde{D}_{-\frac{1}{2}}\right)\left\lvert g\right\rvert_{\sigma}+C\bar{\delta}\tilde{D}_{-\frac{1}{2}}(\left\lvert\nabla_x\mathbf{v}^{\ast}\right\rvert+\left\lvert\partial_t\mathbf{v}^{\ast}\right\rvert). \end{align}\] To calculate \(\partial_{x_1}(\mathcal{S}^{m}_{\Psi})^i\) and \(\partial_{x_1}(\mathcal{S}^{m}_{H})\), we note that the following fact holds: \[\begin{align} &\left\lvert\partial_{x_1} \int_{\mathbb{R}^3} \xi_i \xi_j L_M^{-1}\Pi d\xi\right\rvert=R\left\lvert\partial_{x_1} \int_{\mathbb{R}^3} \theta B_{ij}\left( \frac{\xi-u}{\sqrt{R\theta}} \right)\frac{1}{M} \left(\partial_t G+P_1 \xi \cdot \nabla_x G -Q(G,G)\right) d\xi\right\rvert \notag\\ \le&C\sum_{\left\lvert\gamma\right\rvert=1}\left[\left\lvert g\right\rvert_{\sigma}\left\lvert\partial^{\gamma}g\right\rvert_{\sigma}+\left(\bar{\delta}^{\frac{1}{2}} \tilde{D}_{-\frac{1}{2}}+\left\lvert\nabla_x \mathbf{v}\right\rvert\right)\left( \bar{\delta}\tilde{D}_{-1}+\left\lvert\partial^{\gamma}g\right\rvert_{\sigma}+\left\lvert g\right\rvert_2\left\lvert g\right\rvert_{\sigma}+\bar{\delta}\tilde{D}_{-\frac{1}{2}}\left\lvert g\right\rvert_\sigma \right)\right]\notag\\ &+C\bar{\delta}\tilde{D}_{-\frac{1}{2}}(\left\lvert\nabla_x^2\mathbf{v}^{\ast}\right\rvert+\left\lvert\partial_t\nabla_x\mathbf{v}^{\ast}\right\rvert)+C\sum_{\left\lvert\alpha\right\rvert=2}\left\lvert\partial^{\alpha}g\right\rvert_{\sigma}.\label{non-fluid-s-s-1} \end{align}\tag{148}\] Since the treatment of \(\partial_{x_i} \int_{\mathbb{R}^3} \xi_i \left\lvert\xi\right\rvert^2 L_M^{-1}\Pi d\xi\) is similar, we obtain \[\begin{align} &\left\lvert\partial_{x_1}(\mathcal{S}^{m}_{\Psi})^i\right\rvert+\left\lvert\partial_{x_1}(\mathcal{S}^{m}_{H})\right\rvert\le C\sum_{\left\lvert\alpha\right\rvert=2}\left\lvert\partial^{\alpha}g\right\rvert_{\sigma}+C\sum_{\left\lvert\gamma\right\rvert=1}\bigg[\left\lvert g\right\rvert_{\sigma}\left\lvert\partial^{\gamma}g\right\rvert_{\sigma}+\bar{\delta}\tilde{D}_{-\frac{1}{2}}(\left\lvert\nabla_x^2\mathbf{v}^{\ast}\right\rvert+\left\lvert\partial_t\nabla_x\mathbf{v}^{\ast}\right\rvert)\notag\\ &\qquad\qquad\qquad\qquad\quad+\left(\bar{\delta}^{\frac{1}{2}} \tilde{D}_{-\frac{1}{2}}+\left\lvert\nabla_x \mathbf{v}\right\rvert\right)\left( \bar{\delta}\tilde{D}_{-1}+\left\lvert\partial^{\gamma}g\right\rvert_{\sigma}+\left\lvert g\right\rvert_2\left\lvert g\right\rvert_{\sigma}+\bar{\delta}\tilde{D}_{-\frac{1}{2}}\left\lvert g\right\rvert_\sigma \right)\bigg]. \end{align}\] Thus ?? and ?? have been proved.
For the calculation of \((\mathcal{S}^{m}_{H})_{4}\) and \(\partial_{x_1}(\mathcal{S}^{m}_{H})_{4}\), we have \[\begin{align} (\mathcal{S}^{m}_{H})_{4}=\sum_{i=1}^3 u_i (\mathcal{S}^{m}_{\Psi})^i + \mathbf{D}_0\int_{\mathbb{R}^3} (u-\bar u) \cdot \xi\xi_1L_{\bar M}^{-1}\bar \Pi_1 d\xi. \end{align}\] Then, by applying the a prior assumption 69 , we obtain ?? and ?? .
At this point, we then provide an estimate of the macroscopic part, namely ?? and ?? . By Hölder’s inequality for the definition of \(\mathcal{S}_{\check \Phi}^{f}\) 144 , we have \[\begin{align} \notag \left\lvert\partial_{x_1}^k\mathcal{S}_{\check{\Phi}}^{f}\right\rvert\le C\bar{\delta}\tilde{D}_{-\frac{3+k}{2}}+\mathbf{D}^{(k)}. \end{align}\] Thus, ?? has been proved.
Below, we will mainly focus on the estimate of \(\mathcal{S}_{\check{\Psi}i}^{f}\) and \(\mathcal{S}_{\check{H}i}^{f}\), which consists of two parts via viscosity and flux. Taking \[\label{add46s4p1} \kappa(\tilde{\theta})\left( \partial_{x_1} \mathring{\zeta}-\frac{\partial_{x_1}^2 \check{H}}{\tilde{\rho}} \right)\tag{149}\] as an example, we present the calculation of the viscosity part. We remark that from the calculation of the term 149 above, it can be found that the transformation 143 plays an important role in the viscosity part. We present the following identities needed for calculating this term: \[\begin{align} &\mathring{\zeta}= \mathbf{D}_0\left[\frac{\mathbb{E}}{\rho}-\frac{\tilde{\mathbb{E}}}{\tilde{\rho}}-\left( \frac{\left\lvert m\right\rvert^2}{2\rho^2} - \frac{\left\lvert\tilde{m}\right\rvert^2}{2\tilde{\rho}^2} \right)\right],\qquad \frac{\mathring{\mathbb{E}}}{\mathring{\rho}}-\frac{\tilde{\mathbb{E}}}{\tilde{\rho}}=\frac{\mathring{h}}{\mathring{\rho}}-\frac{\mathring{\phi}\tilde{\theta}}{\mathring{\rho}}-\frac{\left\lvert\tilde{u}\right\rvert^2\mathring{\phi}}{2\mathring{\rho}},\tag{150}\\ &\frac{\mathring{m}_i}{\mathring{\rho}}-\frac{\tilde{m}_i}{\tilde{\rho}}=\frac{\mathring{\varphi}_i}{\mathring{\rho}}-\frac{\tilde{m}_i\mathring{\phi}}{\tilde{\rho}\mathring{\rho}},\qquad \frac{\mathring{m}_i^2}{\mathring{\rho}^2}-\frac{\tilde{m}_i^2}{\tilde{\rho}^2}=\frac{\tilde{\rho}\mathring{m}_i+\mathring{\rho}\tilde{m}_i}{\mathring{\rho}^2\tilde{\rho}} \mathring{\varphi}_i- \frac{\tilde{m}_i(\tilde{\rho}\mathring{m}_i+\mathring{\rho}\tilde{m}_i) }{\mathring{\rho}^2 \tilde{\rho}^2}\mathring{\phi}.\tag{151} \end{align}\] Combining the definition of \(\check{\Phi}\) 143 and 150 , the term 149 can be written as \[\begin{align} \label{lem-2-1-3} &\eqref{add46s4p1} =\kappa(\tilde{\theta})\partial_{x_1}\mathbf{D}_0\left[ \frac{\mathbb{E}}{\rho}-\frac{\mathring{\mathbb{E}}}{\mathring{\rho}}+\frac{\left\lvert\mathring{m}\right\rvert^2}{2\mathring{\rho}^2}-\frac{\left\lvert m\right\rvert^2}{2\rho^2} \right]-\kappa(\tilde{\theta})\partial_{x_1}\left[\frac{\left\lvert\mathring{m}\right\rvert^2}{2\mathring{\rho}^2}-\frac{\left\lvert\tilde{m}\right\rvert^2}{2\tilde{\rho}^2}+\frac{\left\lvert\tilde{u}\right\rvert^2\mathring{\phi}}{2\mathring{\rho}} \right]\notag \\ &\qquad\qquad\qquad\qquad\qquad+\kappa(\tilde{\theta})\left[ \partial_{x_1}\left( \frac{\mathring{h}}{\mathring{\rho}} \right) - \frac{\partial_{x_1}^2H}{\tilde{\rho}}-\partial_{x_1}\left( \frac{\mathring{\phi}\tilde{\theta}}{\mathring{\rho}} \right)+\frac{\partial_{x_1}^2(\tilde{\theta}\Phi)}{\tilde{\rho}}\right]=\mathcal{L}_1+\mathcal{L}_2+\mathcal{L}_3. \end{align}\tag{152}\] To estimate 152 , by 151 , it holds that \[\begin{align} &\left\lvert\mathcal{L}_1\right\rvert\lesssim \left\lvert\mathbf{v}_{\neq}\right\rvert^2+\left\lvert\mathbf{v}_{\neq}\right\rvert\left\lvert\nabla_x \mathbf{v}_{\neq}\right\rvert,\notag \\ &\left\lvert\mathcal{L}_2\right\rvert+\left\lvert\mathcal{L}_3\right\rvert\le C\bar{\delta}^{\frac{1}{2}}\tilde{D}_{-1}\left\lvert\check{\mathbf{V}}\right\rvert+C\bar{\delta}^{\frac{1}{2}}\tilde{D}_{-\frac{1}{2}}\left(\left\lvert\partial_{x_1}\check{\mathbf{V}}\right\rvert+ \left\lvert\partial_{x_1}^2\check{\mathbf{V}}\right\rvert\right)+C\left\lvert\partial_{x_1}^2\check{\mathbf{V}}\right\rvert\left\lvert\partial_{x_1}\check{\mathbf{V}}\right\rvert+C\left\lvert\partial_{x_1}\check{\mathbf{V}}\right\rvert^2. \notag \end{align}\] For the flux terms in \(\mathcal{S}_{\check{\Psi}i}^{f}\) and \(\mathcal{S}_{\check{H}}^{f}\), we take the most complex term \[\label{add46sec446sphp} \mathbf{D}_0\left[ \frac{5}{3}\tilde{\theta}\partial_{x_1} \Psi_{1} -\left( \frac{m_1 \mathbb{E}}{\rho}+\frac{m_1 p}{\rho}-\frac{\tilde{m}_1 \tilde{\mathbb{E}}}{\tilde{\rho}} - \frac{\tilde{m}_1 \tilde{p}}{\tilde{\rho}} \right)\right]\tag{153}\] as an example to present the calculation. We compute \[\begin{align} &\mathring{p}-\tilde{p}=\frac{2}{3}\mathring{h} - \frac{2}{3} \mathbf{D}_0\left[\left( \frac{\varphi_1^2}{2\rho} + \frac{\tilde{m}_1 \varphi_1}{\rho} - \frac{\tilde{m}_1^2\phi}{\rho \tilde{\rho}} \right) +\frac{\varphi_2^2}{2 \rho} + \frac{\varphi_3^2}{2 \rho} \right],\tag{154}\\ &\frac{\mathring{m}_1 \mathring{\mathbb{E}}}{\mathring{\rho}} - \frac{\tilde{m}_1 \tilde{\mathbb{E}}}{\tilde{\rho}} = \frac{\tilde{\mathbb{E}}\mathring{\varphi}_1}{\mathring{\rho}}+\frac{\tilde{m}_1 \mathring{h}}{\mathring{\rho}} -\frac{\tilde{m}_1 \tilde{\mathbb{E}}\mathring{\phi}}{\mathring{\rho}\tilde{\rho}} + \frac{\mathring{\varphi}_1 \mathring{h}}{\mathring{\rho}} , \tag{155}\\ &\frac{\mathring{m}_1 \mathring{p}}{\mathring{\rho}} - \frac{\tilde{m}_1 \tilde{p}}{\tilde{\rho}} = \frac{\tilde{m}_1 (\mathring{p}-\tilde{p})}{\mathring{\rho}} + \frac{\tilde{p}\mathring{\varphi}_1}{\mathring{\rho}} - \frac{\tilde{m}_1 \tilde{p} \mathring{\phi}}{\mathring{\rho}\tilde{\rho}} + \frac{\mathring{\varphi}_1 (\tilde{p}-\tilde{p})}{\mathring{\rho}}. \tag{156} \end{align}\] Using 155 and 156 , it follows that \[\begin{align} \notag \eqref{add46sec446sphp} =\mathcal{L}_4+\mathcal{L}_5+\mathcal{L}_6, \end{align}\] where \[\begin{align} &\mathcal{L}_4=\frac{5}{3}\tilde{\theta}\partial_{x_1}\Psi_1-\frac{\tilde{\mathbb{E}}\mathring{\varphi}_1}{\mathring{\rho}}-\frac{\tilde{p}\mathring{\varphi}_1}{\mathring{\rho}},\quad\mathcal{L}_5=\mathbf{D}_0\left( \frac{\mathring{m}_1 \mathring{\mathbb{E}}}{\mathring{\rho}} -\frac{m_1\mathbb{E}}{\rho}\right)+\mathbf{D}_0\left( \frac{\mathring{m}_1 \mathring{p}}{\mathring{\rho}} - \frac{m_1p}{\rho}\right),\\ &\mathcal{L}_6=\frac{\tilde{m}_1 \tilde{\mathbb{E}}\mathring{\phi}}{\mathring{\rho}\tilde{\rho}}-\frac{\tilde{m}_1 \mathring{h}}{\mathring{\rho}}-\frac{\mathring{\varphi}_1\mathring{h}}{\mathring{\rho}}-\frac{\tilde{m}_1(\mathring{p}-\tilde{p})}{\mathring{\rho}}-\frac{\mathring{\varphi}_1(\mathring{p}-\tilde{p})}{\mathring{\rho}}+\frac{\tilde{m}_1 \tilde{p}\mathring{\phi}}{\mathring{\rho}\tilde{\rho}}. \end{align}\] By 154 , one has \[\begin{align} \notag \left\lvert\mathcal{L}_5\right\rvert+\left\lvert\mathcal{L}_6\right\rvert \le C\bar{\delta}^{\frac{1}{2}}\tilde{D}_{-1}\left\lvert\check{\mathbf{V}}\right\rvert+C\bar{\delta}^{\frac{1}{2}}\tilde{D}_{-\frac{1}{2}}\left\lvert\partial_{x_1}\check{\mathbf{V}}\right\rvert+C\left\lvert\partial_{x_1}\check{\mathbf{V}}\right\rvert^2+C\left\lvert\mathbf{v}_{\neq}\right\rvert^2. \end{align}\] From the relationships among \(\tilde{\theta}\), \(\tilde{\mathbb{E}}\) and \(\tilde{p}\) 53 , we have \[\begin{align} \left\lvert\mathcal{L}_4\right\rvert&=\left\lvert\frac{5}{3}\tilde{\mathbb{E}}\partial_{x_1}\Psi_1\left( \frac{1}{\tilde{\rho}}-\frac{1}{\mathring{\rho}} \right)-\frac{5}{3}\frac{\left\lvert\tilde{m}\right\rvert^2}{2\tilde{\rho}^2}\partial_{x_1}\Psi_1+\frac{\left\lvert\tilde{m}\right\rvert^2}{3 \tilde{\rho}} \frac{\mathring{\varphi}_1}{\mathring{\rho}}\right\rvert \notag\\ &\le C\bar{\delta}^{\frac{1}{2}} \tilde{D}_{-1}\left\lvert\partial_{x_1}\check{\mathbf{V}}\right\rvert+\left\lvert\partial_{x_1}\check{\mathbf{V}}\right\rvert^2.\notag \end{align}\] For \(i=1,2,3,4\), the remaining terms in \(\mathcal{S}_{\check{\Psi}i}^{f}\) can be calculated using a method similarly for obtaining those estimates as above. We thus have \[\begin{align} \label{lem-2-1-12} \sum_{i=1}^3\left\lvert\mathcal{S}_{\check{\Psi}i}^{f}\right\rvert+\left\lvert\mathcal{S}_{\check H}^f\right\rvert\leq C\bar{\delta}^{\frac{1}{2}} \tilde{D}_{-\frac{3}{2}} +\mathbf{D}^{(0)}+\mathbf{T}^{(0)}+\mathbf{T}^{(1)}+\mathbf{Z}^{(0)}+\mathbf{Z}^{(1)}. \end{align}\tag{157}\] Using the same argument as for obtaining 157 , one has \[\begin{align} \notag \sum_{i=1}^3\left\lvert\partial_{x_1}\mathcal{S}_{\check{\Psi}i}^{f}\right\rvert+\left\lvert\partial_{x_1}\mathcal{S}_{\check H}^f\right\rvert\le C\bar{\delta}^{\frac{1}{2}} \tilde{D}_{-2}+\mathbf{D}^{(1)}+\mathbf{T}^{(1)}+\mathbf{T}^{(2)}+\mathbf{Z}^{(1)}+\mathbf{Z}^{(2)}. \end{align}\] We have finished the proof of ?? . Thus, we have completed the proof of Lemma 17. ◻
Now we can give the \(H^2\) estimates for system 144 .
Lemma 18. Recall the definition of \(\check{\mathbf{V}}\) 55 . Under the same assumptions of Proposition 10, it holds that \[\begin{align} &{\frac{d}{dt}}\left(\left\lVert\check{\mathbf{V}}\right\rVert_{L^2}^2+\sum_{l=1}^4\mathcal{X}^l\right) +\left( \left\lVert\partial_{x_1}\check{\Psi}\right\rVert_{L^2}^2 + \left\lVert\partial_{x_1}\check{H}\right\rVert_{L^2}^2 \right) \le C\bar{\delta}(1+t)^{-\frac{3}{2}}+C\check{\delta}\left\lVert\partial_{x_1}\check{\Phi}\right\rVert_{L^2}^2+\eta\left\lVert\partial_t \check{\mathbf{V}}\right\rVert_{L^2}^2 \notag\\ &\quad\qquad+C\check{\delta}\left[ (1+t)^{-1}\left\lVert\check{\mathbf{V}}\right\rVert_{L^2}^2+\left\lVert\partial_{x_1}^2\check{\mathbf{V}}\right\rVert_{L^2}^2+\left\lVert\partial_t \mathbf{v}\right\rVert_{L^2}^2+\left\lVert\nabla_x \mathbf{v}\right\rVert_{L^2}^2\right]+ C_\eta\sum_{\left\lvert\gamma\right\rvert\leq 1} \left\lVert\partial^{\gamma}g\right\rVert_{\sigma}^2,\label{estimate-of-V-1}\\ &{\frac{d}{dt}}\left\lVert\partial_{x_1}\check{\mathbf{V}}\right\rVert_{L^2}^2 +\left( \left\lVert\partial_{x_1}^2\check{\Psi}\right\rVert_{L^2}^2 + \left\lVert\partial_{x_1}^2\check{H}\right\rVert_{L^2}^2 \right) \le C\bar{\delta}(1+t)^{-\frac{5}{2}}+C\check{\delta}\left\lVert\partial_{x_1}^2\Phi\right\rVert_{L^2}^2 + C_\eta\sum_{\left\lvert\gamma\right\rvert=1} \left\lVert\partial^{\gamma}g\right\rVert_{\sigma}^2\notag\\ &\quad\qquad+C\check{\delta}\left[\sum_{i=0}^1(1+t)^{-i-1}\left\lVert\partial_{x_1}^{1-i}\check{\mathbf{V}}\right\rVert_{L^2}^2+(1+t)^{-1}\left\lVert g\right\rVert_{\sigma}^2+\left\lVert\nabla_x \mathbf{v}\right\rVert_{L^2}^2+\left\lVert\partial_t\mathbf{v}^{\ast}\right\rVert_{L^2}^2\right],\label{estimate-of-V-2}\\ &{\frac{d}{dt}}\left(\left\lVert\partial_{x_1}^2\left(\check{\Phi},\sum_{i=2,3}\check{\Psi}_i,\check{H}\right)\right\rVert_{L^2}^2 + \left\lVert\partial_{x_1}\left( \tilde{\theta}\partial_{x_1} \check{\Psi}_1\right)\right\rVert_{L^2}^2 \right) + \left\lVert\partial_{x_1}^3\left(\check{\Psi},\check{H}\right)\right\rVert_{L^2}^2 \notag\\ \le& C_\eta\sum_{\left\lvert\alpha\right\rvert=2} \left\lVert\partial^{\alpha}g\right\rVert_{\sigma}^2+C\check{\delta}\left[\left\lVert\partial_{x_1}^3\check{\Phi}\right\rVert_{L^2}^2+\sum_{i=0}^2(1+t)^{-i-1}\left\lVert\partial_{x_1}^{2-i}\check{\mathbf{V}}\right\rVert_{L^2}^2+(1+t)^{-2}\left\lVert g\right\rVert_{\sigma}^2+\sum_{\left\lvert\gamma\right\rvert=1} \left\lVert\partial^{\gamma}g\right\rVert_{\sigma}^2\right] \notag\\ &+C\bar{\delta}\left((1+t)^{-2}\left\lVert\nabla_x \mathbf{v}\right\rVert_{L^2}^2+\left\lVert\partial_t \mathbf{v}\right\rVert_{L^2}^2\right)+C\bar{\delta}(\left\lVert\nabla_x^2\mathbf{v}\right\rVert_{L^2}^2+\left\lVert\partial_t\nabla_x \mathbf{v}^{\ast}\right\rVert_{L^2}^2)+C\bar{\delta}(1+t)^{-\frac{5}{2}}.\label{estimate-of-V-3} \end{align}\] {#eq: sublabel=eq:estimate-of-V-1,eq:estimate-of-V-2,eq:estimate-of-V-3}
Proof. We first prove ?? . Multiplying 144 by \(\left(\frac{2\check{\Phi}}{3\tilde{\theta}},\check{\Psi}_1,\check{\Psi}_2,\check{\Psi}_3,\frac{\check{H}}{\tilde{\theta}}\right)\), one has \[\begin{align} &\frac{d}{dt}\left(\frac{1}{3}\left\lVert\frac{\check{\Phi}}{\tilde{\theta}^{\frac{1}{2}}}\right\rVert_{L^2}^2+\frac{1}{2}\left\lVert\check{\Psi}\right\rVert_{L^2}^2+\frac{1}{2}\left\lVert\frac{\check{H}}{\tilde{\theta}^{\frac{1}{2}}}\right\rVert_{L^2}^2 \right)+\left\lVert\sqrt{\frac{4\mu(\tilde{\theta})}{3\tilde{\rho}}} \partial_{x_1}\check{\Psi}_1\right\rVert_{L^2}^2\\ &\quad+\sum_{i=2}^3\left\lVert\sqrt{\frac{\mu(\tilde{\theta})}{\tilde{\rho}}} \partial_{x_1}\check{\Psi}_i\right\rVert_{L^2}^2+\left\lVert\sqrt{\frac{\kappa(\tilde{\theta})}{\tilde{\theta}\tilde{\rho}}}\partial_{x_1}\check{H}\right\rVert_{L^2}^2=J_{a1}+J_{a2}+J_{a3}, \end{align}\] where \[\begin{align} J_{a1}=&-\sum_{i=1}^3\sum_{j=1}^3\int_{\mathbb{R}}(\mathcal{S}^{m}_{\Psi})_{ja}^i \check{\Psi}_i dx - \sum_{i=1}^3\int_{\mathbb{R}} \frac{\left((\mathcal{S}^{m}_{H})_{ia}+(\mathcal{S}^{m}_{H})_{4}\right)\check{H}}{\tilde{\theta}} dx_1, \\ J_{a2}=&\sum_{i=1}^3\int_{\mathbb{R}}\left(\mathcal{S}_{\check{\Psi}i}^{f} +\mathcal{S}^{m}_{\check{\Psi}i}\right) \check{\Psi}_i dx_1+\frac{2}{3}\int_{\mathbb{R}} \frac{\mathcal{S}_{\check{\Phi}}^{f}}{\tilde{\theta}} \check{\Phi}dx_1+\int_{\mathbb{R}} \frac{\left(\mathcal{S}_{\check{H}}^{f}+\mathcal{S}^{m}_{\check{H}}\right) \check{H}}{\tilde{\theta}} dx_1 ,\\ J_{a3}=&\frac{1}{2}\int_{\mathbb{R}} \partial_{x_1}^2\left( \frac{\kappa(\tilde{\theta})}{\tilde{\theta}\tilde{\rho}}\right)\check{H}^2 dx_1+\int_{\mathbb{R}}\partial_{x_1}^2\left( \frac{2\mu(\tilde{\theta})}{3\tilde{\rho}}\right)\check{\Psi}_{1}^2 dx_1 + \frac{1}{2}\sum_{i=2}^3 \int_{\mathbb{R}}\partial_{x_1}^2\left( \frac{\mu(\tilde{\theta})}{\tilde{\rho}}\right)\check{\Psi}_{i}^2 dx_1\\ &-\frac{1}{3}\int_{\mathbb{R}} \frac{\partial_t \tilde{\theta}}{\tilde{\theta}^2} \check{\Phi}^2 dx_1 - \frac{1}{2} \int_{\mathbb{R}} \frac{\partial_t \tilde{\theta}}{\tilde{\theta}^2} \check{H}^2dx_1. \end{align}\] Directly, one has \[\begin{align} \left\lvert J_{a3}\right\rvert&\leq C\bar{\delta}\left\lVert\tilde{D}_{-1}\right\rVert_{L^\infty} \left\lVert\check{\mathbf{V}}\right\rVert_{L^2}^2\leq C \bar{\delta}(1+t)^{-1}\left\lVert\check{\mathbf{V}}\right\rVert_{L^2}^2. \label{ja13-a} \end{align}\tag{158}\] We now give a more refined estimate of \(J_{a1}\). By the definition of \((\mathcal{S}^{m}_{\Psi})_{1a}^i\) 132 and applying integration by parts, one has \[\begin{align} \label{math-M-1a} \int_{\mathbb{R}} (\mathcal{S}^{m}_{\Psi})_{1a}^i \check{\Psi}_i d{x_1}=&R\underbrace{\partial_t\int_{\mathbb{R}} \mathbf{D}_0\int_{\mathbb{R}^3} \theta B_{1i}\left( \frac{\xi-u}{\sqrt{R\theta}}\right) \frac{\sqrt{\mu}}{M} g d\xi \check{\Psi}_i dx_1 }_{\mathcal{X}^i}-R\underbrace{\int_{\mathbb{R}} \mathbf{D}_0\int_{\mathbb{R}^3} \partial_t\left(\theta B_{1i}\left( \frac{\xi-u}{\sqrt{R\theta}}\right) \frac{\sqrt{\mu}}{M}\right) g d\xi \check{\Psi}_i dx_1}_{\mathcal{X}_1} \notag \\ &-R\underbrace{\int_{\mathbb{R}} \mathbf{D}_0\int_{\mathbb{R}^3} \theta B_{1i}\left( \frac{\xi-u}{\sqrt{R\theta}}\right) \frac{\sqrt{\mu}}{M} g d\xi \partial_t\check{\Psi}_i dx_1 }_{\mathcal{X}_2}. \end{align}\tag{159}\] Applying Hölder’s inequality and the a priori assumptions 69 , we obtain \[\begin{align} \mathcal{X}_1 \leq &\int_{\Omega} \left\lvert\partial_t (\tilde{\rho},\tilde{u},\tilde{\theta})\right\rvert \left\lvert g\right\rvert_{\sigma} \left\lvert\check{\Psi}_i\right\rvert dx + \int_{\Omega} \left\lvert\partial_t(\phi,\psi,\zeta)\right\rvert \left\lvert g\right\rvert_{\sigma} \left\lvert\check{\Psi}_i\right\rvert dx \notag \\ \leq&C (\bar{\delta}^{\frac{1}{2}}+\chi) \left[ (1+t)^{-1} \left\lVert\check{\Psi}_i\right\rVert_{L^2}^2 +\left\lVert\partial_t(\phi,\psi,\zeta)\right\rVert_{L^2}^2 + \left\lVert g\right\rVert_{\sigma}^2 \right],\tag{160} \\ \mathcal{X}_2 \leq & C_\eta \left\lVert g\right\rVert_{\sigma}^2+\eta\left\lVert\partial_t \check{\Psi}_i\right\rVert_{L^2}^2. \tag{161} \end{align}\] Combining 159 , 160 and 161 , one has \[\begin{align} \label{math-M-1a-1} \int_{\mathbb{R}} (\mathcal{S}^{m}_{\Psi})_{1a}^i \check{\Psi}_i dx_1-\mathcal{X}^i \leq C\check{\delta}\left[ (1+t)^{-1} \left\lVert\check{\Psi}\right\rVert_{L^2}^2 +\left\lVert\partial_t(\phi,\psi,\zeta)\right\rVert_{L^2}^2 \right] + \eta \left\lVert\partial_t \check{\Psi}\right\rVert_{L^2}^2+C_\eta \left\lVert g\right\rVert_{\sigma}^2 . \end{align}\tag{162}\] Using the same method, one has \[\begin{align} \label{math-M-4a-1} \int_{\mathbb{R}} (\mathcal{S}^{m}_{\Psi})_{2a}^i\check{\Psi}_i dx_1 \leq C \check{\delta}\left[ (1+t)^{-1} \left\lVert\check{\Psi}\right\rVert_{L^2}^2 +\left\lVert\partial_{x_1}(\phi,\psi,\zeta)\right\rVert_{L^2}^2\right] + \eta \left\lVert\partial_{x_1} \check{\Psi}\right\rVert_{L^2}^2+C_\eta \left\lVert g\right\rVert_{\sigma}^2. \end{align}\tag{163}\] And we also have \[\begin{align} \int_{\mathbb{R}}(\mathcal{S}^{m}_{\Psi})_{3a}^i\check{\Psi}_idx_1&=R\int_{\mathbb{R}} \mathbf{D}_0\int_{\mathbb{R}^3} \theta B_{1i}\left( \frac{\xi-u}{\sqrt{R\theta}}\right) \frac{\sqrt{\mu}}{M} \left[\Gamma\left(g,g \right)+\Gamma\left(\frac{\bar{G}_0}{\sqrt{\mu}},g \right)+\Gamma\left(g,\frac{\bar{G}_0}{\sqrt{\mu}}\right) \right]d\xi \check{\Psi}_1 dx_i\notag\\ &\leq C\bar{\delta}(1+t)^{-1}\left\lVert\check{\Psi}\right\rVert_{L^2}^2+ C(\bar{\delta}+\chi)\left\lVert g\right\rVert_{\sigma}^2,\label{math-M-5a-1} \end{align}\tag{164}\] where we have used the a priori assumptions 69 . Combining 162 , 163 and 164 , one has \[\begin{align} \label{math-M-1-1-1} & \sum_{i=1}^3 \sum_{j=1}^3 \int_{\mathbb{R}}(\mathcal{S}^{m}_{\Psi})_{ja}^i \check{\Psi}_i dx_1-\sum_{i=1}^3\mathcal{X}^i \notag\\ \leq &C\check{\delta}\left[ (1+t)^{-1} \left\lVert\check{\mathbf{V}}\right\rVert_{L^2}^2 +\left\lVert\partial_t \mathbf{v}\right\rVert_{L^2}^2+\left\lVert\nabla_x \mathbf{v}\right\rVert_{L^2}^2\right] +\eta\left( \left\lVert\partial_t \check{\mathbf{V}}\right\rVert_{L^2}^2+\left\lVert\partial_{x_1} \check{\Psi}\right\rVert_{L^2}^2\right)+ C_\eta \left\lVert g\right\rVert_{\sigma}^2 . \end{align}\tag{165}\] For the terms containing \((\mathcal{S}^{m}_{H})_{4}\), by ?? , ?? and the a priori assumptions 69 , we have \[\begin{align} \label{math-M-1-1-1-1} \int_{\mathbb{R}} \left\lvert\frac{(\mathcal{S}^{m}_{H})_{4}\check{H}}{\tilde{\theta}}\right\rvert dx_1 \leq C\bar{\delta}(1+t)^{-\frac{3}{2}}+ C\check{\delta}\left[(1+t)^{-1}\left\lVert\check{H}\right\rVert_{L^2}^2+\sum_{\left\lvert\alpha\right\rvert\le1}\left\lVert\partial^{\alpha}g\right\rVert_{\sigma}^2 +\left\lVert\mathbf{v}\right\rVert_{H^1}^2+\left\lVert\partial_t\mathbf{v}\right\rVert_{L^2}^2\right]. \end{align}\tag{166}\] The remaining terms in \(J_{a1}\) can be treated as for deriving 159 166 . Then we obtain \[\begin{align} \label{math-M-final} J_{a1} + \sum_{l=1}^4 \mathcal{X}^l\leq& C\bar{\delta}(1+t)^{-\frac{3}{2}}+C\check{\delta} \left[ (1+t)^{-1} \left\lVert\check{\mathbf{V}}\right\rVert_{L^2}^2 +\left\lVert\partial_t \mathbf{v}\right\rVert_{L^2}^2+\left\lVert\nabla_x \mathbf{v}\right\rVert_{L^2}^2\right]\notag \\ &+\eta\left( \left\lVert\partial_t \check{\mathbf{V}}\right\rVert_{L^2}^2+\left\lVert\partial_{x_1} (\check{\Psi},\check{H})\right\rVert_{L^2}^2\right)+ C_\eta \sum_{\left\lvert\alpha\right\rvert\le 1}\left\lVert\partial^{\alpha}g\right\rVert_{\sigma}^2 . \end{align}\tag{167}\] By 145 , ?? , ?? and the a priori assumptions 69 , one has \[\begin{align} \label{ja12} J_{a2}&\le C\int_{\mathbb{R}} \left[\bar{\delta}\tilde{D}_{-\frac{3}{2}}+\bar{\delta}\tilde{D}_{-1}\left\lvert\mathbf{v}\right\rvert+\bar{\delta}\tilde{D}_{-\frac{1}{2}}(\left\lvert\nabla_x\mathbf{v}^{\ast}\right\rvert+\left\lvert\partial_t\mathbf{v}^{\ast}\right\rvert)+\mathbf{D}^{(0)}+\mathbf{T}^{(0)}+\mathbf{T}^{(1)}+\mathbf{Z}^{(0)}+\mathbf{Z}^{(1)}\right] \left\lvert\check{\mathbf{V}}\right\rvert dx_1\notag\\ &\leq C\bar{\delta}(1+t)^{-\frac{3}{2}}+C\check{\delta}\left[ (1+t)^{-1}\left\lVert\check{\mathbf{V}}\right\rVert_{L^2}^2+\left\lVert\partial_{x_1}\check{\mathbf{V}}\right\rVert_{H^1}^2+\left\lVert\nabla_x \mathbf{v}\right\rVert_{L^2}^2 +\left\lVert\partial_t \mathbf{v}^{\ast}\right\rVert_{L^2}^2\right] . \end{align}\tag{168}\] Combining 158 , 167 and 168 , we have completed the proof of ?? .
To prove ?? , multiplying \(\partial_{x_1}\)144 by \(\left(\frac{2}{3}\partial_{x_1}\check{\Phi},\tilde{\theta}\partial_{x_1}\check{\Psi}_{1},\tilde{\theta}\partial_{x_1}\check{\Psi}_{2},\tilde{\theta}\partial_{x_1}\check{\Psi}_{3},\partial_{x_1}\check{H}\right)\), one has \[\begin{align} &\frac{d}{dt}\left(\frac{1}{3}\left\lVert\partial_{x_1}\check{\Phi}\right\rVert_{L^2}^2+\frac{1}{2}\left\lVert\partial_{x_1}\check{\Psi}\right\rVert_{L^2}^2+\frac{1}{2}\left\lVert\partial_{x_1}\check{H}\right\rVert_{L^2}^2 \right)+\left\lVert\sqrt{\frac{4\mu(\tilde{\theta})\tilde{\theta}}{3\tilde{\rho}}} \partial_{x_1}^2\check{\Psi}_1\right\rVert_{L^2}^2\\ &\quad+\sum_{i=2}^3\left\lVert\sqrt{\frac{\mu(\tilde{\theta})\tilde{\theta}}{\tilde{\rho}}} \partial_{x_1}^2\check{\Psi}_i\right\rVert_{L^2}^2+\left\lVert\sqrt{\frac{\kappa(\tilde{\theta})}{\tilde{\rho}}}\partial_{x_1}^2\check{H}\right\rVert_{L^2}^2=J_{b1}+J_{b2}+J_{b3}, \end{align}\] where \[\begin{align} J_{b1}=&-\sum_{i=1}^3\sum_{j=1}^3\int_{\mathbb{R}}\tilde{\theta}\partial_{x_1}(\mathcal{S}^{m}_{\Psi})_{ja}^i \partial_{x_1}\check{\Psi}_i dx_1-\sum_{i=1}^3\int_{\mathbb{R}}\left( \partial_{x_1}(\mathcal{S}^{m}_{H})_{ia}+ \partial_{x_1}(\mathcal{S}^{m}_{H})_{4}\right)\partial_{x_1}\check{H}dx_1 , \\ J_{b2}=&\sum_{i=1}^3\int_{\mathbb{R}}\tilde{\theta}\left(\partial_{x_1}\mathcal{S}_{\check{\Psi}i}^{f}+\partial_{x_1} \mathcal{S}^{m}_{\check{\Psi}i} \right) \partial_{x_1}\check{\Psi}_i dx_1+\frac{2}{3}\int_{\mathbb{R}} \partial_{x_1}\mathcal{S}_{\check{\Phi}}^{f} \partial_{x_1}\check{\Phi}dx_1+\int_{\mathbb{R}} \left(\partial_{x_1}\mathcal{S}_{\check{H}}^{f} + \partial_{x_1}\mathcal{S}^{m}_{\check{H}} \right)\partial_{x_1}\check{H}dx_1,\\ J_{b3}=&\frac{1}{2}\sum_{i=1}^3\int_{\mathbb{R}} \partial_t \tilde{\theta}\left\lvert\partial_{x_1}\check{\Psi}_{i}\right\rvert^2 dx_1+\int_{\mathbb{R}}\frac{2}{3}\partial_{x_1}\left( \frac{\partial_{x_1}\tilde{\theta}\mu(\tilde{\theta})}{\tilde{\rho}}\right)\left\lvert\partial_{x_1}\check{\Psi}_1\right\rvert^2 + \sum_{i=2}^1\frac{1}{2}\partial_{x_1}\left( \frac{\partial_{x_1}\tilde{\theta}\mu(\tilde{\theta})}{\tilde{\rho}}\right)\left\lvert\partial_{x_1}\Psi_i\right\rvert^2 dx_1. \end{align}\] Using 132 , 133 , 136 139 , ?? , ?? , the a priori assumptions 69 and integration by parts, one has \[\begin{align} &\left\lvert J_{b1}\right\rvert \le \sum_{i=1}^3\sum_{j=1}^3\int_{\mathbb{R}}\left(\bar{\delta}\tilde{D}_{-\frac{1}{2}}\left\lvert\partial_{x_1}\check{\mathbf{V}}\right\rvert+\left\lvert\partial_{x_1}^2 \check{\mathbf{V}}^{\ast}\right\rvert\right)\left\lvert(\mathcal{S}^{m}_{\Psi})_{ja}^i\right\rvert dx_1+\sum_{i=1}^3\int_{\mathbb{R}}\left( \left\lvert(\mathcal{S}^{m}_{H})_{ia}\right\rvert+ \left\lvert(\mathcal{S}^{m}_{H})_{4}\right\rvert\right)\left\lvert\partial_{x_1}^2\check{\mathbf{V}}^{\ast}\right\rvert dx_1 \notag\\ &\qquad\le C\bar{\delta}(1+t)^{-\frac{5}{2}}+C\check{\delta}(1+t)^{-1}\left(\left\lVert\partial_{x_1}\check{\mathbf{V}}\right\rVert_{L^2}^2 +\left\lVert g\right\rVert_{\sigma}^2\right) +\eta\left\lVert\partial_{x_1}^2 (\check{\Psi},\check{H})\right\rVert_{L^2}^2\notag\\ &\qquad\quad+C\bar{\delta}\left\lVert\nabla_x \mathbf{v}\right\rVert_{L^2}^2 + C\bar{\delta}\left\lVert\partial_t\mathbf{v}^{\ast}\right\rVert_{L^2}^2+C_\eta \sum_{\left\lvert\alpha\right\rvert=1}\left\lVert\partial^{\alpha}g\right\rVert_{\sigma}^2,\tag{169}\\ &\left\lvert J_{b3}\right\rvert\leq C\bar{\delta}(1+t)^{-1} \left\lVert\partial_{x_1}\check{\mathbf{V}}\right\rVert_{L^2}^2. \tag{170} \end{align}\] Then employing ?? , ?? , the a priori assumptions 69 and integration by parts, we obtain \[\begin{align} \label{jb12} J_{b2}&\le C\int_{\mathbb{R}} \left[\bar{\delta}\tilde{D}_{-\frac{3}{2}}+\bar{\delta}\tilde{D}_{-1}\left\lvert\mathbf{v}\right\rvert+\bar{\delta}\tilde{D}_{-\frac{1}{2}}(\left\lvert\nabla_x\mathbf{v}^{\ast}\right\rvert+\left\lvert\partial_t\mathbf{v}^{\ast}\right\rvert)+\mathbf{D}^{(0)}+\mathbf{T}^{(0)}+\mathbf{T}^{(1)}+\mathbf{Z}^{(0)}+\mathbf{Z}^{(1)}\right] \notag\\ &\qquad\qquad\qquad \times\left(\bar{\delta}^{\frac{1}{2}}\tilde{D}_{-\frac{1}{2}}\left\lvert\partial_{x_1}\check{\mathbf{V}}\right\rvert+\left\lvert\partial_{x_1}^2 \check{\mathbf{V}}\right\rvert\right) dx_1\notag\\ &\leq C\bar{\delta}(1+t)^{-\frac{5}{2}}+C\check{\delta}\left[ \sum_{i=0}^2(1+t)^{-2+i}\left\lVert\partial_{x_1}^i\check{\mathbf{V}}\right\rVert_{L^2}^2+\left\lVert\nabla_x \mathbf{v}\right\rVert_{L^2}^2+\left\lVert\partial_t \mathbf{v}^{\ast}\right\rVert_{L^2}^2 \right] . \end{align}\tag{171}\] Combining 169 , 170 and 171 , we have completed the proof of ?? .
To prove ?? , we will rewrite the equation \(\partial_{x_1}^2\)144 in the following form to avoid the limitations of the structural conditions \[\begin{align} \label{T-A-D-2} \left\{ \begin{aligned} &\partial_t \partial_{x_1}^2\check{\Phi}+\partial_{x_1}^2\left( \tilde{\theta}\partial_{x_1}\check{\Psi}_{1}\right)=\partial_{x_1}^2\mathcal{S}_{\check{\Phi}}^{f},\\ &\partial_t\partial_{x_1}\left( \tilde{\theta}\partial_{x_1}\check{\Psi}_{1} \right) + \frac{2}{3} \partial_{x_1}\left(\tilde{\theta}\partial_{x_1}^2\check{H}\right)+\frac{2}{3}\partial_{x_1}\left(\tilde{\theta}\partial_{x_1}^2\check{\Phi}\right) -\partial_{x_1}\left[\tilde{\theta}\partial_{x_1}\left(\frac{4\mu(\tilde{\theta})}{3\tilde{\rho}} \partial_{x_1}^2\check{\Psi}_{1} \right) \right]\\ &\;+\partial_{x_1}\left(\tilde{\theta}\partial_{x_1}(\mathcal{S}^{m}_{\Psi})^1\right)= \partial_{x_1}\left( \tilde{\theta}\partial_{x_1}\mathcal{S}_{\check{\Psi}1}^{f}\right) + \partial_{x_1}\left( \partial_t \tilde{\theta}\partial_{x_1} \check{\Psi}_1 \right),\\ &\partial_t \partial_{x_1}^2\check{\Psi}_i - \partial_{x_1}^2\left(\frac{\mu(\tilde{\theta})}{\tilde{\rho}}\partial_{x_1}^2 \check{\Psi}_i\right) + \partial_{x_1}^2 (\mathcal{S}^{m}_{\Psi})^i = \partial_{x_1}^2 \mathcal{S}_{\check{\Psi}i}^{f} ,\quad \text{for} \quad i=2,3,\\ &\partial_t \partial_{x_1}^2\check{H}+ \frac{2}{3}\partial_{x_1}^2\left(\tilde{\theta}\partial_{x_1}\check{\Psi}_{1}\right) - \partial_{x_1}^2\left(\frac{\kappa(\tilde{\theta})}{\tilde{\rho}}\partial_{x_1}^2 \check{H}\right) + \partial_{x_1}^2 (\mathcal{S}^{m}_{H})= \partial_{x_1}^2\mathcal{S}_{\check{H}}^{f}, \end{aligned}\right. \end{align}\tag{172}\] where we have used \((\mathcal{S}^{m}_{\Psi})^{i}=\sum_{j=1}^3\sum_{l=a,b}(\mathcal{S}^{m}_{\Psi})_{jl}^i+(\mathcal{S}^{m}_{\Psi})_{4b}^i\) 130 and \((\mathcal{S}^{m}_{H})=\sum_{j=1}^3\Big((\mathcal{S}^{m}_{H})_{ja}+(\mathcal{S}^{m}_{H})_{jb}\Big)+(\mathcal{S}^{m}_{H})_{4b}+(\mathcal{S}^{m}_{H})_{4}\) 131 . Multiplying 172 by \(\left(\frac{2}{3}\tilde{\theta}\partial_{x_1}^2\check{\Phi},\partial_{x_1}\left(\tilde{\theta}\partial_{x_1}\check{\Psi}_{1}\right), \partial_{x_1}^2\check{\Psi}_{2},\partial_{x_1}^2\check{\Psi}_{3}, \; \tilde{\theta}\partial_{x_1}^2\check{H}\right)\), one has \[\begin{align} \label{2025-11-12-2} &\frac{d}{dt}\left[\frac{1}{3}\left\lVert\tilde{\theta}^{\frac{1}{2}}\partial_{x_1}^2\check{\Phi}\right\rVert_{L^2}^2+\frac{1}{2}\left(\left\lVert\partial_{x_1}\left(\tilde{\theta}\partial_{x_1}\check{\Psi}_{1}\right)\right\rVert_{L^2}^2+ \sum_{i=2}^{3}\left\lVert\partial_{x_1}^2\check{\Psi}_i\right\rVert_{L^2}^2 +\left\lVert\tilde{\theta}^{\frac{1}{2}}\partial_{x_1}^2\check{H}\right\rVert_{L^2}^2\right)\right]\notag\\ &+\frac{4}{3}\left\lVert\tilde{\theta}\sqrt{\frac{\mu(\tilde{\theta})}{\tilde{\rho}}} \partial_{x_1}^3\check{\Psi}_1\right\rVert_{L^2}^2 +\sum_{i=2}^3\left\lVert\sqrt{\frac{\mu(\tilde{\theta})}{\tilde{\rho}}} \partial_{x_1}^3\check{\Psi}_i\right\rVert_{L^2}^2+\left\lVert\sqrt{\frac{\tilde{\theta}\kappa(\tilde{\theta})}{\tilde{\rho}}}\partial_{x_1}^3\check{H}\right\rVert_{L^2}^2=J_{c1}+J_{c2}+J_{c3}, \end{align}\tag{173}\] where \[\begin{align} J_{c1}=&-\int_{\mathbb{R}}\partial_{x_1}\left(\tilde{\theta}\partial_{x_1} (\mathcal{S}^{m}_{\Psi})^1\right) \partial_{x_1}\left(\tilde{\theta}\partial_{x_1}\check{\Psi}_1\right) dx_1-\sum_{i=2}^3\int_{\mathbb{R}}\partial_{x_1}^2(\mathcal{S}^{m}_{\Psi})^i \partial_{x_1}^2\check{\Psi}_i dx_1-\int_{\mathbb{R}} \tilde{\theta}\partial_{x_1}^2(\mathcal{S}^{m}_{H}) \partial_{x_1}^2\check{H}dx_1 , \\ J_{c2}=&\frac{2}{3}\int_{\mathbb{R}} \tilde{\theta}\partial_{x_1}^2\mathcal{S}_{\check{\Phi}}^{f} \partial_{x_1}^2\check{\Phi}dx_1+\int_{\mathbb{R}} \partial_{x_1}\left(\tilde{\theta}\partial_{x_1}\check{\Psi}_1\right)\partial_{x_1}\left( \tilde{\theta}\partial_{x_1}\mathcal{S}_{\check{\Psi}1}^{f} \right) +\sum_{i=2}^3\int_{\mathbb{R}} \partial_{x_1}^2\mathcal{S}_{\check{\Psi}i}^{f} \partial_{x_1}^2\check{\Psi}_i dx_1+\int_{\mathbb{R}}\tilde{\theta}\partial_{x_1}^2\mathcal{S}_{\check{H}}^{f}\partial_{x_1}^2\check{H}dx_1,\\ J_{c3}=&\frac{1}{3}\int_{\mathbb{R}}\partial_t\tilde{\theta}\left\lvert\partial_{x_1}^2 \check{\Phi}\right\rvert^2dx_1+\frac{1}{2}\int_{\mathbb{R}}\partial_t\tilde{\theta}\left\lvert\partial_{x_1}^2 \check{H}\right\rvert^2dx_1+\int_{\mathbb{R}} \left(\tilde{\theta}\partial_{x_1}\check{\Psi}_1\right)_{x_1}\left(\partial_t\tilde{\theta}\partial_{x_1}\check{\Psi}_1 \right)_{x_1}dx_1\\ &+\frac{1}{2}\sum_{i=2}^3\int_{\mathbb{R}} \partial_{x_1}^2 \left( \frac{\mu(\tilde{\theta})}{\tilde{\rho}}\right)\left\lvert\partial_{x_1}^2\check{\Psi}_i\right\rvert^2dx_1+ \frac{1}{2}\int_{\mathbb{R}} \left[ \partial_{x_1}^2\left( \frac{\kappa(\tilde{\theta})}{\tilde{\rho}}\right)\tilde{\theta}+\frac{\kappa(\tilde{\theta})}{\tilde{\rho}}\partial_{x_1}^2\tilde{\theta}\right]\left\lvert\partial_{x_1}^2\check{H}\right\rvert^2dx_1\\ &-\frac{4}{3} \int_{\mathbb{R}}\tilde{\theta}\partial_{x_1} \left(\frac{\mu(\tilde{\theta})}{\tilde{\rho}} \right)\partial_{x_1}^{2}\check{\Psi}_1\partial_{x_1}^2\left(\tilde{\theta}\partial_{x_1}\check{\Psi}_1\right)dx_1-\frac{4}{3} \int_{\mathbb{R}} \frac{\tilde{\theta}\mu(\tilde{\theta})}{\tilde{\rho}} \partial_{x_1}^3 \check{\Psi}_1 \left( 2\partial_{x_1}\tilde{\theta}\partial_{x_1}^2\check{\Psi}_1+\partial_{x_1}^2\tilde{\theta}\partial_{x_1}\check{\Psi}_1\right)dx_1. \end{align}\] Employing ?? and the a priori assumptions 69 , one has \[\begin{align} \left\lvert J_{c1}\right\rvert\leq&C \sum_{i=1}^3\int_{\mathbb{R}} \left( \left\lvert\partial_{x_1}(\mathcal{S}^{m}_{\Psi})^i\right\rvert+\left\lvert\partial_{x_1}(\mathcal{S}^{m}_{H})\right\rvert \right)\left( \bar{\delta}^{\frac{1}{2}}\tilde{D}_{-1}\left\lvert\partial_{x_1}\check{\mathbf{V}}\right\rvert +\bar{\delta}^{\frac{1}{2}}\tilde{D}_{-\frac{1}{2}}\left\lvert\partial_{x_1}^2\check{\mathbf{V}}\right\rvert+\left\lvert\partial_{x_1}^3(\check{\Psi},\check{H})\right\rvert\right) dx_1 \notag\\ \leq& C\check{\delta}\left[\sum_{i=0}^1(1+t)^{-2+i}\left\lVert\partial_{x_1}^{1+i}\check{\mathbf{V}}\right\rVert_{L^2}^2+(1+t)^{-2}(\left\lVert\nabla_x\mathbf{v}\right\rVert_{L^2}^2+\left\lVert\partial_t\mathbf{v}\right\rVert_{L^2}^2)+(1+t)^{-2}\left\lVert g\right\rVert_{\sigma}^2+\sum_{\left\lvert\gamma\right\rvert=1}\left\lVert\partial^{\gamma}g\right\rVert_{\sigma}^2\right]\notag\\ &+\eta\left\lVert\partial_{x_1}^3(\check{\Psi},\check{H})\right\rVert_{L^2}^2+C_\eta \sum_{\left\lvert\alpha\right\rvert=2}\left\lVert\partial^{\alpha}g\right\rVert_{\sigma}^2+C\bar{\delta}\Big(\left\lVert\nabla_x^2 \mathbf{v}\right\rVert_{L^2}^2+\left\lVert\partial_t\nabla\mathbf{v}^{\ast}\right\rVert_{L^2}^2+(1+t)^{-\frac{5}{2}}\Big),\tag{174}\\ \left\lvert J_{c3}\right\rvert\leq&C\check{\delta}\left[(1+t)^{-3}\left\lVert\check{\mathbf{V}}\right\rVert_{L^2}^2+ (1+t)^{-2}\left\lVert\partial_{x_1}\check{\mathbf{V}}\right\rVert_{L^2}^2+ (1+t)^{-1}\left\lVert\partial_{x_1}^2\check{\mathbf{V}}\right\rVert_{L^2}^2+\left\lVert\partial_{x_1}^3(\check{\Psi},\check{H})\right\rVert_{L^2}^2\right].\tag{175} \end{align}\] Then employing 145 , ?? and integration by parts, we obtain \[\begin{align} \label{jc12} J_{c2}&\le C \int_{\mathbb{R}} \left[\bar{\delta}\tilde{D}_{-2}+\mathbf{D}^{(1)}+\mathbf{T}^{(1)}+\mathbf{T}^{(2)}+\mathbf{Z}^{(1)}+\mathbf{Z}^{(2)}\right] \left(\left\lvert\partial_{x_1}^3\check{\mathbf{V}}\right\rvert+\bar{\delta}^{\frac{1}{2}}\tilde{D}_{-\frac{1}{2}}\left\lvert\partial_{x_1}^2\check{\mathbf{V}}\right\rvert+\bar{\delta}^{\frac{1}{2}} \tilde{D}_{-1}\left\lvert\partial_{x_1}\check{\mathbf{V}}\right\rvert\right) dx_1\notag \\[1.5mm] &\leq C\bar{\delta}(1+t)^{-\frac{7}{2}}+C\check{\delta}\left[ \sum_{i=0}^3 (1+t)^{-i}\left\lVert\partial_{x_1}^{3-i}\check{\mathbf{V}}\right\rVert_{L^2}^2+\left\lVert\nabla_x^2\mathbf{v}\right\rVert_{L^2}^2 \right], \end{align}\tag{176}\] where we have used the a \(prioti\) assumptions 69 and \[\begin{align} &\int_{\mathbb{R}} \left\lvert\partial_{x_1}^2\check{\mathbf{V}}\right\rvert^2 \left\lvert\partial_{x_1}^3\check{\mathbf{V}}\right\rvert dx_1 \le C \left\lVert\partial_{x_1}^2 \check{\mathbf{V}}\right\rVert_{L^4}^2 \left\lVert\partial_{x_1}^3\check{\mathbf{V}}\right\rVert_{L^2} \le C \left\lVert\partial_{x_1}^2 \check{\mathbf{V}}\right\rVert_{L^2}^{\frac{3}{2}} \left\lVert\partial_{x_1}^3 \check{\mathbf{V}}\right\rVert_{L^2}^\frac{3}{2}\\ &\qquad\le C \chi^{-3}\left\lVert\partial_{x_1}^2\check{\mathbf{V}}\right\rVert_{L^2}^6+C\chi \left\lVert\partial_{x_1}^3 \check{\mathbf{V}}\right\rVert_{L^2}^2 \le C \chi(1+t)^{-1}\left\lVert\partial_{x_1}^2\check{\mathbf{V}}\right\rVert_{L^2}^2+C \chi\left\lVert\partial_{x_1}^3 \check{\mathbf{V}}\right\rVert_{L^2}^2, \\ &\int_{\mathbb{R}} \left\lvert\partial_{x_1}\check{\mathbf{V}}\right\rvert\left\lvert\partial_{x_1}^2\check{\mathbf{V}}\right\rvert \left\lvert\partial_{x_1}^3\check{\mathbf{V}}\right\rvert dx_1 \le C \left\lVert\partial_{x_1} \check{\mathbf{V}}\right\rVert_{L^{\infty}}\left\lVert\partial_{x_1}^2 \check{\mathbf{V}}\right\rVert_{L^2} \left\lVert\partial_{x_1}^3\check{\mathbf{V}}\right\rVert_{L^2} \\ &\qquad\le C\chi\left\lVert\partial_{x_1}^3 \check{\mathbf{V}}\right\rVert_{L^2}^2+ C\chi(1+t)^{-1}\left\lVert\partial_{x_1}^2\check{\mathbf{V}}\right\rVert_{L^2}^2. \end{align}\] Combining (173 )–(176 ) , we have completed the proof of ?? . Hence we have finished the proof of Lemma 18. ◻
Remark 19. In deriving the energy estimate 173 , we have utilized the fact that \[\begin{align} &\int_{\mathbb{R}} \partial_{x_1}^2\left( \tilde{\theta}\partial_{x_1}\check{\Psi}_{1}\right) \tilde{\theta}\partial_{x_1}^2 \check{\Phi}dx_1 + \int_{\mathbb{R}} \partial_{x_1} \left(\tilde{\theta}\partial_{x_1}\check{\Psi}_1 \right) \partial_{x_1}\left(\tilde{\theta}\partial_{x_1}^2\check{\Phi}\right)dx_1=0,\\ & \int_{\mathbb{R}} \partial_{x_1}^2\left( \tilde{\theta}\partial_{x_1}\check{\Psi}_{1}\right) \tilde{\theta}\partial_{x_1}^2 \check{H}dx_1 + \int_{\mathbb{R}} \partial_{x_1} \left(\tilde{\theta}\partial_{x_1}\check{\Psi}_1 \right) \partial_{x_1}\left(\tilde{\theta}\partial_{x_1}^2\check{H}\right)dx_1=0. \end{align}\] As a result, the term \(\bar{\delta}\int_{\mathbb{R}} \tilde{D}_{-\frac{1}{2}} \left[(\partial_{x_1}^2\check{\Phi})^2+\left\lvert\partial_{x_1}^2 \check{\Psi}\right\rvert^2 + (\partial_{x_1}^2\check{H})^2 \right] dx_1\) does not appear in the energy estimate for system 172 , which is the main reason why system 172 is used for the second-order derivative estimate.
Recall the definition of \((\check{\mathbf{V}},\mathbf{v},\mathbf{v}^{\ast})\) 55 . Next, we estimate \(\left\lVert\partial_t \check{\mathbf{V}}\right\rVert_{L^2}^2\) and \(\left\lVert\partial_t \mathbf{v}\right\rVert_{L^2}^2\).
Lemma 20. Under the same assumptions of Proposition 10, it holds that \[\begin{align} &\left\lVert\partial_t\mathbf{v}\right\rVert_{L^2}\leq C\bigg[\left\lVert\nabla_x \mathbf{v}\right\rVert_{L^2}+\left\lVert\nabla_x^2\mathbf{v}^{\ast}\right\rVert_{L^2}+\bar{\delta}\left\lVert\partial_t\nabla_x\mathbf{v}^{\ast}\right\rVert_{L^2}^2+\sum_{1\leq\left\lvert\alpha\right\rvert\leq2}\left\lVert\partial^{\alpha}g\right\rVert_{\sigma} \\ &\qquad\qquad\qquad\quad+\check{\delta}(1+t)^{-\frac{1}{2}}\left(\left\lVert g\right\rVert_{\sigma}+\left\lVert\mathbf{v}\right\rVert_{L^2} \right) + \bar{\delta}(1+t)^{-\frac{5}{4}}\bigg],\\ &\left\lVert\partial_t\check{\mathbf{V}}\right\rVert_{L^2}\leq C\left(\left\lVert\partial_{x_1}\check{\mathbf{V}}\right\rVert_{H^1}^2+\sum_{\left\lvert\gamma\right\rvert\leq1}\left\lVert\partial^{\gamma}g\right\rVert_{\sigma}\right)+C\check{\delta}\left[(1+t)^{-1}\left\lVert\check{\mathbf{V}}\right\rVert_{L^2}+\left\lVert\nabla_x \mathbf{v}\right\rVert_{L^2}+\left\lVert\partial_t\mathbf{v}^{\ast}\right\rVert_{L^2}^2\right]+ C\bar{\delta}(1+t)^{-\frac{5}{4}}. \end{align}\]
Proof. The desired estimates above can be directly obtained from equations 118 and 144 and using the a priori assumptions 69 . For brevity of presentation, details of the proof are omitted. ◻
Finally, we estimate \(\left\lVert\partial_{x_1}^{k+1}\Phi\right\rVert_{L^2}^2\).
Lemma 21. Under the same assumptions of Proposition 10, for \(k=0,1\), it holds that \[\begin{align} \label{H-D-Phi-lem46pp} &\frac{d}{dt}\left( \left\lVert\partial_{x_1}^{k+1}\check{\Phi}\right\rVert_{L^2}^2+\int_{\mathbb{R}} \partial_{x_1}^{k+1}\check{\Phi}\partial_{x_1}^k\check{\Psi}_1 dx_1 \right) + c\left\lVert\partial_{x_1}^{k+1}\check{\Phi}\right\rVert_{L^2}^2\notag \\ \le& C\left\lVert\partial_{x_1}^{k+1}(\check{\Psi},\check{H})\right\rVert_{L^2}^2+ C\bar{\delta}(1+t)^{-\frac{3+2k}{2}}+C\sum_{k\le\left\lvert\alpha\right\rvert\le k+1}\left\lVert\partial^{\alpha}g\right\rVert_{\sigma}^2+C\bar{\delta}\left\lvert k\right\rvert(1+t)^{-1}(\left\lVert\nabla_x\mathbf{v}\right\rVert_{L^2}^2+\left\lVert\partial_t\mathbf{v}^{\ast}\right\rVert_{L^2}^2)\notag\\ &+C\check{\delta}\left[ \sum_{i=0}^k (1+t)^{i-k-1}\left\lVert\partial_{x_1}^i \check{\mathbf{V}}\right\rVert_{L^2}^2+(1+t)^{-k-1}\left\lVert g\right\rVert_{\sigma}^2+\left\lVert\nabla_x^{k+1}\mathbf{v}\right\rVert_{L^2}^2+\left\lVert\partial_t\nabla_x^k\mathbf{v}^{\ast}\right\rVert_{L^2}^2+\left\lVert\partial_{x_1}^{k+2}\check{\mathbf{V}}\right\rVert_{L^2}^2 \right]. \end{align}\qquad{(5)}\]
Proof. Multiplying \(\partial_{x_1}^k\)144 \(_2\) by \(\partial_{x_1}^{k+1}\check{\Phi}\), one has \[\begin{align} \label{H-D-Phi-1} &\frac{d}{dt}\int_{\mathbb{R}} \partial_{x_1}^k\check{\Psi}_1\partial_{x_1}^{k+1}\check{\Phi}dx_1 + \frac{2}{3}\left\lVert\partial_{x_1}^{k+1}\check{\Phi}\right\rVert_{L^2}^2-\frac{4}{3}\int_{\mathbb{R}}\frac{\mu(\tilde{\theta})}{\tilde{\rho}} \partial_{x_1}^{k+2}\check{\Psi}_1\partial_{x_1}^{k+1}\check{\Phi}dx_1 + \int_{\mathbb{R}} \partial_t \partial_{x_1}^k \check{\Phi}\partial_{x_1}^{k+1}\check{\Psi}_1 dx_1\notag\\ =&\frac{4}{3}\sum_{i=0}^{k-1}\int_{\mathbb{R}}\partial_{x_1}^{k-i}\left( \frac{\mu(\tilde{\theta})}{\tilde{\rho}} \right)\partial_{x_1}^{i+2}\check{\Psi}_1 \partial_{x_1}^{k+1}\check{\Phi}dx_1 -\frac{2}{3}\int_{\mathbb{R}}\partial_{x_1}^{k+1}\check{H}\partial_{x_1}^{k+1}\check{\Phi}dx_1 \notag\\ &-\int_{\mathbb{R}}\partial_{x_1}^k (\mathcal{S}^{m}_{\Psi})^1\partial_{x_1}^{k+1}\check{\Phi}dx_1 +\int_{\mathbb{R}} \partial_{x_1}^k \mathcal{S}_{\check{\Psi}1}^{f} \partial_{x_1}^{k+1}\check{\Phi}dx_1. \end{align}\tag{177}\] Using 144 \(_1\), we then obtain \[\begin{align} \label{H-D-Phi-2} &-\frac{4}{3}\int_{\mathbb{R}}\frac{\mu(\tilde{\theta})}{\tilde{\rho}} \partial_{x_1}^{k+2}\check{\Psi}_1\partial_{x_1}^{k+1}\check{\Phi}dx_1 \notag\\ =&\frac{4}{3}\int_{\mathbb{R}}\frac{\mu(\tilde{\theta})}{\tilde{\rho}} \partial_{x_1}^{k+1} \left( \frac{\partial_t\check{\Phi}}{\tilde{\theta}} \right) \partial_{x_1}^{k+1}\check{\Phi}dx_1 -\frac{4}{3}\int_{\mathbb{R}}\frac{\mu(\tilde{\theta})}{\tilde{\rho}} \partial_{x_1}^{k+1} \left( \frac{\mathcal{S}_{\check{\Phi}}^{f}}{\tilde{\theta}} \right) \partial_{x_1}^{k+1}\check{\Phi}dx_1 \notag\\ =&\frac{2}{3}\frac{d}{dt}\left(\int_{\mathbb{R}} \frac{\mu(\tilde{\theta})}{\tilde{\theta}\tilde{\rho}} \left\lvert\partial_{x_1}^{k+1} \check{\Phi}\right\rvert ^2 dx_1\right)-\frac{2}{3}\int_{\mathbb{R}} \frac{d}{dt}\left(\frac{\mu(\tilde{\theta})}{\tilde{\theta}\tilde{\rho}} \right)\left\lvert\partial_{x_1}^{k+1} \check{\Phi}\right\rvert ^2 dx_1 -\frac{4}{3}\int_{\mathbb{R}}\frac{\mu(\tilde{\theta})}{\tilde{\rho}} \partial_{x_1}^{k+1} \left( \frac{\mathcal{S}_{\check{\Phi}}^{f}}{\tilde{\theta}} \right) \partial_{x_1}^{k+1}\check{\Phi}dx_1 \notag\\ &+ \frac{4}{3}\sum_{i=0}^k \int_{\mathbb{R}}\frac{\mu(\tilde{\theta})}{\tilde{\rho}} \partial_{x_1}^{k+1-i}\left( \frac{1}{\tilde{\theta}} \right) \left[\partial_{x_1}^i \mathcal{S}_{\check{\Phi}}^{f} -\partial_{x_1}^i \left( \tilde{\theta}\partial_{x_1}\check{\Psi}_{1}\right) \right]\partial_{x_1}^{k+1} \check{\Phi}dx_1. \end{align}\tag{178}\] And using 144 \(_1\) again, one has \[\begin{align} \label{H-D-Phi-3} \int_{\mathbb{R}}\partial_t \partial_{x_1}^k \check{\Phi}\partial_{x_1}^{k+1}\check{\Psi}_1 dx_1 = -\int_{\mathbb{R}} \partial_{x_1}^k\left(\tilde{\theta}\partial_{x_1}\check{\Psi}_{1} \right) \partial_{x_1}^{k+1}\check{\Psi}_1 dx_1 + \int_{\mathbb{R}}\partial_{x_1}^k\mathcal{S}_{\check{\Phi}}^{f} \partial_{x_1}^{k+1}\check{\Psi}_1 dx_1. \end{align}\tag{179}\] Combining 177 , 178 and 179 , we obtain \[\begin{align} \label{H-D-Phi-4} &\frac{2}{3}\frac{d}{dt}\left\lVert \sqrt{\frac{\mu(\tilde{\theta})}{\tilde{\theta}\tilde{\rho}}} \partial_{x_1}^{k+1} \check{\Phi}\right\rVert_{L^2}^2 + \frac{d}{dt}\int_{\mathbb{R}} \partial_{x_1}^k\check{\Psi}_1\partial_{x_1}^{k+1}\check{\Phi}dx_1 + \frac{2}{3}\left\lVert\partial_{x_1}^{k+1}\check{\Phi}\right\rVert_{L^2}^2=J^{(k)}_{d1}+J^{(k)}_{d2}+J^{(k)}_{d3}, \end{align}\tag{180}\] where \[\begin{align} J_{d1}^{(0)}=& -\int_{\mathbb{R}} (\mathcal{S}^{m}_{\Psi})^1\partial_{x_1}\check{\Phi}dx_1, \qquad \qquad \qquad J_{d1}^{(1)}= -\int_{\mathbb{R}} \partial_{x_1} (\mathcal{S}^{m}_{\Psi})^1\partial_{x_1}^2\check{\Phi}dx_1 ,\\ J_{d2}^{(0)}=& \int_{\mathbb{R}} \mathcal{S}_{\check{\Psi}1}^{f} \partial_{x_1}\check{\Phi}dx_1 +\frac{4}{3}\int_{\mathbb{R}}\frac{\mu(\tilde{\theta})}{\tilde{\rho}} \partial_{x_1} \left( \frac{\mathcal{S}_{\check{\Phi}}^{f}}{\tilde{\theta}} \right) \partial_{x_1}\check{\Phi}dx_1 - \frac{4}{3} \int_{\mathbb{R}}\frac{\mu(\tilde{\theta})}{\tilde{\rho}} \partial_{x_1}\left( \frac{1}{\tilde{\theta}} \right)\mathcal{S}_{\check{\Phi}}^{f}\partial_{x_1} \check{\Phi}dx_1 - \int_{\mathbb{R}}\mathcal{S}_{\check{\Phi}}^{f} \partial_{x_1}\check{\Psi}_1 dx_1 ,\\ J_{d2}^{(1)}=& \int_{\mathbb{R}} \partial_{x_1}\mathcal{S}_{\check{\Psi}1}^{f} \partial_{x_1}^{2}\check{\Phi}dx_1 - \int_{\mathbb{R}}\partial_{x_1}\mathcal{S}_{\check{\Phi}}^{f} \partial_{x_1}^{2}\check{\Psi}_1 dx_1 \\ &+\frac{4}{3}\int_{\mathbb{R}}\frac{\mu(\tilde{\theta})}{\tilde{\rho}} \partial_{x_1}^{2} \left( \frac{\mathcal{S}_{\check{\Phi}}^{f}}{\tilde{\theta}} \right) \partial_{x_1}^{2}\check{\Phi}dx_1 - \frac{4}{3}\sum_{i=0}^1 \int_{\mathbb{R}}\frac{\mu(\tilde{\theta})}{\tilde{\rho}} \partial_{x_1}^{2-i}\left( \frac{1}{\tilde{\theta}} \right)\partial_{x_1}^i \mathcal{S}_{\check{\Phi}}^{f}\partial_{x_1}^{2} \check{\Phi}dx_1,\\ J_{d3}^{(0)}=& -\frac{2}{3}\int_{\mathbb{R}}\partial_{x_1}\check{H}\partial_{x_1}\check{\Phi}dx_1 +\frac{2}{3} \int_{\mathbb{R}}\partial_t \left( \frac{\mu(\tilde{\theta})}{\tilde{\theta}\tilde{\rho}} \right) \left\lvert\partial_{x_1}\check{\Phi}\right\rvert^2 dx_1 +\int_{\mathbb{R}} \tilde{\theta}\left\lvert\partial_{x_1}\check{\Psi}_{1}\right\rvert^2 dx_1 \\ &+ \frac{4}{3}\int_{\mathbb{R}}\frac{\mu(\tilde{\theta})}{\tilde{\rho}} \partial_{x_1}\left( \frac{1}{\tilde{\theta}} \right) \tilde{\theta}\partial_{x_1} \check{\Psi}_{1} \partial_{x_1} \check{\Phi}dx_1,\\ J_{d3}^{(1)}=&\frac{4}{3}\int_{\mathbb{R}}\partial_{x_1}\left( \frac{\mu(\tilde{\theta})}{\tilde{\rho}} \right)\partial_{x_1}^{2}\check{\Psi}_1 \partial_{x_1}^{2}\check{\Phi}dx_1 -\frac{2}{3}\int_{\mathbb{R}}\partial_{x_1}^{2}\check{H}\partial_{x_1}^{2}\check{\Phi}dx_1 +\frac{2}{3} \int_{\mathbb{R}}\partial_t \left( \frac{\mu(\tilde{\theta})}{\tilde{\theta}\tilde{\rho}} \right) \left\lvert\partial_{x_1}^{2}\check{\Phi}\right\rvert^2 dx_1 \\ &+\int_{\mathbb{R}} \partial_{x_1}\left(\tilde{\theta}\partial_{x_1}\check{\Psi}_{1} \right) \partial_{x_1}^{2}\check{\Psi}_1 dx_1 + \frac{4}{3}\sum_{i=0}^1 \int_{\mathbb{R}}\frac{\mu(\tilde{\theta})}{\tilde{\rho}} \partial_{x_1}^{2-i}\left( \frac{1}{\tilde{\theta}} \right) \partial_{x_1}^i \left( \tilde{\theta}\partial_{x_1}\check{\Psi}_{1}\right) \partial_{x_1}^{2} \check{\Phi}dx_1. \end{align}\] By 132 , 133 , 134 , ?? , ?? and the a priori assumptions 69 , we have \[\begin{align} J_{d1}^{(0)}+J_{d3}^{(0)} \leq& \frac{1}{1600}\left\lVert\partial_{x_1}\check{\Phi}\right\rVert_{L^2}^2+ C \bar{\delta}(1+t)^{-\frac{3}{2}}+C\left\lVert\partial_{x_1}\left( \check{\Psi},\check{H}\right)\right\rVert_{L^2}^2 + C \sum_{\left\lvert\alpha\right\rvert\le 1}\left\lVert\partial^{\alpha}g\right\rVert_{\sigma}^2\notag\\ &+C\bar{\delta}\left\lVert\partial_t\mathbf{v}^{\ast}\right\rVert_{L^2}^2+C\bar{\delta}\left\lVert\nabla_x\mathbf{v}\right\rVert_{L^2}^2,\notag \\ J_{d1}^{(1)}+J_{d3}^{(1)} \leq& \frac{1}{1600}\left\lVert\partial_{x_1}^2\check{\Phi}\right\rVert_{L^2}^2 + C \bar{\delta}(1+t)^{-\frac{5}{2}}+C\left\lVert\partial_{x_1}^2\left( \check{\Psi},\check{H}\right)\right\rVert_{L^2}^2+C\bar{\delta}(\left\lVert\partial_t\nabla_x\mathbf{v}^{\ast}\right\rVert_{L^2}^2+\left\lVert\nabla_x^2\mathbf{v}\right\rVert_{L^2}^2)\notag \\ &+C\check{\delta}(1+t)^{-1}\left[ \left\lVert\partial_{x_1}\check{\mathbf{V}}\right\rVert_{L^2}^2 +\left\lVert\nabla_x \mathbf{v}\right\rVert_{L^2}^2+\left\lVert\partial_t\mathbf{v}^{\ast}\right\rVert_{L^2}^2+(1+t)^{-1} \left\lVert g\right\rVert_{\sigma}^2\right]+C\sum_{1\le \left\lvert\alpha\right\rvert\le 2}\left\lVert\partial^{\alpha}g\right\rVert_{\sigma}^2.\notag \end{align}\] Employing 145 , ?? and the a priori assumptions 69 , one has \[\begin{align} J_{d2}^{(0)}& \leq C \int_{\mathbb{R}} \left\lvert\partial_{x_1}\check{\mathbf{V}}\right\rvert\left( \bar{\delta}\tilde{D}_{-\frac{3}{2}}+\bar{\delta}^{\frac{1}{2}}\tilde{D}_{-1}\left\lvert\mathbf{v}\right\rvert+\mathbf{D}^{(0)}+\mathbf{D}^{(1)}+\mathbf{T}^{(0)}+\mathbf{T}^{(1)}+\mathbf{Z}^{(0)}+\mathbf{Z}^{(1)}\right) dx_1 \notag\\ & \le C\bar{\delta}(1+t)^{-\frac{5}{2}}+C\check{\delta}\left[\left\lVert\partial_{x_1}\check{\mathbf{V}}\right\rVert_{H^1}^2 + \left\lVert\nabla_x \mathbf{v}\right\rVert_{L^2}^2+(1+t)^{-2}\left\lVert\check{\mathbf{V}}\right\rVert_{L^2}^2 \right], \notag \\ J_{d2}^{(1)}& \leq C \int_{\mathbb{R}} \left\lvert\partial_{x_1}^2\check{\mathbf{V}}\right\rvert\left( \bar{\delta}\tilde{D}_{-2}+\mathbf{D}^{(1)}+\mathbf{D}^{(2)}+\mathbf{T}^{(1)}+\mathbf{T}^{(2)}+\mathbf{Z}^{(1)}+\mathbf{Z}^{(2)}\right) dx_1 \notag\\ & \le C\bar{\delta}(1+t)^{-\frac{7}{2}}+C\check{\delta}\left[\left\lVert\partial_{x_1}^2\check{\mathbf{V}}\right\rVert_{H^1}^2 + \left\lVert\nabla_x^2 \mathbf{v}\right\rVert_{L^2}^2+ \sum_{i=0}^1(1+t)^{-2-i}\left\lVert\partial_{x_1}^{1-i}\check{\mathbf{V}}\right\rVert_{L^2}^2 \right]. \notag \end{align}\] Plugging all the above estimates to 180 gives ?? . Then we have completed the proof of Lemma 21. ◻
Combining Lemma 18, Lemma 20, and Lemma 21, we have completed the proof of Theorem 16.
We first notice \[\int_{\mathbb{T}^2}(\phi_{\neq},\psi_{\neq},\zeta_{\neq})dx_2dx_3=0.\] Thus, it can be seen that the Poincaré’s inequality holds for the non-zero modes: \[\begin{align} \label{20254664608-1} \|(\phi_{\neq},\psi_{\neq},\zeta_{\neq})\|_{L^2} \leq C \| (\nabla^k\phi_{\neq},\nabla^k\psi_{\neq},\nabla^k\zeta_{\neq})\|_{L^2}. \end{align}\tag{181}\] Taking \(\mathbf{D}_{\neq}\) 33 for the perturbation system 118 , one has \[\begin{align} \label{eqs201} \begin{cases} \partial_t\phi_{\neq}+\mathring{u}\cdot \nabla_x \phi_{\neq}+\mathring{\rho}\operatorname{div}_x \psi_{\neq}=\mathcal{S}_{\phi_{\neq}}, \\ \partial_t\psi_{\neq}+ \mathring{u}\cdot \nabla_x \psi_{\neq}+\frac{2}{3\tilde{\rho}}\left(\mathring{\theta}\nabla_x \phi_{\neq} + \mathring{\rho}\nabla_x \zeta_{\neq} \right)= \frac{\mu(\tilde{\theta})}{\tilde{\rho}}\left(\Delta_x\psi_{\neq}+\frac{1}{3}\nabla_x\text{div}_x\psi_{\neq}\right) +\mathcal{S}_{\psi_{\neq}},\\ \partial_t \zeta_{\neq}+ \mathring{u}\cdot \nabla_x \zeta_{\neq}+\frac{2}{3} \mathring{\theta}\operatorname{div} \psi_{\neq}=\frac{\kappa(\tilde{\theta})}{\tilde{\rho}}\Delta_x\zeta_{\neq} +\mathcal{S}_{\zeta_{\neq}}, \end{cases} \end{align}\tag{182}\] where \[\begin{align} &\mathcal{S}_{\phi_{\neq}}= \mathring{u}\cdot\nabla_x \phi_{\neq} - \mathbf{D}_{\neq}\left( u \cdot \nabla_x\phi\right) + \mathring{\rho}\text{div}_x\psi_{\neq}- \mathbf{D}_{\neq}\left(\rho\text{div}_x \psi \right) -\psi_{\neq} \cdot \nabla_x \tilde{\rho}- \phi_{\neq} \text{div}_x \tilde{u},\\ &\mathcal{S}_{\psi_{\neq}}= \mathring{u}\cdot \nabla_x \psi_{\neq}-\mathbf{D}_{\neq}\left( u \cdot \nabla_x\psi \right) + \frac{2}{3\tilde{\rho}}\left(\mathring{\theta}\nabla_x \phi_{\neq} + \mathring{\rho}\nabla_x \zeta_{\neq} \right) \\ &\qquad\quad- \frac{\nabla_x p_{\neq}}{\tilde{\rho}} - \psi_{\neq} \cdot \nabla \tilde{u}+ \frac{1}{\tilde{\rho}}\mathbf{D}_{\neq}\left(\frac{\phi \nabla_x p}{\rho} \right) + (\mathcal{S}_{\psi}^{f2})_{\neq}+(\mathcal{S}_{\psi}^{m})_{\neq},\\ &\mathcal{S}_{\zeta_{\neq}}= \mathring{u}\cdot \nabla_x \zeta_{\neq}-\mathbf{D}_{\neq}\left( u \cdot \nabla_x\zeta \right) + \frac{2}{3} \mathring{\theta}\text{div}_x\psi_{\neq} - \frac{2}{3} \mathbf{D}_{\neq}\left( \theta \text{div}_x \psi \right) - \psi_{\neq} \cdot \nabla_x\tilde{\theta}- \frac{2}{3}\zeta_{\neq}\text{div}_x \tilde{u}+ (\mathcal{S}_{\zeta}^{f2})_{\neq}+(\mathcal{S}_{\zeta}^{m})_{\neq}. \end{align}\] Recall the definition of \((\mathbf{v},{\mathbf{v}}^{\ast})\) 55 , instant energy functionals \(\mathcal{E}_i\) 56 61 and dissipation energy functionals \(\mathcal{D}_i\) 62 68 . Then, we present the \(H^1\) energy estimate for the non-zero mode for macroscopic part \((\phi_{\neq},\psi_{\neq},\zeta_{\neq})\).
Theorem 22. Under the same assumptions of Proposition 10, it holds that \[\begin{align} \frac{d}{dt}\|\mathbf{v}_{\neq}\|_{H^1}^2+c\|(\nabla_x \mathbf{v}_{\neq},\nabla_x^2{\mathbf{v}}^{\ast}_{\neq})\|_{L^2}^2 \leq C \check{\delta}\left[(1+t)^{-2}\mathcal{D}_2+\mathcal{D}_3 +(1+t)^{-1}\left\lVert\partial_t\nabla_x\mathbf{v}^{\ast}\right\rVert_{L^2}^2 \right]+C_\eta\sum_{\left\lvert\alpha\right\rvert=2}\left\lVert\partial^{\alpha}g\right\rVert_{\sigma}^2. \end{align}\]
The proof of Theorem 22 will be decomposed into two parts including Lemma 23 and Lemma 24 below.
Lemma 23. Under the same assumptions of Proposition 10, it holds that \[\begin{align} \notag &\frac{d}{dt}\|\mathbf{v}_{\neq}\|_{H^1}^2+c\|(\nabla_x \mathbf{v}_{\neq},\nabla_x^2{\mathbf{v}}^{\ast}_{\neq})\|_{L^2_x}^2 \leq C_\eta (\left\lVert\nabla_x \mathcal{S}_{\phi_{\neq}}\right\rVert_{L^2}^2+\left\lVert\mathcal{S}_{\psi_{\neq}}\right\rVert_{L^2}^2+\left\lVert\mathcal{S}_{\zeta_{\neq}}\right\rVert_{L^2}^2). \end{align}\]
Proof. Step 1. Multiplying 182 \(_1\) by \(\frac{2}{3}\frac{\mathring{\theta}}{\mathring{\rho}\tilde{\rho}}\phi_{\neq}\), 182 \(_2\) by \(\psi_{\neq}\), 182 \(_3\) by \(\frac{\mathring{\rho}}{\mathring{\theta}\tilde{\rho}}\zeta_{\neq}\), and then summing all of them, one has \[\begin{align} \label{eqs210} \partial_t\left(\frac{\mathring{\theta}}{3\mathring{\rho}\tilde{\rho}}\phi_{\neq}^2+\frac{|\psi_{\neq}|^2}{2}+\frac{\mathring{\rho}}{2\mathring{\theta}\tilde{\rho}}\zeta_{\neq}^2\right)+\frac{\mu(\tilde{\theta})}{\tilde{\rho}}|\nabla_x \psi_{\neq}|^2+\frac{\mu(\tilde{\theta})}{3\tilde{\rho}}|\text{div}_x\psi_{\neq}|^2+\frac{\mathring{\rho}\kappa(\tilde{\theta})}{\mathring{\theta}\tilde{\rho}^2}|\nabla_x \zeta_{\neq}|^2=\mathcal{J}_0+\text{div}_x(\cdots), \end{align}\tag{183}\] where \[\begin{align} \mathcal{J}_0:=&\left[\left(\frac{\mathring{\theta}}{\mathring{\rho}\tilde{\rho}}\right)_t+\nabla_x\cdot\left(\frac{\mathring{\theta}\mathring{u}}{\mathring{\rho}\tilde{\rho}}\right)\right]\frac{\phi_{\neq}^2}{3}+\left[\left(\frac{\mathring{\rho}}{\mathring{\theta}\tilde{\rho}}\right)_t+\nabla_x\cdot\left(\frac{\mathring{\rho}\mathring{u}}{\mathring{\theta}\tilde{\rho}}\right)\right]\frac{\zeta_{\neq}^2}{2}+\frac{\partial_{x_1} \mathring{u}_1}{2} \left\lvert\psi_{\neq}\right\rvert^2 \notag\\ &-\partial_{x_1}\left(\frac{\mu(\tilde{\theta})}{\tilde{\rho}} \right)\left(\sum_{j=1}^3\partial_{x_1}\psi_{j\neq}\psi_{j\neq}+\frac{1}{3}\text{div}_x\psi_{\neq}\psi_{1\neq}\right)+\partial_{x_1}^2 \left(\frac{\kappa(\tilde{\theta})\mathring{\rho}}{2\tilde{\rho}^2\mathring{\theta}}\right)\left\lvert\zeta_{\neq}\right\rvert^2 \notag \\ &+\frac{2\mathring{\theta}}{3\mathring{\rho}\tilde{\rho}}\phi_{\neq} \mathcal{S}_{\phi_{\neq}}+ \frac{\mathring{\rho}}{\mathring{\theta}\tilde{\rho}} \zeta_{\neq} \mathcal{S}_{\zeta_{\neq}} + \psi_{\neq} \cdot \mathcal{S}_{\psi_{\neq}}+\frac{2\psi_{1\neq}}{3}\left[ \partial_{x_1}\left(\frac{\mathring{\theta}}{\tilde{\rho}}\right)\phi_{\neq}+\partial_{x_1}\left(\frac{\mathring{\rho}}{\tilde{\rho}}\right)\zeta_{\neq} \right]. \end{align}\] According to the a priori assumptions 69 , Poincaré’s inequality 181 and Hölder’s inequality, one has \[\begin{align} \label{eqs212} \int_{\mathbb{R}\times \mathbb{T}^2} \mathcal{J}_0 dx \leq \eta \left\lVert(\nabla_x \phi_{\neq},\nabla_x \psi_{\neq} ,\nabla_x \zeta_{\neq})\right\rVert_{L^2}^2 + C_\eta (\left\lVert\nabla_x \mathcal{S}_{\phi_{\neq}}\right\rVert_{L^2}^2+\left\lVert\mathcal{S}_{\psi_{\neq}}\right\rVert_{L^2}^2+\left\lVert\mathcal{S}_{\zeta_{\neq}}\right\rVert_{L^2}^2). \end{align}\tag{184}\] Integrating 183 on \(\Omega\) and combining 184 , it yields \[\begin{align} \label{eqs213} \frac{d}{dt} \left\lVert\mathbf{v}_{\neq}\right\rVert_{L^2}^2+\left\lVert\nabla_x {\mathbf{v}}^{\ast}_{\neq}\right\rVert_{L^2}^2 \leq \eta\left\lVert\nabla_x\phi_{\neq}\right\rVert^2_{L^2}+C_\eta(\left\lVert\nabla_x \mathcal{S}_{\phi_{\neq}}\right\rVert_{L^2}^2+\left\lVert\mathcal{S}_{\psi_{\neq}}\right\rVert_{L^2}^2+\left\lVert\mathcal{S}_{\zeta_{\neq}}\right\rVert_{L^2}^2) . \end{align}\tag{185}\]
Step 2. We still need to estimate \(\|\nabla_x\phi_{\neq}\|_{L^2}^2\). Taking 182 \(_2\times\frac{\tilde{\rho}}{\mu(\tilde{\theta})}\nabla_x\phi_{\neq}+\nabla_x{\eqref{eqs201}_1}\times\frac{4}{3\mathring{\rho}}\nabla_x\phi_{\neq},\) one has \[\begin{align} \label{eqs220} &\partial_t\left(\frac{2}{3\mathring{\rho}}|\nabla_x\phi_{\neq}|^2+\frac{\tilde{\rho}}{\mu(\tilde{\theta})}\psi_{\neq}\cdot\nabla_x\phi_{\neq}\right)+\frac{2\mathring{\theta}}{3\mu(\tilde{\theta})}|\nabla_x\phi_{\neq}|^2=\mathcal{J}_1+\text{div}_x(\cdots), \end{align}\tag{186}\] where \[\begin{align} \mathcal{J}_1=& \partial_t \left( \frac{\tilde{\rho}}{\mu(\tilde{\theta})} \right) \psi_\neq \cdot \nabla_x\phi_{\neq} + \partial_{x_1} \left( \frac{\tilde{\rho}}{\mu(\tilde{\theta})} \right)\psi_{1\neq}\mathring{u}\cdot\nabla_x \phi_{\neq}+ \partial_{x_1} \left( \frac{\tilde{\rho}}{\mu(\tilde{\theta})} \right)\psi_{1\neq} \mathring{\rho}\text{div}_x \psi_{\neq} \notag\\ &+\frac{\tilde{\rho}}{\mu(\tilde{\theta})} \text{div}_x \psi_{\neq } \mathring{u}\cdot\nabla_x\phi_{\neq} +\frac{\tilde{\rho}\mathring{\rho}}{\mu(\tilde{\theta})} \left\lvert\text{div}_x\psi_{\neq}\right\rvert^2-\frac{2}{3\mathring{\rho}} \partial_{x_1} \phi_{\neq} \partial_{x_1} \mathring{u}\cdot \nabla_x \phi_{\neq}-\frac{4 \partial_{x_1} \mathring{\rho}}{3 \mathring{\rho}} \partial_{x_1} \phi_{\neq}\text{div}_x \psi_{\neq} \notag\\ &-\frac{\tilde{\rho}}{\mu(\tilde{\theta})} \mathring{u}\cdot \nabla_x \psi_{\neq} \cdot\nabla_x\phi_{\neq} - \frac{2\mathring{\rho}}{3\mu(\tilde{\theta})}\nabla_x\zeta_{\neq} \cdot \nabla_x\phi_{\neq} + \frac{\tilde{\rho}}{\mu(\tilde{\theta})} \mathcal{S}_{\psi_{\neq}} \cdot \nabla_x \phi_{\neq} + \frac{4}{3\mathring{\rho}} \nabla_x \phi_{\neq} \cdot \nabla_x \mathcal{S}_{\phi_{\neq}} \notag\\ &-\frac{\tilde{\rho}}{\mu(\tilde{\theta})} \text{div}_x \psi_{\neq} \mathcal{S}_{\phi_{\neq}}-\partial_{x_1} \left( \frac{\tilde{\rho}}{\mu(\tilde{\theta})} \right) \psi_{1\neq} \mathcal{S}_{\phi_{\neq}}. \end{align}\] In order to obtain 186 , we need to use 182 \(_1\) to calculate \(\frac{\tilde{\rho}}{\mu(\tilde{\theta})} \partial_t \psi_{\neq} \nabla_x \phi_{\neq}\) by \[\begin{align} &\frac{\tilde{\rho}}{\mu(\tilde{\theta})} \partial_t \psi_{\neq} \cdot \nabla_x \phi_{\neq}=\partial_t\left(\frac{\tilde{\rho}}{\mu(\tilde{\theta})} \psi_{\neq} \cdot \nabla_x \phi_{\neq} \right) - \partial_t\left(\frac{\tilde{\rho}}{\mu(\tilde{\theta})}\right) \psi_{\neq} \cdot \nabla_x \phi_{\neq} - \frac{\tilde{\rho}}{\mu(\tilde{\theta})} \psi_{\neq} \cdot \nabla_x \partial_t \phi_{\neq} \\ =&\partial_t\left(\frac{\tilde{\rho}}{\mu(\tilde{\theta})} \psi_{\neq} \cdot \nabla_x \phi_{\neq} \right) - \partial_t\left(\frac{\tilde{\rho}}{\mu(\tilde{\theta})}\right) \psi_{\neq} \cdot \nabla_x \phi_{\neq} + \partial_{x_1}\left(\frac{\tilde{\rho}}{\mu(\tilde{\theta})}\right) \psi_{1\neq} \partial_t \phi_{\neq}+ \frac{\tilde{\rho}}{\mu(\tilde{\theta})} \text{div}_x\psi_{\neq} \partial_t \phi_{\neq} + \text{div}_x(\cdots) \\ =&\partial_t\left(\frac{\tilde{\rho}}{\mu(\tilde{\theta})} \psi_{\neq} \cdot \nabla_x \phi_{\neq} \right) - \partial_t\left(\frac{\tilde{\rho}}{\mu(\tilde{\theta})}\right) \psi_{\neq} \cdot \nabla_x \phi_{\neq} - \partial_{x_1}\left(\frac{\tilde{\rho}}{\mu(\tilde{\theta})}\right) \psi_{1\neq} \mathring{u}\cdot \nabla_x \phi_{\neq} - \partial_{x_1}\left(\frac{\tilde{\rho}}{\mu(\tilde{\theta})}\right) \psi_{1\neq} \mathring{\rho}\text{div}_x \phi_{\neq}\\ &- \frac{\tilde{\rho}}{\mu(\tilde{\theta})} \text{div}_x\psi_{\neq} \mathring{u}\cdot \nabla_x \phi_{\neq} -\frac{\tilde{\rho}\mathring{\rho}}{\mu(\tilde{\theta})}\left\lvert\text{div}_x\psi_{\neq}\right\rvert^2 + \partial_{x_1}\left(\frac{\tilde{\rho}}{\mu(\tilde{\theta})}\right) \psi_{1\neq}\mathcal{S}_{\phi_{\neq}} + \frac{\tilde{\rho}}{\mu(\tilde{\theta})} \text{div}_x\psi_{\neq}\mathcal{S}_{\phi_{\neq}}+ \text{div}_x(\cdots). \end{align}\] According to the a priori assumptions 69 , Poincaré’s inequality 181 and applying Hölder’s inequality, one has \[\begin{align} \label{eqs222} \int_{\Omega}\mathcal{J}_1dx\le \eta\|\nabla_x\phi_{\neq}\|_{L^2}^2 + C_\eta \left(\left\lVert\nabla_x \psi_{\neq}\right\rVert_{L^2}^2+\left\lVert \nabla_x\zeta_{\neq}\right\rVert_{L^2}^2 + \left\lVert\nabla_x \mathcal{S}_{\phi_{\neq}}\right\rVert_{L^2}^2 + \left\lVert\mathcal{S}_{\psi_{\neq}}\right\rVert_{L^2}^2\right). \end{align}\tag{187}\] Integrating 186 on \(\Omega\) and combining 187 , it yields \[\begin{align} \label{eqs223} \frac{d}{dt}\int_{\Omega}&\left(\frac{2}{3\mathring{\rho}}|\nabla_x\phi_{\neq}|^2+\frac{\tilde{\rho}}{\mu(\tilde{\theta})}\psi_{\neq}\cdot\nabla_x\phi_{\neq}\right)dx+\int_{\Omega}\frac{\mathring{\theta}}{3\mu(\tilde{\theta})}|\nabla_x\phi_{\neq}|^2dx \notag\\ &\le C_\eta \left(\left\lVert\nabla_x \psi_{\neq}\right\rVert_{L^2}^2+\left\lVert \nabla_x\zeta_{\neq}\right\rVert_{L^2}^2 + \left\lVert\nabla_x \mathcal{S}_{\phi_{\neq}}\right\rVert_{L^2}^2 + \left\lVert\mathcal{S}_{\psi_{\neq}}\right\rVert_{L^2}^2\right). \end{align}\tag{188}\]
Step 3. We estimate \(\left\lVert\nabla_x^2\psi_{\neq}\right\rVert_{L^2}^2\) and \(\left\lVert\nabla_x^2\zeta_{\neq}\right\rVert_{L^2}^2\). Taking \(\nabla_x\eqref{eqs201}_1\times\frac{2\mathring{\theta}}{3\mathring{\rho}\tilde{\rho}}\nabla_x\phi_{\neq}+\nabla_x\eqref{eqs201}_2\times\nabla_x\psi_{\neq}+\nabla_x\eqref{eqs201}_3\times\frac{\mathring{\rho}}{\mathring{\theta}\tilde{\rho}}\nabla_x\zeta_{\neq},\) one has \[\begin{align} \label{eqs230-1} \partial_t\Big(\frac{\mathring{\theta}}{3\mathring{\rho}\tilde{\rho}}\left\lvert\nabla_x\phi_{\neq}\right\rvert&^2+\frac{|\nabla_x\psi_{\neq}|^2}{2}+\frac{\mathring{\rho}}{2\mathring{\theta}\tilde{\rho}}\left\lvert\nabla_x\zeta_{\neq}\right\rvert^2\Big)+\frac{\mu(\tilde{\theta})}{\tilde{\rho}}|\Delta_x\psi_{\neq}|^2\notag\\ &+\frac{\mu(\tilde{\theta})}{3\tilde{\rho}}|\nabla_x\text{div}_x\psi_{\neq}|^2 +\frac{\mathring{\rho}\kappa(\tilde{\theta})}{\mathring{\theta}\tilde{\rho}^2}|\Delta_x\zeta_{\neq}|^2=\mathcal{J}_2+\text{div}_x(\cdots), \end{align}\tag{189}\] where \[\begin{align} \mathcal{J}_{2}=&\left[\left(\frac{\mathring{\theta}}{\mathring{\rho}\tilde{\rho}}\right)_t+\nabla_x\cdot\left(\frac{\mathring{\theta}\mathring{u}}{\mathring{\rho}\tilde{\rho}}\right)\right]\frac{\left\lvert\nabla_x\phi_{\neq}\right\rvert^2}{3}+\left[\left(\frac{\mathring{\rho}}{\mathring{\theta}\tilde{\rho}}\right)_t+\nabla_x\cdot\left(\frac{\mathring{\rho}\mathring{u}}{\mathring{\theta}\tilde{\rho}}\right)\right]\frac{\left\lvert\nabla_x\zeta_{\neq}\right\rvert^2}{2}+\frac{\partial_{x_1} \mathring{u}_1}{2} \left\lvert\nabla_x\psi_{\neq}\right\rvert^2 \notag\\ &-\frac{2\mathring{\theta}\partial_{x_1}\mathring{\rho}}{3\mathring{\rho}\tilde{\rho}} \text{div}_x \psi_{\neq} \partial_{x_1} \phi_{\neq}-\frac{2\mathring{\rho}\partial_{x_1} \mathring{\theta}}{3\mathring{\theta}\tilde{\rho}} \text{div}_x \psi_{\neq} \partial_{x_1} \zeta_{\neq} - \frac{2\mathring{\theta}}{3\mathring{\rho}\tilde{\rho}} \partial_{x_1} \mathring{u}\cdot \nabla_x \phi_{\neq} \partial_{x_1} \phi_{\neq} - \partial_{x_1} \mathring{u}\cdot \nabla_x \psi_{\neq} \cdot\partial_{x_1} \psi_{\neq} \notag\\ &-\frac{\mathring{\rho}}{\mathring{\theta}\tilde{\rho}} \partial_{x_1} \mathring{u}\cdot \nabla_x \zeta_{\neq} \partial_{x_1} \zeta_{\neq} - \frac{2}{3} \partial_{x_1}\left( \frac{\mathring{\theta}}{\tilde{\rho}} \right)\partial_{x_1}\psi_{\neq} \cdot \nabla_x \phi_{\neq} + \frac{2}{3} \partial_{x_1}\left( \frac{\mathring{\theta}}{\tilde{\rho}} \right) \nabla_x \psi_{1\neq} \cdot \nabla_x \phi_{\neq}+\frac{2\mathring{\theta}}{3\mathring{\rho}\tilde{\rho}}\nabla_x\phi_{\neq}\cdot \nabla_x \mathcal{S}_{\phi_{\neq}} \notag\\ &-\frac{2}{3} \partial_{x_1}\left( \frac{\mathring{\rho}}{\tilde{\rho}} \right) \partial_{x_1} \psi_{\neq} \cdot \nabla_x \zeta_{\neq} + \frac{2}{3} \partial_{x_1}\left( \frac{\mathring{\rho}}{\tilde{\rho}} \right) \nabla_x \psi_{1\neq} \cdot \nabla_x \zeta_{\neq} - \frac{1}{3}\partial_{x_1}\left(\frac{\mu(\tilde{\theta})}{\tilde{\rho}} \right) \text{div}_x \psi_{\neq} \Delta \psi_{1\neq} + \nabla_x\psi_{\neq} \cdot \nabla_x\mathcal{S}_{\psi_{\neq}} \notag\\ &- \frac{1}{3}\partial_{x_1}\left(\frac{\mu(\tilde{\theta})}{\tilde{\rho}} \right) \text{div}_x \psi_{\neq} \partial_{x_1} \text{div}_x \psi_{\neq} - \frac{\kappa(\tilde{\theta})}{\tilde{\rho}} \partial_{x_1} \left(\frac{\mathring{\rho}}{\mathring{\theta}\tilde{\rho}} \right) \partial_{x_1} \zeta_{\neq} \Delta_x \zeta_{\neq} + \frac{\mathring{\rho}}{\mathring{\theta}\tilde{\rho}} \nabla_x\zeta_{\neq} \cdot \nabla_x\mathcal{S}_{\zeta_{\neq}}. \end{align}\] According to the a priori assumptions 69 , Poincaré’s inequality 181 and applying Hölder’s inequality, one has \[\begin{align} \label{eqs232} \int_{\Omega}\mathcal{J}_2dx\le \eta \left(\|\nabla_x\phi_{\neq}\|_{L^2}^2 + \left\lVert\nabla_x^2 \psi_{\neq}\right\rVert_{L^2}^2+\left\lVert \nabla_x^2\zeta_{\neq}\right\rVert_{L^2}^2 \right)+ C_\eta (\left\lVert\nabla_x \mathcal{S}_{\phi_{\neq}}\right\rVert_{L^2}^2+\left\lVert\mathcal{S}_{\psi_{\neq}}\right\rVert_{L^2}^2+\left\lVert\mathcal{S}_{\zeta_{\neq}}\right\rVert_{L^2}^2). \end{align}\tag{190}\] Integrating 189 on \(\Omega\) and combining 190 , it yields \[\begin{align} \label{eqs233} \frac{d}{dt}& \left\lVert\nabla_x \mathbf{v}_{\neq}\right\rVert_{L^2}^2 + \left\lVert\nabla_x^2 {\mathbf{v}}^{\ast}_{\neq}\right\rVert_{L^2}^2 \leq\eta \|\nabla_x\phi_{\neq}\|_{L^2}^2 + C_\eta (\left\lVert\nabla_x \mathcal{S}_{\phi_{\neq}}\right\rVert_{L^2}^2+\left\lVert\mathcal{S}_{\psi_{\neq}}\right\rVert_{L^2}^2+\left\lVert\mathcal{S}_{\zeta_{\neq}}\right\rVert_{L^2}^2). \end{align}\tag{191}\] Combining 185 , 188 and 191 , we have completed the proof of Lemma 23. ◻
Lemma 24. Under the same assumptions of Proposition 10, it holds that \[\begin{align} &\left\lVert\nabla_x \mathcal{S}_{\phi_{\neq}}\right\rVert_{L^2}^2\le C \check{\delta}\| (\nabla_x\phi_{\neq},\nabla_x^2\psi_{\neq},\nabla_x^2\zeta_{\neq} )\|_{L^2}^2, \\ &\left\lVert\mathcal{S}_{\psi_{\neq}}\right\rVert_{L^2}^2+\left\lVert\mathcal{S}_{\zeta_{\neq}}\right\rVert_{L^2}^2\leq C \sum_{\left\lvert\alpha\right\rvert= 2} \|\partial^{\alpha} g \|_{\sigma}^2+ C \bar{\delta} \left( \|\nabla_x\phi_{\neq} \|_{L^2}^2+(1+t)^{-2}\left( \left\lVert\nabla_x \mathbf{v}\right\rVert_{L^2}^2+\left\lVert g\right\rVert_{\sigma}^2 \right)+\sum_{\left\lvert\gamma\right\rvert=1}\left\lVert\partial^{\gamma}g\right\rVert_{\sigma}^2 \right)\notag\\ &\qquad\qquad\qquad\qquad\qquad\;\;+C\bar{\delta}(1+t)^{-1}(\left\lVert\nabla_x^2\mathbf{v}^{\ast}\right\rVert_{L^2}^2+\left\lVert\partial_t\nabla_x \mathbf{v}^{\ast}\right\rVert_{L^2}^2). \end{align}\]
Proof. We first make an estimation for \(\mathcal{S}_{\psi_{\neq}}\) 182 . With respect to \(\mathcal{S}_{\psi_{\neq}}\) 182 , we first calculate the part excluding \((\mathcal{S}_{\psi}^{f2})_{\neq}\) and \((\mathcal{S}_{\psi}^{m})_{\neq}\). Among the parts other than \((\mathcal{S}_{\psi}^{f2})_{\neq}\) and \((\mathcal{S}_{\psi}^{m})_{\neq}\), the term \[\label{add46lem946p1} \frac{2}{3\tilde{\rho}}\left(\mathring{\theta}\nabla_x \phi_{\neq} + \mathring{\rho}\nabla_x \zeta_{\neq} \right) - \frac{\nabla_x p_{\neq}}{\tilde{\rho}}\tag{192}\] is the most difficult one to calculate. Therefore, for this part, we only provide an estimate of 192 , and the calculations for the remaining terms are similar. In fact, it holds \[\begin{align} \label{fluid-part-1} \left\lVert\eqref{add46lem946p1}\right\rVert_{L^2} &\leq\left\lVert \frac{2}{3\tilde{\rho}} \left( \partial_{x_1} \mathring{\phi}\zeta_{\neq} + \partial_{x_1} \mathring{\theta}\phi_{\neq} \right) \right\rVert_{L^2}+\sum_{i=1}^3\left\lVert\frac{2}{3\tilde{\rho}} \left( \partial_{xi}\phi_{\neq} \zeta_{\neq} + \partial_{xi}\zeta_{\neq}\phi_{\neq} \right)\right\rVert_{L^2} \notag\\ &\leq C\chi \left\lVert(\nabla_x\phi_{\neq},\nabla_x^2 \psi_{\neq},\nabla_x^2 \zeta_{\neq} )\right\rVert_{L^2}, \end{align}\tag{193}\] where we have used the a priori assumptions 69 . For the fluid part in \((\mathcal{S}_\psi^{f2})_\neq\) 121 , we only carry out the calculation for \[\label{add46lem946p2} \mathbf{D}_{\neq}\left[\mu(\tilde{\theta}) \left( \frac{1}{\rho}-\frac{1}{\tilde{\rho}}\right) \left(\Delta_x u + \frac{1}{3} \nabla_x \text{div}_x u \right)\right],\tag{194}\] and the estimations for the remaining parts can be obtained in the same way. By the a priori assumptions 69 , one has \[\begin{align} \label{fluid-part-2} \left\lVert\eqref{add46lem946p2}\right\rVert_{L^2} \leq&\left\lVert\mu(\tilde{\theta}) \mathbf{D}_{\neq}\left( \frac{1}{\rho}-\frac{1}{\tilde{\rho}}\right) \left(\Delta_x \mathring{u}+ \frac{1}{3} \nabla_x \text{div}_x \mathring{u}\right)\right\rVert_{L^2} + \left\lVert\mu(\tilde{\theta}) \mathbf{D}_0\left( \frac{\phi}{\rho\tilde{\rho}}\right) \left(\Delta_x \psi_{\neq} + \frac{1}{3} \nabla_x \text{div}_x \psi_{\neq} \right)\right\rVert_{L^2} \notag\\ &+\left\lVert[\mu(\tilde{\theta})\mathbf{D}_{\neq}\left[\left( \frac{1}{\rho}-\frac{1}{\tilde{\rho}}\right)_{\neq} \left(\Delta_x \psi_{\neq} + \frac{1}{3} \nabla_x \text{div}_x \psi_{\neq} \right)\right]\right\rVert_{L^2} \notag\\ \leq& C\chi \left\lVert(\nabla_x\phi_{\neq},\nabla_x^2 \psi_{\neq},\nabla_x^2 \zeta_{\neq})\right\rVert_{L^2}. \end{align}\tag{195}\] For the non-fluid part \((\mathcal{S}_{\psi}^{m})_{\neq}\) 122 , we only deal with \[\label{add46lem946p3} \mathbf{D}_{\neq}\left[ \frac{1}{\rho} \int \xi \otimes \xi \cdot \nabla_{x} \left(L_{M}^{-1}\Pi-L_{\bar{M}}^{-1}\bar{\Pi}_1\right) d \xi \right].\tag{196}\] The calculation of the other term is much simpler. It holds \[\begin{align} \label{non-fluid-1} \eqref{add46lem946p3} =& \mathbf{D}_0\left(\frac{1}{\rho}\right) \mathbf{D}_{\neq}\int \xi \otimes \xi \cdot \nabla_{x} L_{M}^{-1}\Pi d \xi + \mathbf{D}_{\neq}\left(\frac{1}{\rho}\right) \mathbf{D}_0\int \xi \otimes \xi \cdot \nabla_{x} \left(L_{M}^{-1}\Pi-L_{\bar{M}}^{-1}\bar{\Pi}_1\right) d \xi \notag \\ &+\mathbf{D}_{\neq}\left[\left(\frac{1}{\rho}\right)_{\neq} \mathbf{D}_{\neq}\int \xi \otimes \xi \cdot \nabla_{x} L_{M}^{-1}\Pi d \xi\right]. \end{align}\tag{197}\] From 148 and the a priori assumptions 69 , one has \[\begin{align} &\left\lVert\mathbf{D}_{\neq}\left(\frac{1}{\rho}\right) \mathbf{D}_0\int \xi \otimes \xi \cdot \nabla_{x} \left(L_{M}^{-1}\Pi-L_{\bar{M}}^{-1}\bar{\Pi}_1\right) d \xi\right\rVert_{L^2}\leq C \check{\delta} \left(\left\lVert\nabla_x \phi_{\neq}\right\rVert_{L^2}+\sum_{1\leq\left\lvert\alpha\right\rvert\leq2}\left\lVert\partial^\alpha g\right\rVert_{\sigma}\right)\tag{198},\\ &\left\lVert\mathbf{D}_{\neq}\int_{\mathbb{R}^3} \xi \otimes \xi \cdot \nabla_x L_M^{-1} \Pi d\xi\right\rVert_{L^2}^2 \leq C \sum_{\left\lvert\alpha\right\rvert=2}\left\lVert\partial^{\alpha}g\right\rVert_{\sigma}^2+C\check{\delta}\left[ (1+t)^{-2}\left( \left\lVert\nabla_x \mathbf{v}\right\rVert_{L^2}^2+\left\lVert g\right\rVert_{\sigma}^2 \right)+\sum_{\left\lvert\gamma\right\rvert=1}\left\lVert\partial^{\gamma}g\right\rVert_{\sigma}^2 \right] \notag\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad +C\bar{\delta}(1+t)^{-1}(\left\lVert\nabla_x^2\mathbf{v}^{\ast}\right\rVert_{L^2}^2+\left\lVert\partial_t\nabla_x \mathbf{v}^{\ast}\right\rVert_{L^2}^2)\tag{199}. \end{align}\] Combining 193 199 , one has \[\begin{align} \label{F-m-111} \left\lVert\mathcal{S}_{\psi_{\neq}}\right\rVert_{L^2}^2\leq& C \sum_{\left\lvert\alpha\right\rvert= 2} \|\partial^{\alpha} g \|_{\sigma}^2+ C \bar{\delta} \left( \|\nabla_x\phi_{\neq} \|_{L^2}^2+(1+t)^{-2}\left( \left\lVert\nabla_x \mathbf{v}\right\rVert_{L^2}^2+\left\lVert g\right\rVert_{\sigma}^2 \right)+\sum_{\left\lvert\gamma\right\rvert=1}\left\lVert\partial^{\gamma}g\right\rVert_{\sigma}^2 \right)\notag\\ &+C\bar{\delta}(1+t)^{-1}(\left\lVert\nabla_x^2\mathbf{v}^{\ast}\right\rVert_{L^2}^2+\left\lVert\partial_t\nabla_x \mathbf{v}^{\ast}\right\rVert_{L^2}^2). \end{align}\tag{200}\] The estimates of \(\mathcal{S}_{\phi_{\neq}}\) and \(\mathcal{S}_{\zeta_{\neq}}\) 182 are the same as those of \(\mathcal{S}_{\psi_{\neq}}\). Thus, using the same argument as 200 , we obtain \[\begin{align} &\left\lVert\nabla_x \mathcal{S}_{\phi_{\neq}}\right\rVert_{L^2}^2\le C \check{\delta}\| (\nabla_x\phi_{\neq},\nabla_x^2\psi_{\neq},\nabla_x^2\zeta_{\neq}) \|_{L^2}^2,\notag \\ &\left\lVert\mathcal{S}_{\psi_{\neq}}\right\rVert_{L^2}^2+\|\mathcal{S}_{\zeta_{\neq}}\|_{L^2_x}^2\leq C \sum_{\left\lvert\alpha\right\rvert= 2} \|\partial^{\alpha} g \|_{\sigma}^2+ C \bar{\delta} \left( \|\nabla_x\phi_{\neq} \|_{L^2}^2+(1+t)^{-2}\left( \left\lVert\nabla_x \mathbf{v}\right\rVert_{L^2}^2+\left\lVert g\right\rVert_{\sigma}^2 \right)+\sum_{\left\lvert\gamma\right\rvert=1}\left\lVert\partial^{\gamma}g\right\rVert_{\sigma}^2 \right)\notag\\ &\qquad\qquad\quad\qquad\qquad\quad+C\bar{\delta}(1+t)^{-1}(\left\lVert\nabla_x^2\mathbf{v}^{\ast}\right\rVert_{L^2}^2+\left\lVert\partial_t\nabla_x \mathbf{v}^{\ast}\right\rVert_{L^2}^2).\label{fluid-and-non-fluid-2} \end{align}\tag{201}\] Combining 200 and 201 , we have completed the proof of Lemma 24. ◻
Combining Lemma 20, Lemma 23 and Lemma 24, we have completed the proof of Theorem 22.
To close the energy estimate, we need to provide the estimates of higher-order derivatives for the fluid part. Recall the definition of \((\mathbf{v},{\mathbf{v}}^{\ast})\) 55 and dissipation energy functionals \(\mathcal{D}_i\) 62 68 . Then we have the following result.
Theorem 25. Under the same assumptions of Proposition 10, it holds that \[\begin{align} & \frac{d}{dt} \int_{\Omega} \nabla_x \psi \cdot\nabla_x^2 \phi dx+ \tilde{c}\left\lVert \nabla_x^2 \phi\right\rVert_{L^2}^2\leq C\bar{\delta}(1+t)^{-\frac{5}{2}}+C\check{\delta}\sum_{i=1}^3(1+t)^{-3+i}\mathcal{D}_i+ C_\eta \left( \left\lVert\nabla_x^2{\mathbf{v}}^{\ast}\right\rVert_{L^2}^2 + \sum_{\left\lvert\alpha\right\rvert=2}\left\lVert\partial^{\alpha}g\right\rVert_{\sigma}^2 \right) ,\\ & \frac{d}{dt} \left(\left\lVert\nabla_x^2\mathbf{v}\right\rVert_{L^2}^2+\tilde{c}\int_{\Omega} \nabla_x^2 \psi \cdot\nabla_x^3 \phi dx\right)+ \tilde{c}\left\lVert \nabla_x^3 \mathbf{v}\right\rVert_{L^2}^2\leq C\bar{\delta}(1+t)^{-\frac{5}{2}}+C\check{\delta}\sum_{i=1}^3(1+t)^{-3+i}\mathcal{D}_i+C_\eta \sum_{\left\lvert\alpha\right\rvert=3}\left\lVert\partial^{\alpha}g\right\rVert_{\sigma}^2. \end{align}\]
The proof of Theorem 25 will be decomposed into two parts via Lemma 26 and Lemma 27 below. To derive estimates for the high-order derivatives (involving time derivatives) of macroscopic quantities and the high-order derivatives of density, we rewrite the equation 9 as \[\begin{align} \label{perturbation-ori} \begin{cases} \partial_t \phi = \mathcal{S}_{0},\\ \partial_t \psi + \frac{2}{3} \frac{\tilde{\theta}}{\tilde{\rho}} \nabla_x\phi=\mathcal{S},\\ \partial_t \zeta=\mathcal{S}_4, \end{cases} \end{align}\tag{202}\] where \[\begin{align} &\mathcal{S}_{0}:=-\partial_t\tilde{\rho}+\operatorname{div}_x(\rho u), \\ &\mathcal{S}:=(\mathcal{S}_1,\mathcal{S}_2,\mathcal{S}_3)^{t}=- \partial_t\tilde{u}- u \cdot \nabla_x u -\frac{\nabla_x p}{\rho}+ \frac{2}{3} \frac{\tilde{\theta}}{\tilde{\rho}} \nabla_x\phi -\frac{1}{\rho}\int \xi \otimes \xi \cdot \nabla_x G d \xi, \\ &\mathcal{S}_4:=-\partial_t\tilde{\theta}-u \cdot \nabla_x \theta - \theta \text{div}_x u -\frac{1}{2\rho}\int_{\mathbb{R}^3} |\xi|^{2} \xi \cdot \nabla_x G d \xi +\frac{u}{\rho}\cdot\int \xi \otimes \xi \cdot \nabla_x G d \xi. \end{align}\]
Lemma 26. Under the same assumptions of Proposition 10, it holds that \[\begin{align} &\left\lVert \partial_t \nabla_x \mathbf{v}\right\rVert_{L^2}^2 \lesssim\left[\bar{\delta}(1+t)^{-\frac{5}{2}}+ \left\lVert\nabla_x^2 \mathbf{v}\right\rVert_{L^2}^2 + \sum_{\left\lvert\alpha\right\rvert=2}\left\lVert\partial^{\alpha}g\right\rVert_{\sigma}^2+\check{\delta} \left( \sum_{\left\lvert\gamma\right\rvert=1} \left\lVert\partial^{\gamma }g\right\rVert_{\sigma}^2 +\sum_{i=1}^2 (1+t)^{-i} \left\lVert\nabla_x^{2-i}\mathbf{v}\right\rVert_{L^2}^2 \right)\right],\label{high-der-ori-1} \\ &\left\lVert \partial_t^2 \mathbf{v}\right\rVert_{L^2}^2 \lesssim\left[\bar{\delta}(1+t)^{-\frac{5}{2}}+ \left\lVert\partial_t\nabla_x \mathbf{v}\right\rVert_{L^2}^2 + \sum_{\left\lvert\alpha\right\rvert=2}\left\lVert\partial^{\alpha}g\right\rVert_{\sigma}^2+\check{\delta} \left( \sum_{\left\lvert\gamma\right\rvert=1} \left\lVert\partial^{\gamma }g\right\rVert_{\sigma}^2 +\sum_{i=1}^2 (1+t)^{-i} \left\lVert\nabla_x^{2-i}\mathbf{v}\right\rVert_{L^2}^2 \right)\right],\label{high-der-ori-1-s} \\ &\left\lVert \partial_t \nabla_x^2 \mathbf{v}\right\rVert_{L^2}^2 \lesssim\left[\bar{\delta}(1+t)^{-\frac{5}{2}}+ \left\lVert\nabla_x^3 \mathbf{v}\right\rVert_{L^2}^2 + \sum_{\left\lvert\alpha\right\rvert=3}\left\lVert\partial^{\alpha}g\right\rVert_{\sigma}^2 + \check{\delta} \left( \sum_{1\leq\left\lvert\gamma\right\rvert\leq 2} \left\lVert\partial^{\gamma }g\right\rVert_{\sigma}^2 +\sum_{i=1}^3 (1+t)^{-i} \left\lVert\nabla_x^{3-i}\mathbf{v}\right\rVert_{L^2}^2 \right)\right],\label{high-der-ori-2} \\ &\left\lVert \partial_t^2 \nabla_x \mathbf{v}\right\rVert_{L^2}^2 \lesssim\left[\bar{\delta}(1+t)^{-\frac{5}{2}}+ \left\lVert\partial_t\nabla_x^2 \mathbf{v}\right\rVert_{L^2}^2 + \sum_{\left\lvert\alpha\right\rvert=3}\left\lVert\partial^{\alpha}g\right\rVert_{\sigma}^2 + \check{\delta} \left( \sum_{1\leq\left\lvert\gamma\right\rvert\leq 2} \left\lVert\partial^{\gamma }g\right\rVert_{\sigma}^2 +\sum_{i=1}^3 (1+t)^{-i} \left\lVert\nabla_x^{3-i}\mathbf{v}\right\rVert_{L^2}^2 \right)\right],\label{high-der-ori-2-s} \\ &\left\lVert \partial_t^3 \mathbf{v}\right\rVert_{L^2}^2 \lesssim\left[\bar{\delta}(1+t)^{-\frac{5}{2}}+ \left\lVert\partial_t^2\nabla_x \mathbf{v}\right\rVert_{L^2}^2 + \sum_{\left\lvert\alpha\right\rvert=3}\left\lVert\partial^{\alpha}g\right\rVert_{\sigma}^2 + \check{\delta} \left( \sum_{1\leq\left\lvert\gamma\right\rvert\leq 2} \left\lVert\partial^{\gamma }g\right\rVert_{\sigma}^2 +\sum_{i=1}^3 (1+t)^{-i} \left\lVert\nabla_x^{3-i}\mathbf{v}\right\rVert_{L^2}^2 \right)\right],\label{high-der-ori-2-ss} \\ &\partial_t \int_{\Omega} \nabla_x \psi \cdot\nabla_x^2 \phi dx+ \tilde{c}\left\lVert \nabla_x^2 \phi\right\rVert_{L^2}^2 \notag\\ &\leq C \left( \left\lVert\nabla_x^2{\mathbf{v}}^{\ast}\right\rVert_{L^2}^2 + \sum_{\left\lvert\alpha\right\rvert=2}\left\lVert\partial^{\alpha}g\right\rVert_{\sigma}^2 \right) +C\bar{\delta}(1+t)^{-\frac{5}{2}}+ C\check{\delta} \left( \sum_{\left\lvert\gamma\right\rvert=1} \left\lVert\partial^{\gamma }g\right\rVert_{\sigma}^2 +\sum_{i=1}^2 (1+t)^{-i} \left\lVert\nabla_x^{2-i}\mathbf{v}\right\rVert_{L^2}^2 \right),\label{high-der-ori-3} \\ &\partial_t \int_{\Omega} \nabla_x^2 \psi \cdot\nabla_x^3 \phi dx+ \tilde{c} \left\lVert \nabla_x^3 \phi\right\rVert_{L^2}^2\notag \\ &\leq C \left( \left\lVert\nabla_x^3{\mathbf{v}}^{\ast}\right\rVert_{L^2}^2 + \sum_{\left\lvert\alpha\right\rvert=3}\left\lVert\partial^{\alpha}g\right\rVert_{\sigma}^2 \right) +C\bar{\delta}(1+t)^{-\frac{5}{2}}+ C\check{\delta} \left( \sum_{1\leq\left\lvert\gamma\right\rvert\leq2} \left\lVert\partial^{\gamma }g\right\rVert_{\sigma}^2 +\sum_{i=1}^3 (1+t)^{-i} \left\lVert\nabla_x^{3-i}\mathbf{v}\right\rVert_{L^2}^2 \right).\label{high-der-ori-4} \end{align}\] {#eq: sublabel=eq:high-der-ori-1,eq:high-der-ori-1-s,eq:high-der-ori-2,eq:high-der-ori-2-s,eq:high-der-ori-2-ss,eq:high-der-ori-3,eq:high-der-ori-4}
Proof. Recall \((\rho,u,\theta)=(\tilde{\rho},\tilde{u},\tilde{\theta})+(\phi,\psi,\zeta)\) and \(G=\bar{G}_0+\sqrt{\mu}g\), then we have \[\begin{align} \notag &\sum_{i=0}^4 \left\lvert\mathcal{S}_i\right\rvert\le C\bar{\delta}^{\frac{1}{2}}\left( \tilde{D}_{-1}+\tilde{D}_{-\frac{1}{2}}\left\lvert\mathbf{v}\right\rvert\right)+C\left\lvert\mathbf{v}\right\rvert\left\lvert\nabla_x \mathbf{v}\right\rvert+C\left\lvert\nabla_x {\mathbf{v}}^{\ast}\right\rvert+C\sum_{\left\lvert\alpha\right\rvert=1}\left(1+\left\lvert\mathbf{v}\right\rvert\right)\left\lvert\partial^{\alpha}g\right\rvert_{\sigma}. \end{align}\] Then employing the a priori assumptions 69 , one has \[\begin{align} &\sum_{i=0}^4 \left\lVert\nabla_x \mathcal{S}_i\right\rVert_{L^2}^2 \leq C\bar{\delta}(1+t)^{-\frac{5}{2}}+C \left( \left\lVert\nabla_x^2{\mathbf{v}}^{\ast}\right\rVert_{L^2}^2 + \sum_{\left\lvert\alpha\right\rvert=2}\left\lVert\partial^{\alpha}g\right\rVert_{\sigma}^2 \right)\notag \\ &\qquad\qquad+C\check{\delta} \left(\left\lVert\nabla_x^2 \phi\right\rVert_{L^2}^2+ \sum_{\left\lvert\gamma\right\rvert=1} \left\lVert\partial^{\gamma }g\right\rVert_{\sigma}^2 +\sum_{i=1}^2 (1+t)^{-i} \left\lVert\nabla_x^{2-i}\mathbf{v}\right\rVert_{L^2}^2 \right) ,\tag{203} \\ &\sum_{i=0}^4 \left\lVert\nabla_x^2 \mathcal{S}_i\right\rVert_{L^2}^2 \leq C\bar{\delta}(1+t)^{-\frac{5}{2}} +C \left( \left\lVert\nabla_x^3 {\mathbf{v}}^{\ast}\right\rVert_{L^2}^2 + \sum_{\left\lvert\alpha\right\rvert=3}\left\lVert\partial^{\alpha}g\right\rVert_{\sigma}^2 \right) \notag\\ &\qquad\qquad+C\check{\delta} \left(\left\lVert\nabla_x^3 \phi\right\rVert_{L^2}^2+ \sum_{1\leq\left\lvert\gamma\right\rvert\leq2} \left\lVert\partial^{\gamma }g\right\rVert_{\sigma}^2 +\sum_{i=1}^3 (1+t)^{-i} \left\lVert\nabla_x^{3-i}\mathbf{v}\right\rVert_{L^2}^2 \right) . \tag{204} \end{align}\] By system 202 , 203 and 204 , one has \[\begin{align} &\left\lVert\partial_t \nabla_x \mathbf{v}\right\rVert_{L^2}^2 \leq C\left( \sum_{i=0}^4\left\lVert\nabla_x \mathcal{S}_i\right\rVert_{L^2}^2 + \left\lVert\nabla_x\left( \frac{\tilde{\theta}}{\tilde{\rho}} \nabla_x \phi \right)\right\rVert_{L^2}^2 \right) \notag\\ \leq& C\bar{\delta}(1+t)^{-\frac{5}{2}}+C \left( \left\lVert\nabla_x^2\mathbf{v}\right\rVert_{L^2}^2 + \sum_{\left\lvert\alpha\right\rvert=2}\left\lVert\partial^{\alpha}g\right\rVert_{\sigma}^2 \right)+C\check{\delta} \left( \sum_{\left\lvert\gamma\right\rvert=1} \left\lVert\partial^{\gamma }g\right\rVert_{\sigma}^2 +\sum_{i=1}^2 (1+t)^{-i} \left\lVert\nabla_x^{2-i}\mathbf{v}\right\rVert_{L^2}^2 \right), \\ &\left\lVert\partial_t \nabla_x^2 \mathbf{v}\right\rVert_{L^2}^2 \leq C\left( \sum_{i=1}^4\left\lVert\nabla_x^2 \mathcal{S}_i\right\rVert_{L^2}^2 + \left\lVert\nabla_x^2\left( \frac{\tilde{\theta}}{\tilde{\rho}} \nabla_x \phi \right)\right\rVert_{L^2}^2 \right)\notag\\ \leq& C\bar{\delta}(1+t)^{-\frac{5}{2}} +C \left( \left\lVert\nabla_x^3 \mathbf{v}\right\rVert_{L^2}^2 + \sum_{\left\lvert\alpha\right\rvert=3}\left\lVert\partial^{\alpha}g\right\rVert_{\sigma}^2 \right) +C\check{\delta} \left( \sum_{1\leq\left\lvert\gamma\right\rvert\leq2} \left\lVert\partial^{\gamma }g\right\rVert_{\sigma}^2 +\sum_{i=1}^3 (1+t)^{-i} \left\lVert\nabla_x^{3-i}\mathbf{v}\right\rVert_{L^2}^2 \right) . \end{align}\] Thus, we have proven ?? and ?? . Using the same method, we can prove ?? , ?? and ?? .
Next, we will provide the proofs of ?? and ?? . Applying \(\partial_{i}\) to 202 \(_2\) and \(\partial_{ij}\) to 202 \(_2\), \(i,j=1,2,3\), respectively, we can obtain \[\begin{align} &\partial_t \partial_{i} \psi_l+\frac{2}{3} \frac{\tilde{\theta}}{\tilde{\rho}} \partial_{il} \phi=\partial_{x_i}\mathcal{S}_{l}- \frac{2}{3}\partial_{i}\left( \frac{\tilde{\theta}}{\tilde{\rho}} \right) \partial_{j} \phi, \tag{205} \\ &\partial_t \partial_{ij} \psi_l+\frac{2}{3} \frac{\tilde{\theta}}{\tilde{\rho}} \partial_{ijl}\phi=\partial_{ij}\mathcal{S}_{l}- \frac{2}{3}\partial_{ij}\left( \frac{\tilde{\theta}}{\tilde{\rho}} \partial_{j} \phi\right)+ \frac{2}{3} \frac{\tilde{\theta}}{\tilde{\rho}} \partial_{ijl}\phi \tag{206}, \end{align}\] where \(l=1,2,3\). Multiplying 205 and 206 by \(\partial_{il}\phi\) and \(\partial_{ijl}\phi\) respectively, then integrating with respect to \(\Omega\), and by taking note of \[\begin{align} &\int_{\Omega} \partial_t \partial_{i} \psi_l \partial_{il} \phi dx = \partial_t\int_{\Omega} \partial_{i} \psi_l \partial_{il} \phi dx + \int_{\Omega} \partial_{i l}\psi_l \partial_t\partial_{x_i}\phi dx =\partial_t\int_{\Omega} \partial_{i} \psi_l \partial_{il} \phi dx + \int_{\Omega} \partial_{i l}\psi_l \partial_{i}\mathcal{S}_{0} dx ,\\ &\int_{\Omega} \partial_t \partial_{ij} \psi_l \partial_{ijl}\phi dx = \partial_t\int_{\Omega} \partial_{ij} \psi_l \partial_{ijl}\phi dx + \int_{\Omega} \partial_{i j l}\psi_l \partial_t\partial_{ij}\phi dx =\partial_t\int_{\Omega} \partial_{ij} \psi_l \partial_{ijl}\phi dx + \int_{\Omega} \partial_{i j l}\psi_l \partial_{i j}\mathcal{S}_{0} dx, \end{align}\] we obtain \[\begin{align} & \partial_t\int_{\Omega} \nabla_x \psi \cdot \nabla_x^2 \phi dx + \tilde{c}\left\lVert\nabla_x^2 \phi\right\rVert_{L^2}^2 \le C \sum_{i=0}^3 \left\lVert\nabla_x \mathcal{S}_i\right\rVert_{L^2}^2 + C\left\lVert\partial_{x_1} \left(\frac{\tilde{\theta}}{\tilde{\rho}}\right) \nabla_x \phi\right\rVert_{L^2}^2, \tag{207} \\ & \partial_t\int_{\Omega} \nabla_x^2 \psi \cdot \nabla_x^3 \phi dx + \tilde{c}\left\lVert\nabla_x^3 \phi\right\rVert_{L^2}^2 \le C\sum_{i=0}^3 \left\lVert\nabla_x^2 \mathcal{S}_i\right\rVert_{L^2}^2 + C\bar{\delta}\left(\left\lVert\tilde{D}_{-1}\nabla_x\mathbf{v}\right\rVert_{L^2}^2+\left\lVert\tilde{D}_{-\frac{1}{2}}\nabla_x^2\mathbf{v}\right\rVert_{L^2}^2\right). \tag{208} \end{align}\] Combining 207 and 203 , we can obtain ?? . And combining with 208 and 204 , we have ?? . Then we have completed the proof of Lemma 26. ◻
Applying \(\partial_{ij}\) to system 118 , one has \[\begin{align} \label{perturbation-2} \begin{cases} \partial_t\partial_{ij}\phi+\tilde{\rho}\operatorname{div}_x \partial_{ij} \psi=\partial_{ij}\mathcal{S}_{\phi}^{f}+\tilde{\mathcal{S}}_0, \\ \partial_t\partial_{ij}\psi+\frac{2\tilde{\theta}}{3\tilde{\rho}} \nabla_x\partial_{ij} \phi + \frac{2}{3} \nabla_x \partial_{ij}\zeta-\frac{\mu(\tilde{\theta})}{\tilde{\rho}}\partial_{ij}\left(\Delta_x \psi+\frac{1}{3} \nabla_x \operatorname{div}_x \psi\right)=\partial_{ij}\left(\mathcal{S}_{\psi}^{f1}+\mathcal{S}_{\psi}^{f2}+\mathcal{S}_{\psi}^{m}\right)+\tilde{\mathcal{S}},\\ \partial_t\partial_{ij}\zeta+\frac{2}{3} \tilde{\theta}\operatorname{div}_x \partial_{ij}\psi-\frac{\kappa(\tilde{\theta})}{\tilde{\rho}}\Delta_x \partial_{ij} \zeta=\partial_{ij}\left(\mathcal{S}_{\zeta}^{f1}+\mathcal{S}_{\zeta}^{f2}+\mathcal{S}_{\zeta}^{m}\right)+\tilde{\mathcal{S}}_4, \end{cases} \end{align}\tag{209}\] where \[\begin{align} &\tilde{\mathcal{S}}_0=\tilde{\rho}\operatorname{div}_x \partial_{ij} \psi-\partial_{ij} \left( \tilde{\rho}\operatorname{div}_x \psi \right), \tag{210} \\ & \tilde{\mathcal{S}}=\frac{2\tilde{\theta}}{3\tilde{\rho}} \nabla_x\partial_{ij} \phi -\frac{2}{3}\partial_{ij}\left(\frac{\tilde{\theta}}{\tilde{\rho}} \nabla_x\phi\right) +\partial_{ij}\left[\frac{\mu(\tilde{\theta})}{\tilde{\rho}}\left(\Delta_x \psi+\frac{1}{3} \nabla_x \operatorname{div}_x \psi\right)\right]-\frac{\mu(\tilde{\theta})}{\tilde{\rho}}\partial_{i j}\left(\Delta_x \psi+\frac{1}{3} \nabla_x \operatorname{div}_x \psi\right), \tag{211} \\ &\tilde{\mathcal{S}}_4=\frac{2}{3} \tilde{\theta}\operatorname{div}_x \partial_{ij}\psi-\frac{2}{3}\partial_{ij}\left( \tilde{\theta}\operatorname{div}_x \psi \right)-\frac{\kappa(\tilde{\theta})}{\tilde{\rho}}\Delta_x \partial_{ij} \zeta+\partial_{ij}\left(\frac{\kappa(\tilde{\theta})}{\tilde{\rho}}\Delta_x \zeta\right).\tag{212} \end{align}\]
Lemma 27. Under the same assumptions of Proposition 10, it holds that \[\begin{align} \frac{d}{dt}\left\lVert\nabla_x^2 \mathbf{v}\right\rVert_{L^2}^2 + \left\lVert\nabla_x^3 {\mathbf{v}}^{\ast}\right\rVert_{L^2}^2 \leq &C_\eta \sum_{\left\lvert\alpha\right\rvert=3} \left\lVert\partial^{\alpha}g\right\rVert_{\sigma}^2+C\check{\delta} \left[(1+t)^{-3}\left\lVert\mathbf{v}\right\rVert_{L^2}^2+(1+t)^{-1}\left\lVert\nabla_x \mathbf{v}\right\rVert_{L^2}^2+\left\lVert\nabla_x^2 \mathbf{v}\right\rVert_{L^2}^2 \right] \\ &+C\check{\delta}\left( (1+t)^{-2} \left\lVert g\right\rVert_{\sigma}^2+\sum_{1\leq \left\lvert\gamma\right\rvert\leq2} \left\lVert\partial^{\gamma}g\right\rVert_{\sigma}^2 \right)+C\bar{\delta}(1+t)^{-\frac{5}{2}}+\eta\left\lVert\nabla_x^3\phi\right\rVert_{L^2}. \end{align}\]
Proof. By the definition of \(\tilde{\mathcal{S}}_0\), \(\tilde{\mathcal{S}}=(\tilde{\mathcal{S}}_1,\tilde{\mathcal{S}}_2,\tilde{\mathcal{S}}_3)^{t}\) and \(\tilde{\mathcal{S}}_4\) as in 210 , 211 and 212 , \(\mathcal{S}_{\phi}^f,\mathcal{S}_{\psi}^{f1},\mathcal{S}_{\psi}^{f2},\mathcal{S}_{\zeta}^{f1}\), \(\mathcal{S}_{\zeta}^{f2}\) 119 121 , 123 and 124 , one has \[\begin{align} &\sum_{i=0}^4 \left\lvert\tilde{\mathcal{S}}_i\right\rvert\le C \bar{\delta}^{\frac{1}{2}} \left[\tilde{D}_{-1}\left\lvert\nabla_x \mathbf{v}\right\rvert + \tilde{D}_{-\frac{1}{2}}\left\lvert\nabla_x^2 \mathbf{v}\right\rvert+\tilde{D}_{-\frac{1}{2}}\left\lvert\nabla_x^3 \mathbf{v}\right\rvert\right], \tag{213}\\ &\left\lvert\nabla_x^2\mathcal{S}_{\phi}^f\right\rvert+\left\lvert\nabla_x^2\mathcal{S}_{\psi}^{f1}\right\rvert+\left\lvert\nabla_x^2 \mathcal{S}_{\zeta}^{f1}\right\rvert\le C\bigg[ \left\lvert\mathbf{v}\right\rvert\left\lvert\nabla_x^3 \mathbf{v}\right\rvert+\left\lvert\nabla_x \mathbf{v}\right\rvert\left\lvert\nabla_x^2 \mathbf{v}\right\rvert+\left\lvert\nabla_x \mathbf{v}\right\rvert^3+\left\lvert\mathbf{v}\right\rvert\left\lvert\nabla_x \mathbf{v}\right\rvert\left\lvert\nabla_x^2 \mathbf{v}\right\rvert \notag\\ &\qquad\qquad\quad\qquad\qquad\qquad\qquad+ \bar{\delta}^{\frac{1}{2}}\left(\tilde{D}_{-1}\left\lvert\nabla_x \mathbf{v}\right\rvert + \tilde{D}_{-\frac{1}{2}}\left\lvert\nabla_x^2 \mathbf{v}\right\rvert+\tilde{D}_{-\frac{1}{2}}\left\lvert\nabla_x^3 \mathbf{v}\right\rvert+\tilde{D}_{-\frac{3}{2}}\left\lvert \mathbf{v}\right\rvert\right) +\bar{\delta}\tilde{D}_{-\frac{5}{2}}\bigg],\tag{214}\\ &\left\lvert\nabla_x\mathcal{S}_{\psi}^{f2}\right\rvert+\left\lvert\nabla_x \mathcal{S}_{\zeta}^{f2}\right\rvert\le C\bigg[ \left\lvert\mathbf{v}\right\rvert\left\lvert\nabla_x^3 \mathbf{v}\right\rvert+\left\lvert\nabla_x \mathbf{v}\right\rvert\left\lvert\nabla_x^2 \mathbf{v}\right\rvert+\left\lvert\nabla_x \mathbf{v}\right\rvert^3+\left\lvert\mathbf{v}\right\rvert\left\lvert\nabla_x \mathbf{v}\right\rvert\left\lvert\nabla_x^2 \mathbf{v}\right\rvert \notag\\ &\qquad\qquad\qquad\qquad\quad\;+ \bar{\delta}^{\frac{1}{2}}\left(\tilde{D}_{-1}\left\lvert\nabla_x \mathbf{v}\right\rvert + \tilde{D}_{-\frac{1}{2}}\left\lvert\nabla_x^2 \mathbf{v}\right\rvert+\tilde{D}_{-\frac{3}{2}}\left\lvert \mathbf{v}\right\rvert\right) +\bar{\delta}\tilde{D}_{-2}\bigg].\tag{215} \end{align}\] For the non-fluid part, we take \(\partial_l\left[\frac{1}{\rho}\partial_i \int_{\mathbb{R}^3} \xi_i \xi_j L_M^{-1} \Pi d\xi\right]\) in \(\partial_l\mathcal{S}_{\psi}^{m}\) 122 as an example and perform the calculation. It holds \[\begin{align} & \partial_{l}\left[ \frac{1}{\rho}\partial_i \int_{\mathbb{R}^3} \xi_i \xi_j L_M^{-1} \Pi d\xi \right]=\frac{R}{\rho} \partial_{il} \int_{\mathbb{R}^3} \theta B_{ij} \left( \frac{\xi-u}{\sqrt{R \theta}} \right) \frac{1}{M} \left[\partial_t G+P_1 \left( \xi \cdot \nabla_x G \right) -Q(G,G)\right]d\xi\notag\\ &\qquad\qquad\qquad+\partial_l\left(\frac{R}{\rho}\right) \partial_{i} \int_{\mathbb{R}^3} \theta B_{ij} \left( \frac{\xi-u}{\sqrt{R \theta}} \right) \frac{1}{M} \left[\partial_t G+P_1 \left( \xi \cdot \nabla_x G \right) -Q(G,G)\right]d\xi\label{high-der-s-4}. \end{align}\tag{216}\] Substituting \((\rho,u,\theta)=(\tilde{\rho},\tilde{u},\tilde{\theta})+(\phi,\psi,\zeta)\) and \(G=\bar{G}_0+\sqrt{\mu}g\) into 216 , one has \[\begin{align} &\left\lvert \partial_{l}\left[ \frac{1}{\rho}\partial_i \int_{\mathbb{R}^3} \xi_i \xi_j L_M^{-1} \Pi d\xi \right] \right\rvert\le C\left(\sum_{\left\lvert\alpha\right\rvert=3} \left\lvert\partial^{\alpha}g\right\rvert_{\sigma} + \sum_{\left\lvert\gamma\right\rvert=1}\left\lvert\partial^{\gamma} g\right\rvert_{\sigma}\left\lvert\partial^{\gamma} g\right\rvert_{2}+\sum_{\left\lvert\beta\right\rvert=2}\left\lvert g\right\rvert_{\sigma} \left\lvert\partial^{\beta}g\right\rvert_{\sigma}\right)\\ &\qquad+C\left(\bar{\delta}^{\frac{1}{2}} \tilde{D}_{-1}+\left\lvert\nabla_x \mathbf{v}\right\rvert^2 + \left\lvert\nabla_x^2 \mathbf{v}\right\rvert + \bar{\delta}^{\frac{1}{2}} \tilde{D}_{-\frac{1}{2}} \left\lvert\nabla_x \mathbf{v}\right\rvert \right) \sum_{\left\lvert\gamma\right\rvert=1} \left(\left\lvert\partial^\gamma g\right\rvert_{\sigma}+\bar{\delta}^{\frac{1}{2}} \tilde{D}_{-\frac{1}{2}}\left\lvert g\right\rvert_{\sigma}+ \left\lvert g\right\rvert_{\sigma}\left\lvert g\right\rvert_{2} + \bar{\delta}^{\frac{1}{2}} \tilde{D}_{-1}\right) \\ &\qquad+C\left(\bar{\delta}^{\frac{1}{2}}\tilde{D}_{-\frac{1}{2}}+\left\lvert\nabla_x \mathbf{v}\right\rvert\right) \sum_{\left\lvert\gamma\right\rvert=1}\sum_{\left\lvert\beta\right\rvert=2} \left(\left\lvert\partial^\beta g\right\rvert_{\sigma}+\bar{\delta}^{\frac{1}{2}} \tilde{D}_{-\frac{1}{2}}\left\lvert\partial^\gamma g\right\rvert_{\sigma} + \bar{\delta}^{\frac{1}{2}}\tilde{D}_{-1}\left\lvert g\right\rvert_{\sigma}+ \left\lvert\partial^{\gamma} g\right\rvert_{\sigma}\left\lvert g\right\rvert_{\sigma} \right)\\ &\qquad + C \bar{\delta}D_{-\frac{1}{2}}\left(\left\lvert\nabla_x^3 \mathbf{v}^{\ast}\right\rvert+\left\lvert\partial_t \nabla_x^2 \mathbf{v}^{\ast}\right\rvert \right)+C\bar{\delta}D_{-1}\left\lvert\partial_t\nabla_x\mathbf{v}^{\ast}\right\rvert+C\bar{\delta}D_{-\frac{3}{2}}\left\lvert\partial_t \mathbf{v}^{\ast}\right\rvert. \end{align}\] The calculation of the other terms in \(\nabla_x\mathcal{S}_{\psi}^{m},\nabla_x\mathcal{S}_{\zeta}^{m}\) 122 and 125 is similar to the above calculation, and thus we can obtain \[\begin{align} &\left\lvert\nabla_x\mathcal{S}_{\psi}^{m}\right\rvert+\left\lvert\nabla_x\mathcal{S}_{\zeta}^{m}\right\rvert\le C\left(\sum_{\left\lvert\alpha\right\rvert=3} \left\lvert\partial^{\alpha}g\right\rvert_{\sigma} + \sum_{\left\lvert\gamma\right\rvert=1}\left\lvert\partial^{\gamma} g\right\rvert_{\sigma}\left\lvert\partial^{\gamma} g\right\rvert_{2}+\sum_{\left\lvert\beta\right\rvert=2}\left\lvert g\right\rvert_{\sigma} \left\lvert\partial^{\beta}g\right\rvert_{\sigma}\right)\notag\\ &\qquad+C\left(\bar{\delta}^{\frac{1}{2}} \tilde{D}_{-1}+\left\lvert\nabla_x \mathbf{v}\right\rvert^2 + \left\lvert\nabla_x^2 \mathbf{v}\right\rvert + \bar{\delta}^{\frac{1}{2}} \tilde{D}_{-\frac{1}{2}} \left\lvert\nabla_x \mathbf{v}\right\rvert \right) \sum_{\left\lvert\gamma\right\rvert=1} \left(\left\lvert\partial^\gamma g\right\rvert_{\sigma}+\bar{\delta}^{\frac{1}{2}} \tilde{D}_{-\frac{1}{2}}\left\lvert g\right\rvert_{\sigma}+ \left\lvert g\right\rvert_{\sigma}\left\lvert g\right\rvert_{2} + \bar{\delta}^{\frac{1}{2}} \tilde{D}_{-1}\right)\notag \\ &\qquad+C\left(\bar{\delta}^{\frac{1}{2}}\tilde{D}_{-\frac{1}{2}}+\left\lvert\nabla_x \mathbf{v}\right\rvert\right) \sum_{\left\lvert\gamma\right\rvert=1}\sum_{\left\lvert\beta\right\rvert=2} \left(\left\lvert\partial^\beta g\right\rvert_{\sigma}+\bar{\delta}^{\frac{1}{2}} \tilde{D}_{-\frac{1}{2}}\left\lvert\partial^\gamma g\right\rvert_{\sigma} + \bar{\delta}^{\frac{1}{2}}\tilde{D}_{-1}\left\lvert g\right\rvert_{\sigma}+ \left\lvert\partial^{\gamma} g\right\rvert_{\sigma}\left\lvert g\right\rvert_{\sigma} \right) \notag\\ &\qquad + C \bar{\delta}D_{-\frac{1}{2}}\left(\left\lvert\nabla_x^3 \mathbf{v}^{\ast}\right\rvert+\left\lvert\partial_t \nabla_x^2 \mathbf{v}^{\ast}\right\rvert \right)+C\bar{\delta}D_{-1}\left\lvert\partial_t\nabla_x\mathbf{v}^{\ast}\right\rvert+C\bar{\delta}D_{-\frac{3}{2}}\left\lvert\partial_t \mathbf{v}^{\ast}\right\rvert.\label{high-der-s-5} \end{align}\tag{217}\] Multiply 209 \(_1\), 209 \(_2\), and 209 \(_3\) by \(\frac{5\tilde{\theta}}{3\tilde{\rho}^2}\partial_{ij}\phi\), \(\partial_{ij} \psi\), and \(\frac{5}{2\tilde{\theta}}\partial_{ij} \zeta\), respectively, and then integrate them in region \(\Omega\). By the a priori assumption 69 , applying integration by parts and using Lemma 20, Lemma 26, 213 215 , 217 , we can obtain the following \[\begin{align} &\frac{d}{dt}\left\lVert\nabla_x^2 \mathbf{v}\right\rVert_{L^2}^2 + \tilde{c}\left\lVert\nabla_x^3 {\mathbf{v}}^{\ast}\right\rVert_{L^2}^2\\ \leq& C_\eta\sum_{i=0}^4\left\lVert\tilde{\mathcal{S}}_i\right\rVert_{L^2}^2+\bar{\delta}^{-\frac{1}{2}}\left(\left\lVert\nabla_x^2 \mathcal{S}_{\phi}^f\right\rVert_{L^2}^2+\left\lVert\nabla_x^2 \mathcal{S}_{\psi}^{f1}\right\rVert_{L^2}^2+\left\lVert\nabla_x^2 \mathcal{S}_{\zeta}^{f1}\right\rVert_{L^2}^2 \right) +\bar{\delta}^{\frac{1}{2}}\left\lVert\nabla_x^2\mathbf{v}\right\rVert_{L^2}^2+\eta\left\lVert\nabla_x^3 \mathbf{v}\right\rVert_{L^2}^2\\ &+C_\eta\left( \left\lVert\nabla_x \mathcal{S}_{\psi}^{f2}\right\rVert_{L^2}^2+\left\lVert\nabla_x \mathcal{S}_{\zeta}^{f2}\right\rVert_{L^2}^2+\left\lVert\nabla_x \mathcal{S}_{\psi}^{m}\right\rVert_{L^2}^2+\left\lVert\nabla_x \mathcal{S}_{\zeta}^{m}\right\rVert_{L^2}^2 \right) \\ \leq &C_\eta \sum_{\left\lvert\alpha\right\rvert=3} \left\lVert\partial^{\alpha}g\right\rVert_{\sigma}^2+C\check{\delta} \left[(1+t)^{-3}\left\lVert\mathbf{v}\right\rVert_{L^2}^2+(1+t)^{-1}\left\lVert\nabla_x \mathbf{v}\right\rVert_{L^2}^2 \right] +\bar{\delta}^{\frac{1}{2}}\left\lVert\nabla_x^2\mathbf{v}\right\rVert_{L^2}^2+\eta\left\lVert\nabla_x^3 \mathbf{v}\right\rVert_{L^2}^2\\ &+C\check{\delta}\left( (1+t)^{-2} \left\lVert g\right\rVert_{\sigma}^2+\sum_{1\leq \left\lvert\gamma\right\rvert\leq2} \left\lVert\partial^{\gamma}g\right\rVert_{\sigma}^2 \right)+C\bar{\delta}(1+t)^{-\frac{5}{2}}. \end{align}\] Then we have completed the proof of Lemma 27. ◻
Combining Lemma 26 and Lemma 27, we have completed the proof of Theorem 25.
Recall the definition of \(\mathbf{v}\) 55 and dissipation energy functionals \(\mathcal{D}_{i,\omega}\) and \(\mathcal{D}_i\) 62 68 . Then we have the following result.
Lemma 28. Under the same assumptions of Proposition 10, let \(|\alpha|+|\beta|\leq 3\), then for \(|\beta|\geq 1\) and \(\left\lvert\beta'\right\rvert=1\), one has \[\begin{align} \frac{d}{dt}\left\lVert\omega(\beta)\partial^{\alpha}_{\beta}g\right\rVert^2_2+&\left\lVert\omega(\beta)\partial^{\alpha}_{\beta}g\right\rVert_{\sigma}^2\leq C\bar{\delta}(1+t)^{-\frac{3}{2}-\left\lvert\alpha\right\rvert}+ C\check{\delta}\left[\mathcal{D}_{2,\omega}(t)+(1+t)^{-1}\mathcal{D}_{1,\omega}(t)\right]+\eta\sum_{\left\lvert\beta_1\right\rvert=\left\lvert\beta\right\rvert}\left\lVert\partial_{\beta_1}^{\alpha}g\right\rVert_{\sigma,\omega}^2\notag\\ &\qquad+C_\eta \left(\sum_{|\beta_1|<|\beta|}\left\lVert\partial^\alpha_{\beta_1} g\right\rVert_{\sigma,\omega}^2+\left\lVert\partial^\alpha_{\beta-\beta'}\nabla_xg\right\rVert^2_{\sigma,\omega}+\left\lVert\partial^{\alpha}\nabla_xg\right\rVert_{\sigma,\omega}^2+\left\lVert\nabla_x^{\left\lvert\alpha\right\rvert+1}\mathbf{v}\right\rVert_{L^2}^2\right),\label{54630} \end{align}\qquad{(6)}\] and for \(\beta=0\), \(|\alpha|\leq 2\), one has \[\begin{align} &\frac{d}{dt}\left\lVert\partial^{\alpha}g\right\rVert^2_{2,\omega}+\left\lVert\partial^{\alpha}g\right\rVert_{\sigma,\omega}^2\le C \bar{\delta}(1+t)^{-\frac{3}{2}-\left\lvert\alpha\right\rvert}+ C\check{\delta}\left[\mathcal{D}_{2,\omega}(t)+(1+t)^{-1}\mathcal{D}_{1,\omega}(t)\right]\notag\\ &\qquad\qquad\qquad\qquad\qquad\qquad+C_\eta\sum_{ \left\lvert\gamma_1\right\rvert=\left\lvert\alpha\right\rvert}\left\lVert\partial^{\gamma_1} g\right\rVert_{\sigma}^2+C_\eta\left\lVert\partial^{\alpha}\nabla_x g\right\rVert_{\sigma,\omega}^2+C_\eta\left\lVert\nabla_x^{\left\lvert\alpha\right\rvert+1}\mathbf{v}\right\rVert_{L^2}^2,\label{54630-1}\\ &\frac{d}{dt}\left\lVert\partial^{\alpha}g\right\rVert^2_2+\left\lVert\partial^{\alpha}g\right\rVert_{\sigma}^2 \leq C\bar{\delta}(1+t)^{-\frac{3}{2}-\left\lvert\alpha\right\rvert}+ C\check{\delta}\left[\mathcal{D}_{2}(t)+(1+t)^{-1}\mathcal{D}_{1}(t)\right]+C_\eta\left(\left\lVert\partial^{\alpha}\nabla_xg\right\rVert_{\sigma}^2+\left\lVert\nabla_x^{\left\lvert\alpha\right\rvert+1} \mathbf{v}\right\rVert_{L^2}^2\right).\label{54630-2} \end{align}\] {#eq: sublabel=eq:54630-1,eq:54630-2} Moreover, for \(\left\lvert\alpha\right\rvert+\left\lvert\beta\right\rvert\le 3\), \(\left\lvert\alpha\right\rvert\geq1,\;\left\lvert\beta\right\rvert\geq1\), \(\left\lvert\beta'\right\rvert=1\), one has \[\begin{align} \label{54630-4} \frac{d}{dt}\left\lVert\partial^{\alpha}_{\beta}g\right\rVert^2_{2,\omega}+&\left\lVert\partial^{\alpha}_{\beta}g\right\rVert_{\sigma,\omega}^2 \leq C\bar{\delta}(1+t)^{-\frac{3}{2}-\left\lvert\alpha\right\rvert}+C\check{\delta} \sum_{i=1}^{3}(1+t)^{-3+i}\mathcal{D}_{i,\omega}(t)+\eta\sum_{\left\lvert\beta_1\right\rvert=\left\lvert\beta\right\rvert}\left\lVert\partial_{\beta_1}^{\alpha}g\right\rVert_{\sigma,\omega}^2\nonumber\\ &\qquad+C_\eta\left(\sum_{|\beta_1|<|\beta|}\left\lVert\partial^\alpha_{\beta_1} g\right\rVert_{\sigma,\omega}^2+\left\lVert\partial^\alpha_{\beta-\beta'}\nabla_xg\right\rVert^2_{\sigma,\omega}+\left\lVert\partial^{\alpha}\nabla_xg\right\rVert_{\sigma,\omega}^2+\left\lVert\nabla_x^{\left\lvert\alpha\right\rvert+1}\mathbf{v}\right\rVert_{L^2}^2\right), \end{align}\qquad{(7)}\] and for \(1\leq \left\lvert\alpha\right\rvert\leq 2,\;\left\lvert\beta\right\rvert=0\), one has \[\begin{align} &\frac{d}{dt}\left\lVert\partial^{\alpha}g\right\rVert^{2}_{2,\omega}+\left\lVert\partial^{\alpha}g\right\rVert_{\sigma,\omega}^2 \leq C\left[\bar{\delta}(1+t)^{-\frac{3}{2}-\left\lvert\alpha\right\rvert}+\check{\delta} \sum_{i=1}^{3}(1+t)^{-3+i}\mathcal{D}_{i,\omega}(t) \right]\notag\\ &\qquad\qquad\qquad\qquad\qquad\qquad+C_\eta\left[\sum_{ \left\lvert\gamma_1\right\rvert=\left\lvert\alpha\right\rvert}\left\lVert\partial^{\gamma_1} g\right\rVert_{\sigma}^2+\left\lVert\partial^{\alpha}\nabla_x g\right\rVert_{\sigma,\omega}^2+\left\lVert\nabla_x^{\left\lvert\alpha\right\rvert+1}\mathbf{v}\right\rVert_{L_x^2}^2\right],\label{54630-5}\\ &\frac{d}{dt}\left\lVert\partial^{\alpha}g\right\rVert^2_2+\left\lVert\partial^{\alpha}g\right\rVert_{\sigma}^2 \leq C\bar{\delta}(1+t)^{-\frac{3}{2}-\left\lvert\alpha\right\rvert}+C\check{\delta} \sum_{i=1}^{3}(1+t)^{-3+i}\mathcal{D}_{i}(t)+C_\eta\left\lVert\partial^{\alpha}\nabla_xg\right\rVert_{\sigma}^2+C_\eta\left\lVert\nabla_x^{\left\lvert\alpha\right\rvert+1}\mathbf{v}\right\rVert_{L^2}^2,\label{54630-6} \end{align}\] {#eq: sublabel=eq:54630-5,eq:54630-6} where \(\bar{\delta}=\delta+\varepsilon_0\) and \(\check{\delta}=\chi+\bar{\delta}^{\frac{1}{2}}\).
Proof. We focus on proving ?? and ?? , and the estimates for the remaining parts in the Lemma 28 can be obtained by similar calculations. Applying \(\partial^{\alpha}_{\beta}\), \(|\alpha|+|\beta|\leq 3,\;\;\left\lvert\beta\right\rvert\geq 1\) to 126 , one has \[\begin{align} \label{mic-perturbation-ab} &\partial^{\alpha}_{\beta}\partial_t g + \xi \cdot \nabla_x\partial^{\alpha}_{\beta} g +\sum_{\left\lvert\beta'\right\rvert=1}\partial_{\beta'}\xi \cdot\nabla_x\partial_{\beta-\beta'}^{\alpha}g-\partial^{\alpha}_{\beta}\mathcal{L} g =\partial_{\beta}^{\alpha}\Bigg(\frac{P_0 \left( \xi \cdot \sqrt{\mu}\nabla_x g \right)}{\sqrt{\mu}}\Bigg)+\partial_{\beta}^{\alpha}(\mathcal{S}_{g1}+\mathcal{S}_{g2}+\mathcal{S}_{g3}), \end{align}\tag{218}\] where \[\begin{align} \partial^{\alpha}_{\beta}\mathcal{S}_{g1}:=&-\partial^{\alpha}_{\beta}\Bigg(\frac{P_1 \left( \xi_1 \partial_{x_1} \bar{G}_0 \right)}{\sqrt{\mu}}+\frac{\partial_t \bar{G}_0}{\sqrt{\mu}}\Bigg),\\ \partial^{\alpha}_{\beta}\mathcal{S}_{g2}:=&-\partial_{\beta}^{\alpha}\left\{\frac{1}{\sqrt{\mu}}P_1 \left[\xi_1 \left(\frac{\left\lvert\xi-u\right\rvert^2\partial_{x_1} (\tilde{\theta}-\bar{\theta})}{2 R \theta^2} +\frac{\left( \xi -u\right)\cdot \partial_{x_1} (\tilde{u}-\bar{u})}{R \theta} \right)M \right]\right\}\notag\\ & -\partial_{\beta}^{\alpha}\left\{\frac{1}{\sqrt{\mu}}P_1 \left[\xi \cdot \left( \frac{\left\lvert\xi-u\right\rvert^2\nabla_x \zeta}{2 R \theta^2} +\frac{\left( \xi -u\right)\cdot \nabla_x \psi}{R \theta} \right) M\right] \right\} := \partial^{\alpha}_{\beta}\mathcal{S}_{g21}+\partial^{\alpha}_{\beta}\mathcal{S}_{g22}, \\ \partial^{\alpha}_{\beta}\mathcal{S}_{g3}:=&\partial^{\alpha}_{\beta}\left\{\Gamma\left(g,\frac{M-\mu}{\sqrt{\mu}} \right)+\Gamma\left(\frac{M-\mu}{\sqrt{\mu}},g \right)+\Gamma\left(G,G\right)\right\}. \end{align}\] Since \(|\tilde{\theta}-\bar{\theta}|+\left\lvert\tilde{u}-\bar{u}\right\rvert\lesssim\left\lvert\Xi\right\rvert\), by Corollary 15 and Lemma 40, we obtain \[\begin{align} \label{20254664605-1} &\left\lvert\partial_{\beta}\mathcal{S}_{g1}\right\rvert_{2,\omega}+\left\lvert\partial_{\beta}\mathcal{S}_{g21}\right\rvert_{2,\omega}\leq C \bar{\delta}\left(\tilde{D}_{-1}+\tilde{D}_{-\frac{1}{2}}(\left\lvert\nabla_x\mathbf{v}\right\rvert+\left\lvert\partial_t\mathbf{v}^{\ast}\right\rvert) \right),\notag\\ &\left\lvert\partial_{\beta}\mathcal{S}_{g22}\right\rvert_{2,\omega}\leq C \left\lvert\mathbf{v}\right\rvert\left\lvert\nabla_x \mathbf{v}\right\rvert+C\left\lvert\nabla_x \mathbf{v}\right\rvert+ C\bar{\delta}^{\frac{1}{2}}\tilde{D}_{-\frac{1}{2}}\left\lvert\mathbf{v}\right\rvert. \end{align}\tag{219}\] Multiplying 218 by \(\omega \partial^{\alpha}_{\beta}g\) and then integrating the resulting equation over \(\Omega\times\mathbb{R}^3\), one has \[\begin{align} &\frac{d}{dt}\left\lVert\omega \partial^{\alpha}_{\beta}g\right\rVert_2^{2}-\iint_{\Omega\times\mathbb{R}^3}\omega^2\partial^{\alpha}_{\beta}g\partial^{\alpha}_{\beta}\mathcal{L}gdxd\xi+\sum_{\left\lvert\beta'\right\rvert=1}\iint_{\Omega\times\mathbb{R}^3}\omega^2\partial^{\alpha}_{\beta}g\partial_{\beta'}\xi \cdot\nabla_x\partial_{\beta-\beta'}^{\alpha}gdxd\xi \nonumber\\ =&\iint_{\Omega\times\mathbb{R}^3}\omega^2\Big(\partial_{\beta}^{\alpha}\mathcal{S}_{g1}+\partial_{\beta}^{\alpha}\mathcal{S}_{g2}+\partial_{\beta}^{\alpha}\mathcal{S}_{g3}\Big)\partial^{\alpha}_{\beta}g dxd\xi+\iint_{\Omega\times\mathbb{R}^3}\omega^2\partial^{\alpha}_{\beta}\Big\{\frac{P_0 \left( \xi \cdot \sqrt{\mu}\nabla_x g \right)}{\sqrt{\mu}}\Big\}\partial^{\alpha}_{\beta}g dxd\xi .\label{gab} \end{align}\tag{220}\] For the second term in the left-hand side of 220 , by Lemma 37, we have \[-\langle\partial^\alpha_\beta\mathcal{L}g,\omega^2(\beta)\partial^\alpha_\beta g\rangle\geq |\omega(\beta)\partial^\alpha_\beta g|_\sigma^2-\eta\sum_{|\beta_1|=|\beta|}|\omega(\beta_1)\partial^\alpha_{\beta_1} g|_\sigma^2 -C_\eta\sum_{|\beta_1|<|\beta|}|\omega(\beta_1)\partial^\alpha_{\beta_1} g|_\sigma^2.\] Next, we estimate the other terms in 220 . Note that terms involving \(\mathcal{S}_{g3}\) have been estimated in Lemma 41 and Lemma 42. For \(\alpha=0\), we have \[\begin{align} \iint_{\Omega\times\mathbb{R}^3}\omega^2\partial_{\beta}^{\alpha}\mathcal{S}_{g3}\partial^{\alpha}_{\beta}g dxd\xi\leq C\check{\delta}\left[\|\omega(\beta)\partial_\beta^\alpha g\|_{\sigma}^2+\mathcal{D}_{2,\omega}(t)+ (1+t)^{-1} \mathcal{D}_{1,\omega} \right]+C\bar{\delta}(1+t)^{-\frac{3}{2}}. \end{align}\] For \(\left\lvert\alpha\right\rvert\geq1\), by ?? , ?? , Lemma 41, and Lemma 42, we obtain \[\begin{align} \iint_{\Omega\times\mathbb{R}^3}\omega^2\partial_{\beta}^{\alpha}\mathcal{S}_{g3}\partial^{\alpha}_{\beta}g dxd\xi\leq &C\check{\delta}\Big[\|\omega(\beta)\partial_\beta^\alpha g\|_{\sigma}^2+\sum_{i=1}^3(1+t)^{-3+i}\mathcal{D}_{i,\omega}(t)\Big]+C\bar{\delta}(1+t)^{-\frac{3}{2}-\left\lvert\alpha\right\rvert}.\label{20254664605-4} \end{align}\tag{221}\] Next, we focus on the \(\mathcal{S}_{g22}\). For \(|\alpha|=0\), by 219 , Lemma 20, Lemma 26 and the a priori assumptions 69 , one has \[\begin{align} &\iint_{\Omega\times\mathbb{R}^3}\omega^2\partial_{\beta}^{\alpha}\mathcal{S}_{g22}\partial^{\alpha}_{\beta}g dxd\xi\leq C_\eta\left[\left\lVert\nabla_x \mathbf{v}\right\rVert_{L^2}^2+\check{\delta}(1+t)^{-1}\left\lVert\mathbf{v}\right\rVert_{L^2}^2 \right]+\eta\left\lVert\omega(\beta)\partial_{\beta}g\right\rVert_{\sigma}^2. \end{align}\] Then, for \(|\alpha|=1\), by Gagliardo-Nirenberg inequality, ?? , 219 , Lemma 26 and the a priori assumptions 69 , one has \[\begin{align} &\iint_{\Omega\times\mathbb{R}^3}\omega^2\partial_{\beta}^{\alpha}\mathcal{S}_{g22}\partial^{\alpha}_{\beta}g dxd\xi \notag\\ \leq&C_\eta\left\lVert\partial^{|\alpha|}\nabla_x\mathbf{v}\right\rVert_{L_x^2}^2+C_\eta\left\lVert\partial^{\left\lvert\alpha\right\rvert}\mathbf{v}\right\rVert_{L^4}^4+C\bar{\delta}\left\lVert\tilde{D}_{-\frac{1}{2}}\right\rVert_{L^{\infty}}^2\left\lVert\partial^{\left\lvert\alpha\right\rvert}\mathbf{v}\right\rVert_{L^2}^2+C_\eta\bar{\delta}\left\lVert\tilde{D}_{-1}\right\rVert_{L^{\infty}}^2\left\lVert\mathbf{v}\right\rVert_{L^2}^2+\eta\left\lVert\omega(\beta)\partial_{\beta}^{\alpha}g\right\rVert_{\sigma}^2\nonumber\\ \leq &\eta\left\lVert\omega(\beta)\partial^{\alpha}_{\beta}g\right\rVert_{\sigma}^2+C\check{\delta}\left[(1+t)^{-1}\left\lVert\nabla_x\mathbf{v}\right\rVert_{L^2}^2+(1+t)^{-2}\left\lVert\mathbf{v}\right\rVert_{L^2}^2\right]+C_\eta\left\lVert\nabla_x^{2}\mathbf{v}\right\rVert_{L^2}^2. \end{align}\] Similarly, for \(|\alpha|=2\), by Gagliardo-Nirenberg inequality, ?? , 219 , Lemma 26 and the a priori assumptions 69 , it holds that \[\begin{align} &\iint_{\Omega\times\mathbb{R}^3}\omega^2\partial_{\beta}^{\alpha}\mathcal{S}_{g22}\partial^{\alpha}_{\beta}g dxd\xi \\ \leq &\eta\left\lVert\omega(\beta)\partial^{\alpha}_{\beta}g\right\rVert_{\sigma}^2+C\check{\delta}\left[\left\lVert\nabla_x^2\mathbf{v}\right\rVert_{L^2}^2+(1+t)^{-1}\left\lVert\nabla_x\mathbf{v}\right\rVert_{L^2}^2+(1+t)^{-2}\left\lVert\mathbf{v}\right\rVert_{L^2}^2\right]+C_\eta\left\lVert\nabla_x^{3}\mathbf{v}\right\rVert_{L^2}^2.\notag \end{align}\] For \(\mathcal{S}_{g1}\) and \(\mathcal{S}_{g21}\), by 219 , we only need to treat \(\bar{\delta}\partial^{\alpha}\tilde{D}_{-1}\) and other terms can be estimated as \(\mathcal{S}_{g22}\). It then holds \[\begin{align} \bar{\delta}\int_{\mathbb{R}^3}\partial^{\alpha} \tilde{D}_{-1} \left\lvert\omega \partial_\beta^\alpha g\right\rvert_{\sigma} dx \lesssim C\bar{\delta}\left\lVert\omega\partial^\alpha_{\beta}g\right\rVert_{\sigma}^2+C\bar{\delta}(1+t)^{-\frac{3}{2}-|\alpha|}. \end{align}\] Then we have \[\begin{align} \iint_{\Omega\times\mathbb{R}^3}\omega^2\partial_{\beta}^{\alpha}(\mathcal{S}_{g1}+\mathcal{S}_{g21})\partial^{\alpha}_{\beta}g d\xi dx\leq C\bar{\delta}\left\lVert\omega\partial^\alpha_{\beta}g\right\rVert_{\sigma}^2+C\bar{\delta}(1+t)^{-\frac{3}{2}-|\alpha|}+\iint_{\Omega\times\mathbb{R}^3}\left\lvert\omega^2\partial_\beta^\alpha \mathcal{S}_{g22}\partial_{\beta}^\alpha g\right\rvert d\xi dx. \end{align}\] And for the last term in the first line in 220 , direct calculation yields \[\begin{align} &\sum_{\left\lvert\beta'\right\rvert=1}\iint_{\Omega\times\mathbb{R}^3}\omega^2\partial^{\alpha}_{\beta}g\partial_{\beta'}\xi \cdot\nabla_x\partial_{\beta-\beta'}^{\alpha}gdxd\xi\le \eta\left\lVert\omega\partial^\alpha_{\beta}g\right\rVert^2_{\sigma}+C_\eta\left\lVert\omega(\beta-\beta')\partial^\alpha_{\beta-\beta'}\nabla_xg\right\rVert^2_{\sigma}. \end{align}\] Finally, for the last term in the second line in 220 , for \(\alpha=0\), one has \[\begin{align} &\iint_{\Omega\times\mathbb{R}^3}\omega^2\partial_{\beta}\Big\{\frac{P_0 \left( \xi \cdot \sqrt{\mu}\nabla_x g \right)}{\sqrt{\mu}}\Big\}\partial_{\beta}g dxd\xi\leq\eta\left\lVert\omega(\beta)\partial_{\beta}g\right\rVert^2_{\sigma}+C_\eta\left\lVert\omega(0)\nabla_xg\right\rVert_{\sigma}^2. \end{align}\] For \(\left\lvert\alpha\right\rvert\ge 1\), one has \[\begin{align} &\iint_{\Omega\times\mathbb{R}^3}\omega^2\partial_{\beta}^{\alpha}\Big\{\frac{P_0 \left( \xi \cdot \sqrt{\mu}\nabla_x g \right)}{\sqrt{\mu}}\Big\}\partial^{\alpha}_{\beta}g dxd\xi=\iint_{\Omega\times\mathbb{R}^3}\omega^2\partial^{\alpha}_{\beta}\Big\{\frac{1}{\sqrt{\mu}}\mathop{\sum }\limits_{{j = 0}}^{4}\left\langle {\xi \cdot \sqrt{\mu}\nabla_x g,{\chi}_{j}}\right\rangle {\chi}_{j}\Big\}\partial^{\alpha}_{\beta}g dxd\xi\nonumber\\ \leq&\eta\left\lVert\omega\partial^\alpha_{\beta}g\right\rVert^2_{\sigma}+C\sum_{0\leq|\bar{\alpha}|\leq| \alpha|-1}\left\lVert\partial^{|\alpha|-|\bar{\alpha}|}(\rho,u,\theta)\right\rVert_{L^4}^2\left\lVert\left\lvert\omega(0)\partial^{\bar{\alpha}}\nabla_xg\right\rvert_{\sigma}\right\rVert^2_{L^4}+C_\eta\left\lVert\omega(0)\partial^{{\alpha}}\nabla_xg\right\rVert_{\sigma}^2\nonumber\\ \leq&\eta\left\lVert\omega\partial^\alpha_{\beta}g\right\rVert^2_{\sigma}+\check{\delta}\mathcal{D}_{3,\omega}+C_\eta\left\lVert\omega(0)\partial^{{\alpha}}\nabla_xg\right\rVert_{\sigma}^2. \end{align}\] Plugging all the above estimates back to 220 , we then have finished the proof of ?? and ?? .
The proof for ?? , ?? , ?? and ?? are basically the same as those for ?? and ?? . The only difference lies in the dissipative estimate derived from the linear operator \(\mathcal{L}\): \[\label{20254664606-1} -\langle\partial^\alpha\mathcal{L}g,\omega^2(0)\partial^\alpha g\rangle\geq c_{4}|\omega(0)\partial^\alpha g|_\sigma^2-C_\eta|\chi_{\eta}(\xi)\partial^\alpha g|_2^2,\qquad\quad -\langle\partial^\alpha\mathcal{L}g,\partial^\alpha g\rangle\geq c_{4}|\partial^\alpha g|_\sigma^2.\tag{222}\] Then we have completed the proof of Lemma 28. ◻
Recall the definition of \(\mathbf{v}\) 55 and dissipation energy functionals \(\mathcal{D}_{i,\omega}\) and \(\mathcal{D}_i\) 62 68 . Then we have the following result.
Lemma 29. Under the same assumptions of Proposition 10, it holds that \[\begin{align} &\frac{d}{dt}\sum_{|\alpha|=3}\left\lVert\frac{\partial^\alpha f}{\sqrt{\mu}}\right\rVert_{2,\omega}^2+c\sum_{|\alpha|=3}\|\partial^\alpha{g}\|_{\sigma,\omega}^2\leq C\check{\delta} \sum_{i=1}^{3}(1+t)^{-3+i}\mathcal{D}_{i,\omega}(t)+C\bar{\delta}(1+t)^{-\frac{5}{2}}\notag\\ &\qquad\quad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+C_\eta\sum_{\left\lvert\alpha\right\rvert=3}\left(\|\partial^\alpha{g}\|_\sigma^2+\left\lVert\partial^{\alpha}\mathbf{v}\right\rVert_{L^2}^2 \right),\label{202546084624-1}\\ & \frac{d}{dt}\sum_{|\alpha|=3}\left\lVert\frac{\partial^\alpha f}{\sqrt{\mu}}\right\rVert_2^2+c\sum_{|\alpha|=3}\|\partial^\alpha{g}\|_\sigma^2\leq C\check{\delta} \sum_{i=1}^{3}(1+t)^{-3+i}\mathcal{D}_{i}(t)+C\bar{\delta}(1+t)^{-\frac{5}{2}}.\label{202546084624-2} \end{align}\] {#eq: sublabel=eq:202546084624-1,eq:202546084624-2}
Proof. From 1 , one has \[\begin{align} \label{equ-f} \partial_t \left(\frac{f}{\sqrt{\mu}}\right) + \xi \cdot \nabla_x \left( \frac{f}{\sqrt{\mu}} \right)-\mathcal{L}g=\mathcal{S}_{f}+ \frac{1}{\sqrt{\mu}} {P}_1 \left[\xi_1 \left( \frac{\left\lvert\xi-u\right\rvert^2}{2R\theta^2}\partial_{x_1}\bar{\theta}+ \frac{(\xi-u)\cdot\partial_{x_1}\bar{u}}{R\theta} \right)M\right], \end{align}\tag{223}\] where \[\begin{align} \mathcal{S}_{f}=& \Gamma \left( \frac{M-\mu}{\sqrt{\mu}},g \right) + \Gamma\left( g,\frac{M-\mu}{\sqrt{\mu}} \right)+\Gamma\left(\frac{G}{\sqrt{\mu}},\frac{G}{\sqrt{\mu}}\right). \end{align}\] Applying \(\partial^{\alpha}\) with \(|\alpha|=3\) to 223 , multiplying the results by \(\omega^2(0)\frac{\partial^{\alpha}f}{\sqrt{\mu}}\) and then integrating the resulting equation over \(\Omega\times\mathbb{R}^3\), one has \[\begin{align} &\frac{d}{dt}\left\lVert\omega(0)\frac{\partial^{\alpha}f}{\sqrt{\mu}}\right\rVert^2_{L^2_{\xi,x}}-\iint_{\Omega\times\mathbb{R}^3}\omega^2(0)\partial^{\alpha}g\mathcal{L}\partial^{\alpha}gd\xi dx=\iint_{\Omega\times\mathbb{R}^3}\omega^2(0)\partial^{\alpha}\Big(\frac{M}{\sqrt{\mu}}+\frac{\bar{G}_0}{\sqrt{\mu}}\Big)\mathcal{L}\partial^{\alpha}gd\xi dx\nonumber\\ &\qquad+\iint_{\Omega\times\mathbb{R}^3}\omega^2(0)\frac{\partial^{\alpha}f}{\sqrt{\mu}}\partial^{\alpha}\mathcal{S}_{f}d\xi+\iint_{\Omega\times\mathbb{R}^3}\omega^2(0)\frac{\partial^{\alpha}f}{\sqrt{\mu}}\partial^{\alpha}\left\{\frac{1}{\sqrt{\mu}} {P}_1 \left[\xi_1 \left( \frac{\left\lvert\xi-u\right\rvert^2}{2R\theta^2}\partial_{x_1}\bar{\theta}+ \frac{(\xi-u)\cdot\partial_{x_1}\bar{u}}{R\theta} \right)M\right]\right\}d\xi dx\nonumber\\ &\qquad:=I_{f}^1+I_{f}^2+I_{f}^3. \end{align}\] We first calculate the following \[\begin{align} \label{M1-2} \partial^{\alpha}M =&\mu\left\{\frac{\partial^{\alpha}\rho}{\rho} +\frac{(\xi-u)\cdot\partial^{\alpha}u}{R\theta} +\left(\frac{|\xi-u|^2}{2R\theta}-\frac{3}{2}\right)\frac{\partial^{\alpha}\theta}{\theta}\right\}\nonumber\\ &+(M-\mu)\left\{\frac{\partial^{\alpha}\rho}{\rho} +\frac{(\xi-u)\cdot\partial^{\alpha}u}{R\theta} +\left(\frac{|\xi-u|^2}{2R\theta}-\frac{3}{2}\right)\frac{\partial^{\alpha}\theta}{\theta}\right\}\nonumber\\ &+\sum_{|\alpha_1|=1,2}C_{\alpha}^{\alpha_1}\bigg\{\partial^{\alpha_1}\left(\frac{M}{\rho}\right)\partial^{\alpha-\alpha_1}\rho +\partial^{\alpha_1}\left(M\frac{\xi-u}{R\theta}\right)\cdot\partial^{\alpha-\alpha_1}u\nonumber\\ &+\partial^{\alpha_1}\left(M\frac{|\xi-u|^2}{2R\theta^2}-M\frac{3}{2\theta}\right)\partial^{\alpha-\alpha_1}\theta\bigg\} =:J^\alpha_1+J^\alpha_2+J^\alpha_3. \end{align}\tag{224}\] For \(\iint_{\Omega\times\mathbb{R}^3}\omega^2(0)\frac{\partial^{\alpha}M}{\sqrt{\mu}}\mathcal{L}\partial^{\alpha}gd\xi dx\), we have \[\begin{align} \label{2026-4-25-1} \iint_{\Omega\times\mathbb{R}^3}\omega^2(0)\frac{\partial^{\alpha}M}{\sqrt{\mu}}\mathcal{L}\partial^{\alpha}gd\xi dx\leq C\check{\delta}\left[\mathcal{D}_{3,\omega}(t)+(1+t)^{-1}\mathcal{D}_{2,\omega}\right]+C\bar{\delta}(1+t)^{-\frac{5}{2}}+\eta\left\lVert\partial^{\alpha}g\right\rVert_{\sigma,\omega}^2+C_{\eta}\left\lVert\partial^{\alpha} \mathbf{v}\right\rVert_{L^2}^2. \end{align}\tag{225}\] To provide estimates for \(I_{f}^2\), by Lemma 26 and Lemma 40, we first note that \[\begin{align} \sum_{\left\lvert\alpha\right\rvert=3} \left\lVert\frac{\partial^{\alpha}f}{\sqrt{\mu}}\right\rVert^2_{\sigma,\omega}&\leq C\sum_{\left\lvert\alpha\right\rvert=3}\left( \left\lVert\partial^{\alpha}g\right\rVert_{\sigma,\omega}^2 + \left\lVert\frac{\partial^{\alpha}\bar{G}_0}{\sqrt{\mu}}\right\rVert_{\sigma,\omega}^2 + \left\lVert\frac{\partial^{\alpha}(M-\tilde{M})}{\sqrt{\mu}}\right\rVert_{\sigma,\omega}^2+ \left\lVert\frac{\partial^{\alpha}\tilde{M}}{\sqrt{\mu}}\right\rVert_{\sigma,\omega}^2 \right) \notag\\ &\leq C \sum_{\left\lvert\alpha\right\rvert=3} \left( \left\lVert\partial^{\alpha}g\right\rVert_{\sigma,\omega}^2 + \left\lVert\nabla_x^3 \mathbf{v}\right\rVert_{L^2}^2 \right)+C \check{\delta}\left[ (1+t)^{-\frac{5}{2}}+(1+t)^{-1} \mathcal{D}_{2,\omega} \right]. \end{align}\] Thus, the estimate of \(I_{f}^2\) can be found in Lemma 41 and Lemma 42, which is similar to 221 . Then, we estimate the rest of \(I_{f}^1\) and \(I_{f}^3\). By Lemma 40, for \(\alpha\ge1\), \(\beta\ge0\) and any \(b\ge0\), we observe that \[\begin{align} \left\lvert\langle\xi \rangle^b \partial^\alpha_\beta(\frac{\bar G_0}{\sqrt \mu})\right\rvert_{2,\sigma} \approx \left\lvert\partial_\beta^\alpha\left\{\frac{1}{\sqrt{\mu}} {P}_1 \left[\xi_1 \left( \frac{\left\lvert\xi-u\right\rvert^2}{2R\theta^2}\partial_{x_1}\bar{\theta}+ \frac{(\xi-u)\cdot\partial_{x_1}\bar{u}}{R\theta} \right)M\right]\right\}\right\rvert_{2,\sigma}\approx\bar{\delta}\tilde{D}_{-\frac{1}{2}}\left\lvert\partial_{\beta}^{\alpha}(\frac{M}{\sqrt \mu})\right\rvert_{2,\sigma}. \end{align}\] Using the same method as 225 , it can directly obtain \[\begin{align} \notag \iint_{\Omega\times\mathbb{R}^3}\omega^2(0)\frac{\partial^{\alpha}\bar{G}_0}{\sqrt{\mu}}\mathcal{L}\partial^{\alpha}gd\xi dx+I_{f}^3\leq C\check{\delta}\left[\mathcal{D}_{3,\omega}(t)+(1+t)^{-1}\mathcal{D}_{2,\omega}\right]+C\bar{\delta}\Big((1+t)^{-\frac{5}{2}}+\left\lVert\partial^{\alpha}g\right\rVert_{\sigma,\omega}^2+\left\lVert\partial^{\alpha} \mathbf{v}\right\rVert_{L^2}^2\Big). \end{align}\] And the estimate of the linear operator \(\mathcal{L}\) is consistent with 222 , thereby enabling us to obtain the proof of ?? .
Next, we turn to prove ?? . Due to 224 and \((\rho,u,\theta)=\mathbf{v}+(\tilde{\rho},\tilde{u},\tilde{\theta})\), we have \[\begin{align} &\iint_{\Omega\times\mathbb{R}^3}\Bigg(J^{\alpha}_1\mathcal{L}\partial^{\alpha}g+J^{\alpha}_2\mathcal{L}\partial^{\alpha}g+J^{\alpha}_3\mathcal{L}\partial^{\alpha}g\Bigg)d\xi dx\leq C\check{\delta}\left[\mathcal{D}_{3}(t)+(1+t)^{-1}\mathcal{D}_{2}\right]+C\bar{\delta}(1+t)^{-\frac{5}{2}}+\eta\left\lVert\partial^{\alpha}g\right\rVert_{\sigma}^2, \end{align}\] where we have used the fact that \(J^{\alpha}_1\in \text{ker}\mathcal{L}\). It then follows that \(\iint_{\Omega\times\mathbb{R}^3} J^{\alpha}_1\mathcal{L}\partial^{\alpha}g d\xi dx=0\). The estimate of the remaining terms is similar to the previous calculation, and the proof is omitted for the convenience of reading. Thus, we have completed the proof of Lemma 29. ◻
In this section, we present the differential inequalities for the instant energy functionals \(\mathcal{E}_{i,\omega},\mathcal{E}_i\) (see 56 61 ) and dissipation energy functionals \(\mathcal{D}_{i,\omega},\mathcal{D}_i\) (see 62 68 ), and thereby prove Proposition 10. At first, we present a useful Lemma. For the Landau collision operator, the dissipative norm is not equivalent to the \(L^2\) norm. Motivated by [22], [23], we need the following lemma to perform the time-velocity interpolation technique.
Lemma 30. For any integer \(\alpha\) and positive numbers \(\epsilon>0\) and \(\upsilon>0\), one has \[\begin{align} \left\lvert\partial^{\alpha} f\right\rvert_2^2 \leq \upsilon(1+t)^{\epsilon} \left\lvert\partial^{\alpha} f\right\rvert_{\sigma}^2+e^{-\frac{q}{4}\upsilon^2( 1+t )^{2\epsilon}} \left\lvert e^{\frac{q}{8}\langle\xi\rangle^2} \partial^{\alpha} f\right\rvert_2^2. \end{align}\]
Proof. By the definition of dissipative norm, one has \[\begin{align} \left\lvert\partial^{\alpha} f\right\rvert_{\sigma}^2& \geq \int_{\mathbb{R}^3} \langle\xi\rangle^{-1} \left\lvert\partial^{\alpha} f\right\rvert^2 d\xi=\int_{\left\lvert\xi\right\rvert\leq \upsilon(1+t)^{\epsilon}}+\int_{\left\lvert\xi\right\rvert\geq \upsilon(1+t)^{\epsilon}}\nonumber\\ &\geq\frac{(1+t)^{-\epsilon}}{\upsilon}\int_{\left\lvert\xi\right\rvert\leq \upsilon(1+t)^{\epsilon}}\left\lvert\partial^{\alpha} f\right\rvert^2 d\xi \nonumber\\ &=\frac{(1+t)^{-\epsilon}}{\upsilon}\left\lvert\partial^{\alpha}f\right\rvert_{2}^2-\frac{(1+t)^{-\epsilon}}{\upsilon}\int_{\left\lvert\xi\right\rvert\geq \upsilon (1+t)^{\epsilon}}\left\lvert\partial^{\alpha} f\right\rvert^2, \end{align}\] and \[\begin{align} \int_{\left\lvert\xi\right\rvert\geq \upsilon(1+t)^{\epsilon}}\left\lvert\partial^{\alpha} f\right\rvert^2 d\xi\leq e^{-\frac{q}{4}\upsilon^2 (1+t)^{2\epsilon} } \int_{\mathbb{R}^3} e^{\frac{q}{4}\langle\xi \rangle^2 } \left\lvert\partial^{\alpha} f\right\rvert^2 d\xi. \end{align}\] Then we complete the proof of Lemma 30. ◻
Now we are going to prove the stability and obtain optimal decay rate of the planar entropy wave for the Landau equation. Recall the definition of \((\check{\mathbf{V}},\mathbf{v})\) 55 , instant energy functionals \(\mathcal{E}_{i,\omega},\mathcal{E}_i\) 56 61 and dissipation energy functionals \(\mathcal{D}_{i,\omega},\mathcal{D}_i\) 62 68 . Combining Theorem 16, Theorem 22, Theorem 25, Lemma 26, Lemma 28 and Lemma 29, we have \[\begin{align} &\frac{d}{dt} \mathcal{E}_{1,\omega} + \mathcal{D}_{1,\omega} \leq C\check{\delta}(1+t)^{-1}\mathcal{E}_{1,\omega} + C\bar{\delta}(1+t)^{-\frac{3}{2}}, \tag{226}\\ &\frac{d}{dt}\mathcal{E}_{2,\omega}+\mathcal{D}_{2,\omega}\leq C\check{\delta}\left[(1+t)^{-1}\left\lVert\mathbf{v}\right\rVert_{L^2}^2+(1+t)^{-2}\left\lVert\check{\mathbf{V}}\right\rVert_{L^2}^2 \right]+ C\bar{\delta}(1+t)^{-\frac{3}{2}}.\tag{227} \end{align}\] Integrating 226 and 227 with respect to time \(t\) and combining with the a priori assumptions 69 , we have \[\begin{align} \label{20254664611-1} \mathcal{E}_{2,\omega}\leq C(\delta+\varepsilon_0). \end{align}\tag{228}\] Moreover, we derive the following differential inequalities for the unweighted microscopic quantities \[\begin{align} &\frac{d}{dt}\mathcal{E}_1 + \mathcal{D}_1\leq C\check{\delta}(1+t)^{-1}\mathcal{E}_1 + C\bar{\delta}(1+t)^{-\frac{3}{2}}, \tag{229} \\ &\frac{d}{dt}\mathcal{E}_2 + \mathcal{D}_2\leq C\check{\delta}\left[(1+t)^{-1}\mathcal{D}_1+(1+t)^{-2}\mathcal{E}_1 \right]+ C\bar{\delta}(1+t)^{-\frac{3}{2}}, \tag{230}\\ &\frac{d}{dt}\tilde{\mathcal{E}}_2 +\tilde{\mathcal{D}}_2 \leq C\check{\delta}\left[(1+t)^{-1}\mathcal{D}_1+(1+t)^{-2}\mathcal{E}_1\right]+C\bar{\delta}(1+t)^{-\frac{5}{2}},\tag{231}\\ &\frac{d}{dt}\mathcal{E}_3 + \mathcal{D}_3\leq C\check{\delta}\left[(1+t)^{-1}\mathcal{D}_2+(1+t)^{-2}\mathcal{D}_1+(1+t)^{-3}\mathcal{E}_1 \right]+ C\bar{\delta}(1+t)^{-\frac{5}{2}}.\tag{232} \end{align}\] Recalling \(f=M-\tilde{M}+\sqrt{\mu}g + \bar{G}_0+\tilde{M}\), one has \[\begin{align} \label{decay-3-0-2} \sum_{\left\lvert\alpha\right\rvert=3}\left\lVert \frac{\partial^{\alpha}f}{\sqrt{\mu}}\right\rVert_2^2 &\lesssim\sum_{\left\lvert\alpha\right\rvert=3}\left\lVert \frac{\partial^{\alpha}(M-\tilde{M})}{\sqrt{\mu}}\right\rVert_2^2+\sum_{\left\lvert\alpha\right\rvert=3}\left\lVert \partial^{\alpha}g\right\rVert_2^2+\bar{\delta}(1+t)^{-\frac{5}{2}}\lesssim \left\lVert\nabla_x \mathbf{v}\right\rVert_{H^2}^2 +\sum_{\left\lvert\alpha\right\rvert=3}\left\lVert \partial^{\alpha}g\right\rVert_2^2+\bar{\delta}(1+t)^{-\frac{5}{2}}. \end{align}\tag{233}\] Combining Lemma 30 together with 228 and 233 , and then taking \(\epsilon=\frac{1}{10}\) and \(\upsilon=\frac{1}{2}\), one has \[\begin{align} &\int_0^t \left(\mathcal{E}_2+\tilde{\mathcal{E}}_2\right) d \tau \le C(1+t)^{\frac{1}{10}} \int_0^t \mathcal{D}_1 d\tau +C\bar{\delta},\tag{234}\\ &\int_0^t (1+\tau)\mathcal{E}_3 d \tau \le C(1+t)^{\frac{1}{10}} \int_0^t(1+\tau) \tilde{\mathcal{D}}_2 d\tau +C\bar{\delta}.\tag{235} \end{align}\] By taking \(\int_0^t\)229 \(d\tau\), \(\int_0^t(1+\tau)\)230 \(d\tau\), \(\int_0^t(1+\tau)\)231 \(d\tau\), and \(\int_0^t(1+\tau)^2\)232 \(d\tau\), and using 234 235 , one has \[\begin{align} &\mathcal{E}_1 +\int_0^t \mathcal{D}_1d\tau \leq C\bar{\delta}(1+t)^{\check{\delta}}\leq C\bar{\delta}(1+t)^{\frac{1}{10}}, \tag{236}\\ &(1+t)\mathcal{E}_2+\int_0^t (1+\tau)\mathcal{D}_2 d\tau \le \int_0^t \left(\mathcal{E}_2 +\mathcal{D}_1\right)d\tau + \int_0^t(1+\tau)^{-1} \mathcal{E}_1 d\tau+C\bar{\delta}(1+t)^{\frac{1}{2}} \notag\\ &\qquad\qquad\le C(1+t)^{\frac{1}{10}}\int_0^t \mathcal{D}_1 d\tau + \int_0^t(1+t)^{-1} \mathcal{E}_1 d\tau+C\bar{\delta}(1+t)^{\frac{1}{2}}\leq C\bar{\delta}(1+t)^{\frac{1}{2}},\tag{237}\\ &(1+t)\tilde{\mathcal{E}}_2+\int_0^t(1+\tau)\tilde{\mathcal{D}}_2d\tau\leq\int_0^t \left(\tilde{\mathcal{E}}_2+\mathcal{D}_1\right)d\tau + \int_0^t (1+\tau)^{-1} \mathcal{E}_1 d\tau+C\bar{\delta}\notag\\ &\qquad\qquad\leq C(1+t)^{\frac{1}{10}} \int_0^t \mathcal{D}_1 d\tau+ \int_0^t (1+\tau)^{-1} \mathcal{E}_1 d\tau+C\bar{\delta}\leq C\bar{\delta}(1+t)^{\frac{1}{5}},\tag{238}\\ &(1+t)^2\mathcal{E}_3+\int_0^t (1+\tau)^2\mathcal{D}_3 d\tau \notag \\ &\le C\int_0^t (1+\tau)\left(\mathcal{E}_3+\mathcal{D}_2\right) d\tau + C\int_0^t \mathcal{D}_1d\tau+ C\int_0^t(1+\tau)^{-1} \mathcal{E}_1 d\tau+C\bar{\delta}(1+t)^{\frac{1}{2}} \notag\\ &\le C\int_0^t \left[(1+\tau){\mathcal{D}}_2+ \mathcal{D}_1\right] d\tau+ C(1+t)^{\frac{1}{10}}\int_0^t (1+\tau)\tilde{\mathcal{D}}_2 d\tau + C\int_0^t(1+\tau)^{-1} \mathcal{E}_1 d\tau+C\bar{\delta}(1+t)^{\frac{1}{2}} \notag\\ &\leq C\bar{\delta}(1+t)^{\frac{1}{2}}\tag{239}. \end{align}\] From 236 , 237 , 238 and 239 , it is direct to see \[\begin{align} \label{decay-3} & \mathcal{E}_2 \leq C\bar{\delta}(1+t)^{-\frac{1}{2}},\qquad\quad\tilde{\mathcal{E}}_2\leq C\bar{\delta}(1+t)^{-\frac{4}{5}},\qquad\quad\mathcal{E}_3\leq C\bar{\delta}(1+t)^{-\frac{3}{2}}. \end{align}\tag{240}\] From 240 , we then obtain the long-time behavior of the perturbation quantities of the Landau equation with respect to planar entropy waves as \[\begin{align} \label{20254664611-2} \|\frac{f(t,x,\xi)-M_{[\tilde{\rho},\tilde{u},\tilde{\theta}](t,x)}(\xi)}{\sqrt{\mu}}\|_{L_{x}^{\infty}L_{\xi}^{2}}\leq C(\delta^{\frac{1}{2}}+\varepsilon_0^{\frac{1}{2}})(1+t)^{-\frac{1}{2}}. \end{align}\tag{241}\] Combining 228 , 240 and 241 , we have completed the proof of Proposition 10.
In this section, we present the stretched exponential decay estimates for non-zero modes of solutions to the Landau equation 1 . This section is divided into two parts. The first part provides the dissipation estimates for non-zero mode of macroscopic quantities, and the second part gives the dissipation estimates for non-zero mode of microscopic quantities. By combining these two parts of estimates and applying Lemma 30, we can obtain the stretched exponential decay estimates for non-zero mode. We state the result as follows.
Theorem 31. Under the same assumptions of Theorem 8, it holds that \[\begin{align} \label{add46sec646thm} \left\lVert\frac{(f-M_{[\check{\rho},\check{u},\check \theta]})_{\neq}}{\sqrt{\mu}}\right\rVert_{L_x^\infty L_{\xi}^2}^2\leq C (\delta+\varepsilon_0) e^{-ct^{\frac{2}{3}}}. \end{align}\qquad{(8)}\]
In order to present the proof of the above result more conveniently, we introduce the new perturbation \(\mathbf{f}\) as a perturbation near the global Maxwellian \(\mu\) in the form that \(f=\mu+\sqrt{\mu}\mathbf{f}\). With this notation, it holds that \[\label{add46ff46p1} \mathbf{f}_{\neq}=\frac{f_{\neq}}{\sqrt \mu}=\frac{(f-M_{[\check{\rho},\check{u},\check\theta]})_{\neq}}{\sqrt \mu},\tag{242}\] so we only need to prove \(\left\lVert\mathbf{f}_{\neq}\right\rVert_{L_x^\infty L_\xi^2}\leq C (\delta+\varepsilon_0) e^{-ct^{\frac{2}{3}}}\) corresponding to the desired result ?? . Recall \(f=M+G\) 5 , then we have \[\begin{align} \label{2026-4-26-2} \mathbf{f}=\frac{f-\mu}{\sqrt \mu}=\frac{M-M_{[\check{\rho},\check{u},\check\theta]}+M_{[\check{\rho},\check{u},\check\theta]}-\mu+\bar G_0+\sqrt \mu g}{\sqrt \mu}. \end{align}\tag{243}\] With this observation, by Proposition 10, Corollary 15 and 243 , we obtain \[\begin{align} \label{2026-4-26-1} \left\lVert\left\lvert\mathbf{f}\right\rvert_{2,\omega}\right\rVert_{L^\infty}+\sum_{1\le|\alpha_1|\le3} \left\lVert\partial^{\alpha_1} \mathbf{f}\right\rVert_{2,\omega}+\sum_{\left\lvert\alpha_0\right\rvert\le3}\left\lVert\partial^{\alpha_0}\mathbf{f}_{\neq}\right\rVert_{2,\omega}+\left\lVert\left\lvert\mathbf{f}_{\neq}\right\rvert_{\sigma,\omega}\right\rVert_{L_x^\infty}+\sum_{|\alpha_2|=1}\left\lVert\left\lvert\partial^{\alpha_2}\mathbf{D}_0\mathbf{f}\right\rvert_{\sigma,\omega}\right\rVert_{L_x^\infty} \le C \bar{\delta}. \end{align}\tag{244}\]
For the linearized operator \(\mathcal{L}\) 127 near the global Maxwellian \(\mu\), recall that \(\ker(\mathcal{L})=\text{span} \{\sqrt \mu ,\xi\sqrt \mu,\left\lvert\xi\right\rvert^2\sqrt \mu \}\). We will denote the projection onto the kernel as \(\mathbf{P}\) and for the new perturbation \(\mathbf{f}_{\neq}:=\mathbf{f}-\int_{\mathbf{T}^2}\mathbf{f}dx_2dx_3\) we denote \[\begin{align} \mathbf{P}\mathbf{f}_{\neq}=\Big(\mathbf{a}_{\neq}+\mathbf{b}_{\neq}\cdot\xi +\mathbf{c}_{\neq}(\left\lvert\xi\right\rvert^2-3)\Big)\sqrt{\mu}, \end{align}\] where \[\begin{align} \mathbf{a}_{\neq}=\int_{\mathbb{R}^3} \mathbf{f}_{\neq} \sqrt \mu d\xi,\qquad \mathbf{b}_{i\neq}=\int_{\mathbb{R}^3} \mathbf{f}_{\neq} \xi_i\sqrt{\mu} d\xi,\qquad \mathbf{c}_{\neq}=\frac{1}{6}\int_{\mathbb{R}^3} \mathbf{f}_{\neq}(\left\lvert\xi\right\rvert^2-3)\sqrt{\mu}d\xi. \end{align}\] The high-order moment functions \(\mathbf{E}[h]:=[\mathbf{E}_{ij}(h)]_{3\times3}\) and \(\mathbf{F}[h]:=(\mathbf{F}_1[h],\mathbf{F}_2[h],\mathbf{F}_3[h])\) are defined as \[\begin{align} \label{2026-4-26-7} \mathbf{E}_{ij}[h]=\int_{\mathbb{R}^3} (\xi_i\xi_j-1)\sqrt{\mu} h d\xi,\qquad \mathbf{F}_{i}[h]=\int_{\mathbb{R}^3}(\left\lvert\xi\right\rvert^2-5) \xi_i \sqrt{\mu} h d\xi. \end{align}\tag{245}\] Then we have the hydrodynamic system (see \(e.g.\) [60], [61]) \[\begin{align} &\partial_t \mathbf{a}_{\neq} + \nabla_x \cdot \mathbf{b}_{\neq}=0, \tag{246}\\ &\partial_t \mathbf{b}_{\neq}+\nabla_x \mathbf{a}_{\neq}=-2\nabla_x \mathbf{c}_{\neq}-\nabla_x \cdot \mathbf{E}[(I-\mathbf{P})\mathbf{f}_{\neq}],\tag{247}\\ &\partial_t \mathbf{c}_{\neq}=-\frac{1}{3} \nabla_x \cdot \mathbf{b}_{\neq}-\frac{1}{6} \nabla_x \cdot \mathbf{F}[(I-\mathbf{P})\mathbf{f}_{\neq}] \tag{248},\\ &\partial_t \left( \mathbf{E}[(I-\mathbf{P})\mathbf{f}_{\neq}]+2\mathbf{c}_{\neq} I \right)+\nabla_x\mathbf{b}_{\neq}+(\nabla_x\mathbf{b}_{\neq})^{\perp}=-\mathbf{E}\left[ l+n\right],\tag{249}\\ &\partial_t\mathbf{F}[(I-\mathbf{P})\mathbf{f}_{\neq}] + \nabla_x \mathbf{c}_{\neq} =-\mathbf{F}[l+n],\tag{250} \end{align}\] where \[\begin{align} l=\xi \cdot \nabla_x (I-\mathbf{P})\mathbf{f}_{\neq}-\mathcal{L}(I-\mathbf{P})\mathbf{f}_{\neq},\qquad n=-\mathbf{D}_{\neq}\Gamma(\mathbf{f},\mathbf{f}). \end{align}\] For the \(\mathbf{E}[l],\mathbf{E}[n],\mathbf{F}[l],\mathbf{F}[n]\), we have the following estimate.
Lemma 32. Let \(\mathbf{E}[\cdot],\mathbf{F}[\cdot]\) be defined as in 245 , then there exists a universal constant \(0<\epsilon\le\frac{1}{4}\) such that \[\begin{align} &\left\lvert\mathbf{E}_{ij}[(I-\mathbf{P})\mathbf{f}_{\neq}]\right\rvert+\left\lvert\mathbf{F}_i[(I-\mathbf{P})\mathbf{f}_{\neq}]\right\rvert\le C \left\lvert\mu^{\epsilon}(I-\mathbf{P})\mathbf{f}_{\neq}\right\rvert_2 ,\qquad \left\lvert\mathbf{E}_{ij}[l]\right\rvert+\left\lvert\mathbf{F}_i[l]\right\rvert\le C \left\lvert\mu^{\epsilon}\nabla_x(I-\mathbf{P})\mathbf{f}_{\neq}\right\rvert_2,\\ &\left\lvert\mathbf{E}_{ij}[n]\right\rvert+\left\lvert F_i[n]\right\rvert\le C \bar{\delta}\left( \left\lvert\mu^{\epsilon} \mathbf{f}_{\neq}\right\rvert_2+\left\lvert\mathbf{f}_{\neq}\right\rvert_{\sigma}\right). \end{align}\]
Proof. It should be noted that \[\begin{align} \label{2026-4-27-1} \mathbf{D}_{\neq}\Gamma(\mathbf{f},\mathbf{f})=\mathbf{D}_{\neq}\Gamma(\mathbf{f}_{\neq},\mathbf{f}_{\neq})+\Gamma(\mathbf{D}_0\mathbf{f},\mathbf{f}_{\neq})+\Gamma(\mathbf{f}_{\neq},\mathbf{D}_0\mathbf{f}). \end{align}\tag{251}\] Combining 244 , 251 and ?? , and using the same method as Lemma 4.5 in [60] and Lemma 3.5 in [61], we can prove Lemma 32. For simplicity of presentation, we omit the details of proof here. ◻
Based on the hydrodynamic system 246 249 , the following estimates hold for the non-zero modes of macroscopic quantities.
Lemma 33. Under the same assumptions of Theorem 8, for suitable large constants \(C_{1},C_{2}\) and a universal constant \(C\), it holds that \[\begin{align} &\frac{d}{dt}\mathcal{E}_{mac} + C\Big(\left\lVert\nabla_x \mathbf{a}_{\neq}\right\rVert_{L^2}^2+\left\lVert\nabla_x \mathbf{b}_{\neq}\right\rVert_{L^2}^2+\left\lVert\nabla_x \mathbf{c}_{\neq}\right\rVert_{L^2}^2\Big) \\ \le& (C_{\eta_1}+C_{\eta_2})\left\lVert\mu^{\epsilon}\nabla_x(I-\mathbf{P})\mathbf{f}_{\neq}\right\rVert_2^2+C\bar{\delta}\left( \left\lVert\mu^{\epsilon} (I-\mathbf{P})\mathbf{f}_{\neq}\right\rVert_2^2+\left\lVert(I-\mathbf{P})\mathbf{f}_{\neq}\right\rVert_\sigma^2\right), \end{align}\] where \(\mathcal{E}_{mac}:=\mathbf{b}_{\neq}\cdot\nabla_x \mathbf{a}_{\neq}+C_{1} \left(\mathbf{E}[(I-\mathbf{P})\mathbf{f}_{\neq}]+2\mathbf{c}_{\neq}I \right):[\nabla_x \mathbf{b}_{\neq}+(\nabla_x \mathbf{b}_{\neq})^{\perp}]+C_{2}\mathbf{F}[(I-\mathbf{P})\mathbf{f}_{\neq}]\cdot\nabla_x \mathbf{c}_{\neq}\).
Proof. Multiplying 247 by \(\nabla_x \mathbf{a}_{\neq}\) and using 246 , we have \[\begin{align} \label{2026-4-27-3} &\frac{d}{dt}\int_{\Omega} \mathbf{b}_{\neq} \cdot \nabla_x \mathbf{a}_{\neq} dx + \left\lVert\nabla_x \mathbf{a}_{\neq}\right\rVert_{L^2}^2 \notag\\ =&-2\int_{\Omega} \nabla_x \mathbf{c}_{\neq} \cdot \nabla_x \mathbf{a}_{\neq} dx-\int_{\Omega} \partial_t \mathbf{a}_{\neq} \nabla_x \cdot \mathbf{b}_{\neq} dx -2\int_{\Omega} \nabla_x \cdot\mathbf{E}[(I-\mathbf{P})\mathbf{f}_{\neq}] \cdot \nabla_x \mathbf{a}_{\neq} dx \notag\\ \le& \frac{1}{100} \left\lVert\nabla_x \mathbf{a}_{\neq}\right\rVert_{L^2}^2+ C\left( \left\lVert\nabla_x \mathbf{c}_{\neq}\right\rVert_{L^2}^2+\left\lVert\nabla_x \mathbf{b}_{\neq}\right\rVert_{L^2}^2 \right)+C\left\lVert\mu^{\epsilon} \nabla_x(I-\mathbf{P})\mathbf{f}_{\neq}\right\rVert_2^2. \end{align}\tag{252}\] Multiplying 249 by \(\nabla_x \mathbf{b}_{\neq}+(\nabla_x \mathbf{b}_{\neq})^{\perp}\), and using 247 and Lemma 32, we obtain \[\begin{align} \label{2026-4-27-4} & \frac{d}{dt}\int_{\Omega} (\mathbf{E}(I-\mathbf{P})\mathbf{f}_{\neq}+2\mathbf{c}_{\neq}I):(\nabla_x \mathbf{b}_{\neq}+(\nabla_x \mathbf{b}_{\neq})^{\perp}) dx +\left\lVert\nabla_x b+(\nabla_x \mathbf{b}_{\neq})^{\perp}\right\rVert_{L^2}^2 \notag\\ =&-2\int_{\Omega} \partial_t \mathbf{b}_{\neq} \cdot \left( \nabla_x \cdot \mathbf{E}[(I-\mathbf{P})\mathbf{f}_{\neq}]+2\nabla_x\mathbf{c}_{\neq}I \right) dx-\int_{\Omega} \mathbf{E}[l+n] :(\nabla_x \mathbf{b}_{\neq}+(\nabla_x \mathbf{b}_{\neq})^{\perp}) dx\notag\\ \le & \eta_1 \left\lVert\nabla_x \mathbf{b}_{\neq}+(\nabla_x \mathbf{b}_{\neq})^{\perp}\right\rVert_{L^2}^2+ \eta_1\left\lVert\nabla_x \mathbf{a}_{\neq}\right\rVert_{L^2}^2 + C\bar{\delta}\left(\left\lVert\mu^{\epsilon} \mathbf{f}_{\neq}\right\rVert_2^2 +\left\lVert\mathbf{f}_{\neq}\right\rVert_{\sigma}^2 \right) \notag\\ &+C_{\eta_1}\left(\left\lVert\nabla_x \mathbf{c}_{\neq}\right\rVert_{L^2}^2 +\left\lVert\mu^{\epsilon}\nabla_x(I-\mathbf{P})\mathbf{f}_{\neq}\right\rVert_2^2 \right). \end{align}\tag{253}\] Multiplying 250 by \(\nabla_x \mathbf{b}_{\neq}+(\nabla_x \mathbf{b}_{\neq})^{\perp}\), and using 248 and Lemma 32, one has \[\begin{align} \label{2026-4-27-5} &\frac{d}{dt}\int_{\Omega}\mathbf{F}[(I-\mathbf{P})\mathbf{f}_{\neq}] \cdot \nabla_x \mathbf{c}_{\neq} dx + \left\lVert\nabla_x \mathbf{c}_{\neq}\right\rVert_{L^2}^2 \notag\\ =&-\int_{\Omega} \partial_t\mathbf{c}_{\neq}\nabla_x \cdot\mathbf{F}[(I-\mathbf{P})\mathbf{f}_{\neq}] dx-\int_{\Omega} \mathbf{F}[l+n] \cdot \nabla_x \mathbf{c}_{\neq} dx \notag\\ \le& \eta_2 \left( \left\lVert\nabla_x \mathbf{b}_{\neq}\right\rVert_{L^2}^2 + \left\lVert\nabla_x \mathbf{c}_{\neq}\right\rVert_{L^2}^2 \right) + C_{\eta_2}\left\lVert\mu^{\epsilon} \nabla_x (I-\mathbf{P})\mathbf{f}_{\neq}\right\rVert_2^2 + C \bar{\delta}\left( \left\lVert\mu^{\epsilon} \mathbf{f}_{\neq}\right\rVert_2^2+\left\lVert\mathbf{f}_{\neq}\right\rVert_\sigma^2\right). \end{align}\tag{254}\] Combining 252 , 253 , 254 and using the small strength \(\bar{\delta}\) of entropy wave, we have finished the proof of Lemma 33. ◻
Recall \(f=\mu+\sqrt{\mu}\mathbf{f}\) and Landau equation 1 , then we can write the equation of \(\mathbf{f}_{\neq}\) as \[\begin{align} \label{mic-non-f} \partial_t \mathbf{f}_{\neq}+\xi \cdot\nabla_x \mathbf{f}_{\neq}-\mathcal{L} \mathbf{f}_{\neq}=\mathbf{D}_{\neq}\Gamma(\mathbf{f},\mathbf{f}). \end{align}\tag{255}\] Based on the equation of \(\mathbf{f}_{\neq}\) 255 , we have the following result.
Lemma 34. Under the same assumptions of Theorem 8, it holds that \[\begin{align} \frac{d}{dt} \left\lVert\nabla_x \mathbf{f}_{\neq}\right\rVert_2^2 + c\left\lVert(I-\mathbf{P})\nabla_x\mathbf{f}_{\neq}\right\rVert_\sigma^2 \le C\bar{\delta}\left( \left\lVert\nabla_x \mathbf{a}_{\neq}\right\rVert_{L^2}^2+\left\lVert\nabla_x \mathbf{b}_{\neq}\right\rVert_{L^2}^2+\left\lVert\nabla_x \mathbf{c}_{\neq}\right\rVert_{L^2}^2 \right). \end{align}\]
Proof. Applying \(\partial_{x_i}\) to 255 , we have \[\begin{align} \label{mic-non-f-1} \partial_t \partial_{x_i}\mathbf{f}_{\neq} + \xi\cdot \nabla_x \partial_{x_i}\mathbf{f}_{\neq} - \mathcal{L} \partial_{x_i} \mathbf{f}_{\neq} = \partial_{x_i} \mathbf{D}_{\neq}\Gamma(\mathbf{f},\mathbf{f}). \end{align}\tag{256}\] Multiplying 256 by \(\partial_{x_i} \mathbf{f}_{\neq}\), and applying \(-\iint_{\Omega\times \mathbb{R}^3} h\mathcal{L}h d\xi dx \ge c \left\lVert(I-\mathbf{P})h\right\rVert_{\sigma}^2\), 244 , 251 and ?? , one has \[\begin{align} \frac{d}{dt}\left\lVert\partial_{x_i} \mathbf{f}_{\neq}\right\rVert_2^2 + c\left\lVert(I-\mathbf{P})\partial_{x_i}\mathbf{f}_{\neq}\right\rVert_\sigma^2 \le C\bar{\delta}\left( \left\lVert\partial_{x_i} \mathbf{a}_{\neq}\right\rVert_{L^2}^2+\left\lVert\partial_{x_i} \mathbf{b}_{\neq}\right\rVert_{L^2}^2+\left\lVert\partial_{x_i} \mathbf{c}_{\neq}\right\rVert_{L^2}^2 \right), \end{align}\] where we have used \[\begin{align} &\iint_{\Omega\times\mathbb{R}^3} \mathbf{D}_{\neq}\partial_{x_i} \Gamma (\mathbf{f},\mathbf{f}) \partial_{x_i} \mathbf{f}_{\neq} d\xi dx=\iint_{\Omega\times\mathbb{R}^3} \left[\partial_{x_i} \Gamma(\mathbf{f}_{\neq},\mathbf{f}_{\neq})+\partial_{x_i}\Gamma(\mathbf{D}_0\mathbf{f},\mathbf{f}_{\neq})+\partial_{x_i}\Gamma(\mathbf{f}_{\neq},\mathbf{D}_0\mathbf{f}) \right] \partial_{x_i}\mathbf{f}_{\neq} d\xi dx \\ \le&C\left\lVert\partial_{x_i} \mathbf{f}_{\neq}\right\rVert_\sigma \left[(\left\lVert\left\lvert\mathbf{f}\right\rvert_{\sigma}\right\rVert_{L^\infty}+\left\lVert\left\lvert\mathbf{f}\right\rvert_{2}\right\rVert_{L^\infty})\left\lVert\partial_{x_i} \mathbf{f}_{\neq}\right\rVert_{\sigma}+(\left\lVert\left\lvert\partial_{x_1}\mathbf{D}_0\mathbf{f}\right\rvert_\sigma\right\rVert_{L^\infty}+\left\lVert\left\lvert\partial_{x_1}\mathbf{D}_0\mathbf{f}\right\rvert_2\right\rVert_{L^\infty}) \left\lVert\mathbf{f}_{\neq}\right\rVert_{\sigma} \right]\\ \le& C\bar{\delta}\left( \left\lVert\partial_{x_i} \mathbf{a}_{\neq}\right\rVert_{L^2}^2+\left\lVert\partial_{x_i} \mathbf{b}_{\neq}\right\rVert_{L^2}^2+\left\lVert\partial_{x_i} \mathbf{c}_{\neq}\right\rVert_{L^2}^2 \right)+ C \bar{\delta}\left\lVert(I-\mathbf{P})\partial_{x_i}\mathbf{f}_{\neq}\right\rVert_{\sigma}^2. \end{align}\] Then we have completed the proof of Lemma 34. ◻
Based on Lemma 33 and Lemma 34, we give the proof of Theorem 31.
Proof of Theorem 31.. Combining Lemma 33, Lemma 34 and \(\left\lVert\mathbf{f}_{\neq}\right\rVert_{2}\le \left\lVert\nabla_x \mathbf{f}_{\neq}\right\rVert_{2}\), we have \[\begin{align} \notag &\frac{d}{dt} \left\lVert\nabla_x\mathbf{f}_{\neq}\right\rVert_{2}^2 + c \left\lVert \nabla_x\mathbf{f}_{\neq}\right\rVert_{\sigma}^2 \leq 0. \end{align}\] Further from Lemma 30, by taking \(\upsilon\ge 1\), we deduce that \[\begin{align} \label{2025-11-13-2} \frac{d}{dt} \left\lVert\nabla_x\mathbf{f}_{\neq}\right\rVert_2^2 + \frac{c(1+t)^{-\epsilon}}{\upsilon} \left\lVert\nabla_x\mathbf{f}_{\neq}\right\rVert_2^2 \le \frac{c(1+t)^{-\epsilon}}{\upsilon} e^{-\frac{q}{4}\upsilon^2(1+t)^{2\epsilon}}\left\lVert e^{\frac{q}{8}\langle\xi \rangle^2}\nabla_x\mathbf{f}_{\neq}\right\rVert_2^2. \end{align}\tag{257}\] By 244 and 257 , we have \[\begin{align} \notag \frac{d}{dt}\left\{ \exp\left[\frac{c(1+t)^{1-\epsilon}}{\upsilon(1-\epsilon)} \right]\left\lVert\nabla_x\mathbf{f}_{\neq}\right\rVert_2^2 \right\}\le C (\delta+\varepsilon_0) (1+t)^{-\epsilon} \exp\left[ \frac{c(1+t)^{1-\epsilon}}{\upsilon(1-\epsilon)} - \frac{q^2}{4} \upsilon^2 (1+t)^{2\epsilon}\right]. \end{align}\] Integrating the above inequality from \(0\) to \(t\) and taking \(\epsilon=\frac{1}{3}\) and \(\upsilon\) sufficiently large, we obtain the stretched exponential decay for \(\left\lVert\nabla_x\mathbf{f}_{\neq}\right\rVert_2^2\) by \[\begin{align} \notag \left\lVert\nabla_x\mathbf{f}_{\neq}\right\rVert_2^2 \leq C (\delta+\varepsilon_0) e^{-ct^{\frac{2}{3}}}. \end{align}\] Using the Gagliardo-Nirenberg inequality, \(\left\lVert\mathbf{f}_{\neq}\right\rVert_{2}\le \left\lVert\nabla_x \mathbf{f}_{\neq}\right\rVert_{2}\) and employing 244 , one has \[\begin{align} \notag \left\lVert\mathbf{f}_{\neq}\right\rVert_{L_x^\infty L_\xi^2}^2\le C \left\lVert\mathbf{f}_{\neq}\right\rVert_{2}^\frac{1}{2} \left\lVert\nabla_x^2\mathbf{f}_{\neq}\right\rVert_{2}^\frac{3}{2}\le C \left\lVert\nabla_x\mathbf{f}_{\neq}\right\rVert_{2}^\frac{1}{2} \left\lVert\nabla_x^2\mathbf{f}_{\neq}\right\rVert_{2}^\frac{3}{2} \leq C (\delta+\varepsilon_0) e^{-c t^{\frac{2}{3}}}, \end{align}\] which gives the desired estimate ?? in terms of 242 . Then we have completed the proof of Theorem 31. ◻
In this appendix, we will give some basic estimates that have been used in the previous sections.
To overcome some difficulties due to the terms involving \(L^{-1}_{M}\) and \(\bar{G}\), we need to consider the integrality about the velocity. In this subsection, we first list some properties of the Burnett functions and then give the fast decay about the velocity \(\xi\) of the Burnett functions. Recall the Burnett functions, cf. [33], [62]–[64]: \[\label{5461} \hat{A}_{j}(\xi)=\frac{|\xi|^{2}-5}{2}\xi_{j}\quad and \quad \hat{B}_{ij}(\xi)=\xi_{i}\xi_{j}-\frac{1}{3}\delta_{ij}|\xi|^{2} \quad for \quad i,j=1,2,3.\tag{258}\] Noting that \(\hat{A}_{j}M\) and \(\hat{B}_{ij}M\) are orthogonal to the null space \(\mathcal{N}\) of \(L_{M}\), we can define functions \(A_{j}(\frac{\xi-u}{\sqrt{R\theta}})\) and \(B_{ij}(\frac{\xi-u}{\sqrt{R\theta}})\) such that \(P_{0}A_{j}=0\), \(P_{0}B_{ij}=0\) and \[\label{5462} A_{j}(\frac{\xi-u}{\sqrt{R\theta}})=L^{-1}_{M}[\hat{A}_{j}(\frac{\xi-u}{\sqrt{R\theta}})M]\quad and \quad B_{ij}(\frac{\xi-u}{\sqrt{R\theta}})=L^{-1}_{M}[\hat{B}_{ij}(\frac{\xi-u}{\sqrt{R\theta}})M].\tag{259}\] We shall list some elementary but important properties of the Burnett functions summarized in the following lemma, cf. [33], [62], [64].
Lemma 35. The Burnett functions have the following properties:
\(-\langle \hat{A}_{i}, A_{i}\rangle\) is positive and independent of i;
\(\langle \hat{A}_{i}, A_{j}\rangle=0\) for any \(i\neq j\);\(\langle \hat{A}_{i}, B_{jk}\rangle=0\) for any i, j, k;
\(\langle\hat{B}_{ij},B_{kj}\rangle=\langle\hat{B}_{kl},B_{ij}\rangle=\langle\hat{B}_{ji},B_{kj}\rangle\), which is independent of i, j, for fixed k, l;
\(-\langle \hat{B}_{ij}, B_{ij}\rangle\) is positive and independent of i, j when \(i\neq j\);
\(-\langle \hat{B}_{ii}, B_{jj}\rangle\) is positive and independent of i, j when \(i\neq j\);
\(-\langle \hat{B}_{ii}, B_{ii}\rangle\) is positive and independent of i;
\(\langle \hat{B}_{ij}, B_{kl}\rangle=0\) unless either \((i,j)=(k,l)\) or \((l,k)\), or i=j and k=l;
\(\langle \hat{B}_{ii}, B_{ii}\rangle-\langle \hat{B}_{ii}, B_{jj}\rangle=2\langle \hat{B}_{ij}, B_{ij}\rangle\) holds for any \(i\neq j\).
In terms of Burnett functions, the viscosity coefficient \(\mu(\theta)\) and heat conductivity coefficient \(\kappa(\theta)\) in 15 can be represented by \[\begin{align} \mu(\theta)=&- R\theta\int_{\mathbb{R}^{3}}\hat{B}_{ij}(\frac{\xi-u}{\sqrt{R\theta}}) B_{ij}(\frac{\xi-u}{\sqrt{R\theta}})\,d\xi>0,\quad i\neq j, \nonumber\\ \kappa(\theta)=&-R^{2}\theta\int_{\mathbb{R}^{3}}\hat{A}_{j}(\frac{\xi-u}{\sqrt{R\theta}}) A_{j}(\frac{\xi-u}{\sqrt{R\theta}})\,d\xi>0. \end{align}\] Notice that these coefficients are positive smooth functions depending only on \(\theta\).
The following lemma is borrowed from [53], which is about the fast velocity decay of the Burnett functions.
Lemma 36. Suppose that \(U(\xi)\) is any polynomial of \(\frac{\xi-\hat{u}}{\sqrt{R}\hat{\theta}}\) such that \(U(\xi)\widehat{M}\in(\ker{L_{\widehat{M}}})^{\perp}\) for any Maxwellian \(\widehat{M}=M_{[\widehat{\rho},\widehat{u},\widehat{\theta}]}(\xi)\) as 6 where \(L_{\widehat{M}}\) is as in 13 . For any \(\epsilon\in(0,1)\) and any multi-index \(\beta\), there exists a constant \(C_{\beta}>0\) such that \[|\partial_{\beta}L^{-1}_{\widehat{M}}(U(\xi)\widehat{M})|\leq C_{\beta}(\widehat{v},\widehat{u},\widehat{\theta})\widehat{M}^{1-\epsilon}.\] In particular, under the assumptions of 69 , there exists a constant \(C_{\beta}>0\) such that \[\notag |\partial_{\beta}A_{j}(\frac{\xi-u}{\sqrt{R\theta}})|+|\partial_{\beta}B_{ij}(\frac{\xi-u}{\sqrt{R\theta}})| \leq C_{\beta}M^{1-\epsilon}.\]
Now, we shall turn to recall the refined estimates for the linearized operator \(\mathcal{L}\) and the nonlinear collision terms \(\Gamma(g_1,g_2)\) defined in 127 . They can be proved by a straightforward modification of the arguments used in [18] and [22] and we thus omit their proofs for brevity.
Lemma 37. Let \(\omega=\omega(\beta)\) be defined by 31 . For any \(\eta>0\) small enough, there exists \(C>0\) such that \[\begin{align} \notag -\langle\partial^\alpha_\beta\mathcal{L}g,\omega^2(\beta)\partial^\alpha_\beta g\rangle\geq |\omega(\beta)\partial^\alpha_\beta g|_\sigma^2-\eta\sum_{|\beta_1|=|\beta|}|\omega(\beta_1)\partial^\alpha_{\beta_1} g|_\sigma^2 -C\sum_{|\beta_1|<|\beta|}|\omega(\beta_1)\partial^\alpha_{\beta_1} g|_\sigma^2. \end{align}\] If \(|\beta| = 0\), there exists \(c_{4}>0\) such that \[\notag -\langle\partial^\alpha\mathcal{L}g,w^2(0)\partial^\alpha g\rangle\geq c_{4}|w(0)\partial^\alpha g|_\sigma^2-C|\chi_{\eta}(\xi)\partial^\alpha g|_2^2,\] where \(\chi_\eta(\xi)\) is a general cutoff function depending on \(\eta\).
Lemma 38. Under the same assumptions of Lemma 37, for any \(\epsilon>0\) small enough, one has \[\label{5467} \langle\partial^\alpha \Gamma(g_1,g_2), g_3\rangle\leq C\sum_{|\alpha_1|\leq|\alpha|}|\mu^{\epsilon}\partial^{\alpha_1}g_1|_2| \partial^{\alpha-\alpha_1}g_2|_\sigma| g_3|_\sigma,\qquad{(9)}\] and \[\label{5468} \langle\partial^\alpha_\beta \Gamma(g_1,g_2), \omega^2(\beta) g_3\rangle\leq C\sum_{|\alpha_1|\leq|\alpha|}\sum_{|\bar{\beta}|\leq|\beta_1|\leq|\beta|}|\mu^{\epsilon}\partial^{\alpha_1}_{\bar{\beta}}g_1|_2|\omega(\beta) \partial^{\alpha-\alpha_1}_{\beta-\beta_1}g_2|_{\sigma}|\omega(\beta) g_3|_{\sigma}.\qquad{(10)}\]
Lemma 39. Under the same assumptions of Theorem 9, for any \(\beta'\geq0\) and \(b>0\), one has \[\begin{align} | \langle \xi\rangle^{b}\partial _{\beta'}(\frac{M-\mu}{\sqrt{\mu}})|_{\sigma,\omega}^2+| \langle \xi\rangle^{b}\partial _{\beta'}(\frac{M-\mu}{\sqrt{\mu}})|_{2,\omega}^2 \leq C \bar{\delta},\notag \end{align}\] where \(\bar{\delta}=\delta+\varepsilon_0\).
Proof. For any \(\beta'\geq0\) and any \(b>0\), from 31 , 32 and 69 , there exists a small \(\varepsilon_{1}>0\) such that \[\begin{align} | \langle \xi\rangle^{b}\partial _{\beta'}(\frac{M-\mu}{\sqrt{\mu}})|_{\sigma,\omega}^2+| \langle \xi\rangle^{b}\partial _{\beta'}(\frac{M-\mu}{\sqrt{\mu}})|_{2,\omega}^2\leq C_b\sum_{|\beta'|\leq|\beta''|\leq|\beta'|+1}\int_{{\mathbb{R}}^3}\mu^{-\varepsilon_1} |\partial _{\beta''}(\frac{M-\mu}{\sqrt{\mu}})|^2\,d\xi. \end{align}\] From the definition of \(M\) 6 and the a priori assumptions 69 , one has \[\begin{align} \int_{\mathbb{R}^3}&\mu^{-\varepsilon_1}|\partial _{\beta''}(\frac{M-\mu}{\sqrt{\mu}})|^2 \,d\xi\leq C \bar{\delta}. \end{align}\] Then we have completed the proof of Lemma 39. ◻
Lemma 40. For any \(|\bar{\alpha}|\geq 1\) and \(|\bar{\beta}|\geq 0\), we use the similar expansion as before to get \[| \langle \xi\rangle^{b}\partial _{\bar{\beta}}(\frac{\bar{G}_0}{\sqrt{\mu}})|_{2,\omega}+|\langle \xi\rangle^{b} \partial _{\bar{\beta}}(\frac{\bar{G}_0}{\sqrt{\mu}})|_{\sigma,\omega} \leq C|[\partial_{x_1}{\bar{u}},\partial_{x_1}\bar{\theta}]|\leq C \bar{\delta}D_{-\frac{1}{2}},\] and \[|\langle \xi\rangle^{b} \partial^{\bar{\alpha}}_{\bar{\beta}}(\frac{\bar{G}_0}{\sqrt{\mu}})|_{2,\omega}+| \langle \xi\rangle^{b} \partial^{\bar{\alpha}}_{\bar{\beta}}(\frac{\bar{G}_0}{\sqrt{\mu}})|_{\sigma,\omega} \lesssim\bar{\delta}\left(\tilde{D}_{-\frac{1+\bar\alpha}{2}}+ \sum_{i=1}^{\bar\alpha}\tilde{D}_{-\frac{1+\bar\alpha-i}{2}}\left\lvert\partial^{i}\mathbf{v}^{\ast}\right\rvert+\sum_{i=2}^{\bar\alpha}\sum_{j=1}^{i-1}\tilde{D}_{-\frac{1+\bar\alpha-i}{2}}\left\lvert\partial^j \mathbf{v}^{\ast}\right\rvert\left\lvert\partial^{i-j} \mathbf{v}^{\ast}\right\rvert\right).\] Moreover, let \(\partial^{\bar \alpha}=(\partial_{t}^{\bar \alpha_1}\partial_{x_2}^{\bar \alpha_2}\partial_{x_3}^{\bar \alpha_3})\), then we have \[\begin{align} \label{2026-4-27-a} |\langle \xi\rangle^{b} \partial^{\bar{\alpha}}_{\bar{\beta}}(\frac{\bar{G}_0}{\sqrt{\mu}})|_{2,\omega}+| \langle \xi\rangle^{b} \partial^{\bar{\alpha}}_{\bar{\beta}}(\frac{\bar{G}_0}{\sqrt{\mu}})|_{\sigma,\omega} \lesssim \bar{\delta}\left(\tilde{D}_{-\frac{2+\bar\alpha}{2}}+ \sum_{i=1}^{\bar\alpha}\tilde{D}_{-\frac{1+\bar\alpha-i}{2}}\left\lvert\partial^{i}\mathbf{v}^{\ast}\right\rvert+\sum_{i=2}^{\bar\alpha}\sum_{j=1}^{i-1}\tilde{D}_{-\frac{1+\bar\alpha-i}{2}}\left\lvert\partial^j \mathbf{v}^{\ast}\right\rvert\left\lvert\partial^{i-j} \mathbf{v}^{\ast}\right\rvert\right). \end{align}\qquad{(11)}\] Here \(\bar{\delta}=\delta+\varepsilon_0\) and we also recall the definition of \(\mathbf{v}^{\ast}\) 55 .
Proof. By 258 and 259 , it holds \[\bar{G}_0(t,x,\xi)=\frac{\sqrt{R}\;\partial_{x_1}\bar{\theta}}{\sqrt{\theta}}{A}_1(\frac{\xi-u}{\sqrt{R\theta}}) +\bar{u}_{jx_1}{B}_{1j}(\frac{\xi-u}{\sqrt{R\theta}}),\notag\] which implies that for \(\beta_1=(1,0,0)\), \[\partial_{\beta_1}\bar{G}_0=\frac{\sqrt{R}\;\partial_{x_1}\bar{\theta}}{\sqrt{\theta}}\partial_{\xi_1}{A}_1(\frac{\xi-u}{\sqrt{R\theta}})(\frac{1}{\sqrt{R\theta}}) +\bar{u}_{1x_1}\partial_{\xi_1}{B}_{11}(\frac{\xi-u}{\sqrt{R\theta}})(\frac{1}{\sqrt{R\theta}}).\] Similarly, we also have \[\begin{align} \label{54620} \partial_{x_1}\bar{G}_0= & \frac{\sqrt{R}\;\partial_{x_1}^2\bar{\theta}}{\sqrt{\theta}}\bar{A}_1(\frac{\xi-u}{\sqrt{R\theta}}) - \frac{\sqrt{R}\;\partial_{x_1}\bar{\theta}\partial_{x_1}\theta}{2\sqrt{\theta^3}}\bar{A}_1(\frac{\xi-u}{\sqrt{R\theta}}) \notag\\ &- \frac{\sqrt{R}\;\partial_{x_1}\bar{\theta}}{\sqrt{\theta}}\nabla_\xi \bar{A}_1(\frac{\xi-u}{\sqrt{R\theta}})\cdot \frac{u_{x_1}}{\sqrt{R\theta}} - \frac{\sqrt{R}\; \partial_{x_1}\bar{\theta}\partial_{x_1}\theta}{\sqrt{\theta}}\nabla_\xi \bar{A}_1(\frac{\xi-u}{\sqrt{R\theta}})\cdot\frac{\xi-u}{2\sqrt{R\theta^3}} \notag\\ &+ \partial_{x_1}^2{\bar{u}}_{1}\bar{B}_{11}(\frac{\xi-u}{\sqrt{R\theta}}) - \frac{{\bar{u}}_{1x_1}u_{x_1}}{\sqrt{R\theta}}\cdot \nabla_\xi \bar{B}_{11}(\frac{\xi-u}{\sqrt{R\theta}}) - \frac{{\bar{u}}_{1x_1}\partial_{x_1}\theta(\xi-u)}{2\sqrt{R\theta^3}}\cdot \nabla_\xi \bar{B}_{11}(\frac{\xi-u}{\sqrt{R\theta}}). \end{align}\tag{260}\] And \(\partial_t \bar{G}\) has the similar expression as 260 . For \(\left\lvert\bar{\alpha}\right\rvert>1\) and \(\left\lvert\bar{\beta}\right\rvert\ge0\), we use the similar expansion as before to obtain \[\begin{align} &| \langle \xi\rangle^{b}\partial _{\bar{\beta}}(\frac{\bar{G}_0}{\sqrt{\mu}})|_{2,\omega}+|\langle \xi\rangle^{b} \partial _{\bar{\beta}}(\frac{\bar{G}_0}{\sqrt{\mu}})|_{\sigma,\omega} \leq C|[\partial_{x_1}{\bar{u}},\partial_{x_1}\bar{\theta}]|\leq C \bar{\delta}D_{-\frac{1}{2}},\\ &|\langle \xi\rangle^{b} \partial^{\bar{\alpha}}_{\bar{\beta}}(\frac{\bar{G}_0}{\sqrt{\mu}})|_{2,\omega}+| \langle \xi\rangle^{b} \partial^{\bar{\alpha}}_{\bar{\beta}}(\frac{\bar{G}_0}{\sqrt{\mu}})|_{\sigma,\omega}\le \left\lvert\partial^{\bar\alpha}(\bar u_x,\bar{\theta}_x)\right\rvert+\cdots+\left\lvert\partial^{\bar\alpha}(u,\theta)\right\rvert\left\lvert(\bar u_x,\bar{\theta}_x)\right\rvert \\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\leq C \bar{\delta}\left(\tilde{D}_{-\frac{1+\bar\alpha}{2}}+ \sum_{i=1}^{\bar\alpha}\tilde{D}_{-\frac{1+\bar\alpha-i}{2}}\left\lvert\partial^{i}\mathbf{v}^{\ast}\right\rvert+\sum_{i=2}^{\bar\alpha}\sum_{j=1}^{i-1}\tilde{D}_{-\frac{1+\bar\alpha-i}{2}}\left\lvert\partial^j \mathbf{v}^{\ast}\right\rvert\left\lvert\partial^{i-j} \mathbf{v}^{\ast}\right\rvert\right). \end{align}\] Here we have used Lemma 36 and the fact that \(|\langle \xi\rangle^b w(\bar{\beta})\mu^{-\frac{1}{2}}M^{1-\epsilon}|_2\leq C\) for any \(b\ge0\) and any small \(\epsilon>0\). Using the fact that \(\partial_{x_2}\bar{\theta}=0\), \(\partial_{x_3}\bar{\theta}=0\) and \(\partial_{t}\bar{\theta}\lesssim \bar{\delta}D_{-1}\), we obtain ?? . ◻
In what follows, we prove some linear and nonlinear estimates which have been used in Lemma 28 and Lemma 29. Recall the definition of dissipation energy functionals \(\mathcal{D}_{i,\omega}\) and \(\mathcal{D}_i\) 62 68 . We first consider the estimates of the terms \(\Gamma(g,\frac{M-\mu}{\sqrt{\mu}})\) and \(\Gamma(\frac{M-\mu}{\sqrt{\mu}},g)\).
Lemma 41. Under the same assumptions of Theorem 8, let \(|\alpha|+|\beta|\leq 3\) and \(\omega=\omega(\beta)\) be defined by 31 , then one has \[\begin{align} \label{54629-t} &&\big|(\partial^\alpha_\beta \Gamma(\frac{M-\mu}{\sqrt{\mu}},g), \omega^2(\beta) h)\big| +\big|(\partial^\alpha_\beta \Gamma(g,\frac{M-\mu}{\sqrt{\mu}}),\omega^2(\beta) h)\big| \leq C\check{\delta}\Big(\big\|\omega(\beta)h\big\|_{\sigma}^2+\mathcal{D}_{2,\omega}(t)\Big), \end{align}\qquad{(12)}\] and \[\begin{align} \label{54630-t} &&\big|(\partial^\alpha\Gamma(\frac{M-\mu}{\sqrt{\mu}},g),h)\big| +\big|(\partial^\alpha \Gamma(g,\frac{M-\mu}{\sqrt{\mu}}),h)\big| \leq C\check{\delta}\Big(\|h\|_{\sigma}^2+\mathcal{D}_{2}(t)\Big), \end{align}\qquad{(13)}\] where \(\bar{\delta}:=\delta+\varepsilon_0\) and \(\check{\delta}:=\chi+\bar{\delta}^{\frac{1}{2}}\). Moreover, for \(\left\lvert\alpha\right\rvert\geq1\), one has \[\begin{align} \label{54630-1t} &\Big|\big(\partial^\alpha_\beta \Gamma(\frac{M-\mu}{\sqrt{\mu}},g)+\partial^\alpha_\beta \Gamma(g,\frac{M-\mu}{\sqrt{\mu}}),\omega^2(\beta) h\big)\Big| \leq C\check{\delta}\left(\|\omega(\beta)h\|_{\sigma}^2+\mathcal{D}_{3,\omega}(t) + (1+t)^{-1} \mathcal{D}_{2,\omega}(t) \right), \end{align}\qquad{(14)}\] and \[\begin{align} \label{54630-2t} &\Big|\big(\partial^\alpha \Gamma(\frac{M-\mu}{\sqrt{\mu}},g)+\partial^\alpha \Gamma(g,\frac{M-\mu}{\sqrt{\mu}}), h\big)\Big| \leq C\check{\delta}\left(\|h\|_{\sigma}^2+\mathcal{D}_{3}(t) + (1+t)^{-1} \mathcal{D}_{2}(t) \right). \end{align}\qquad{(15)}\]
Proof. We only consider the first term on the left hand side of ?? while the second term can be handled in the same way. It follows from ?? that \[\begin{align} \label{54631} |(\partial^\alpha_\beta \Gamma(\frac{M-\mu}{\sqrt{\mu}},g), \omega^2(\beta) h)| \leq C\sum_{|\alpha_1|\leq|\alpha|}\sum_{ |\bar{\beta}|\leq|\beta_1|\leq|\beta|} \underbrace{\int_{\Omega}|\mu^{\epsilon}\partial^{\alpha_1}_{\bar{\beta}}(\frac{M-\mu}{\sqrt{\mu}})|_2| \omega(\beta) \partial^{\alpha-\alpha_1}_{\beta-\beta_1}g|_{\sigma}| \omega(\beta)h|_{\sigma}\,dx}_{I_{1}}. \end{align}\tag{261}\] Thus, for any \(\left\lvert\beta'\right\rvert\geq0\) and \(b>0\), we deduce from the estimates in Lemma 39 that \[\begin{align} \label{54632-t} | \langle \xi\rangle^{b}\partial _{\beta'}(\frac{M-\mu}{\sqrt{\mu}})|_{\sigma,\omega}^2+| \langle \xi\rangle^{b}\partial _{\beta'}(\frac{M-\mu}{\sqrt{\mu}})|_{2,\omega}^2 \leq C \check{\delta}. \end{align}\tag{262}\] Note that \(|\alpha_1|\leq|\alpha|\leq 3\) in 261 since we consider \(|\alpha|+|\beta|\leq 3\). If \(|\alpha_1|=0\) , we have from 262 and 63 that \[\begin{align} \label{5-1} I_{1}&\le\int_{\Omega}|\mu^{\epsilon}\partial_{\bar{\beta}}(\frac{M-\mu}{\sqrt{\mu}})|_2| \omega(\beta) \partial^{\alpha}_{\beta-\beta_1}g|_{\sigma}| \omega(\beta)h|_{\sigma}\,dx\leq C\check{\delta}\|\partial^{\alpha}_{\beta-\beta_1}g\|_{\sigma,\omega}\| \omega(\beta)h\|_{\sigma} \leq C\check{\delta}\mathcal{D}_{2,\omega}(t)+C\check{\delta}\| \omega(\beta)h\|_{\sigma}^2. \end{align}\tag{263}\] If \(|\alpha_1|=1\), we have from the a priori assumptions 69 and Lemma 26 that \[\begin{align} I_{1}&\leq C\|\partial^{\alpha_1}[\rho,u,\theta]\|_{L_{x}^\infty} \| \omega(\beta)\partial^{\alpha-\alpha_1}_{\beta-\beta_1}g\|_{\sigma}\| \omega(\beta) h\|_{\sigma} \notag\\ &\leq C\check{\delta}\left( \left\lVert\partial_{\beta-\beta_1}^{\alpha-\alpha_1}g\right\rVert_{\sigma,\omega}+\sum_{2\le\left\lvert\alpha\right\rvert\le3}\left\lVert\partial^{\alpha}g\right\rVert_{\sigma} \right) \left\lVert\omega(\beta)h\right\rVert_{\sigma} \leq C\check{\delta}\mathcal{D}_{2,\omega}(t)+C\check{\delta}\| \omega(\beta)h\|_{\sigma}^2.\notag \end{align}\] If \(|\alpha_1|=2\), we can obtain \[\begin{align} \label{higer-order-N2} I_{1}&\leq C(\|\partial^{\alpha_1}[\rho,u,\theta]\|_{L_x^2}+\sum_{|\alpha'|=1}\|[\partial^{\alpha'}(\rho,u,\theta)]^2\|_{L_x^2}) \Big\||\omega(\beta)\partial^{\alpha-\alpha_1}_{\beta-\beta_1}g|_{\sigma}\Big\|_{L_{x}^{\infty}}\| \omega(\beta) h\|_{\sigma} \leq C\check{\delta}\| \omega(\beta) h\|^{2}_{\sigma}+C\check{\delta}\mathcal{D}_{2,\omega}(t). \end{align}\tag{264}\] For \(|\alpha_1|\geq2\), using a method similar to that of 264 , we can obtain the same conclusion as that of 264 . Hence, for \(\eta_0>0\), \(\delta>0\) and \(\varepsilon_{0}>0\) small enough, we deduce from the above estimates that \[\begin{align} |(\partial^\alpha_\beta[ \Gamma(\frac{M-\mu}{\sqrt{\mu}},g)],\omega^2(\beta) h)| \leq C\check{\delta}\big(\|\omega(\beta)h\|_{\sigma}^2+\mathcal{D}_{2,\omega}(t)\big). \end{align}\] Similar arguments as the above give \[|(\partial^\alpha_\beta [\Gamma(g,\frac{M-\mu}{\sqrt{\mu}})], \omega^2(\beta) h)| \leq C\check{\delta}\big(\|\omega(\beta)h\|_{\sigma}^2+\mathcal{D}_{2,\omega}(t)\big).\] The estimate ?? thus follows from the above two estimates. By ?? and the similar calculations as ?? , we can prove that ?? holds and we omit the details for brevity.
We now proceed to prove ?? . When \(\left\lvert\alpha_1\right\rvert<\left\lvert\alpha\right\rvert\), using the method similar to that of 263 264 , we have \[\begin{align} \notag I_1\leq C\check{\delta}\mathcal{D}_{3,\omega}(t)+ C \check{\delta}\left\lVert\omega(\beta)h\right\rVert_{\sigma}^2. \end{align}\] When \(\left\lvert\alpha_1\right\rvert=\left\lvert\alpha\right\rvert\), more meticulous calculations are required. When \(|\alpha|=|\alpha_1|=1\), one has \[\begin{align} \label{5-3461} I_{1}&\leq C\underbrace{\|\partial^{\alpha}[\tilde{\rho},\tilde{u},\tilde{\theta}]\|_{L_{x}^\infty} \| \omega(\beta)\partial_{\beta-\beta_1}g\|_{\sigma}\| \omega(\beta) h\|_{\sigma}}_{I_{11}} +C\underbrace{\|\partial^{\alpha}[\phi,\psi,\zeta]\|_{L_{x}^\infty} \left\lVert \omega(\beta)\partial_{\beta-\beta_1}g\right\rVert_{\sigma} \| \omega(\beta) h\|_{\sigma}}_{I_{12}}. \end{align}\tag{265}\] For \(I_{11}\), one has \[\begin{align} \label{5-3462} I_{11}\leq C \bar{\delta}^{\frac{1}{2}}(1+t)^{-1} \| \omega(\beta)\partial_{\beta-\beta_1}g\|_{\sigma}^2 +C\bar{\delta}^{\frac{1}{2}}\| \omega(\beta) h\|_{\sigma}^2. \end{align}\tag{266}\] Applying Lemma 20, Lemma 26 and the a priori assumption 69 , one has \[\begin{align} \label{5-3464} I_{12}&\leq\left\lVert\partial^{\alpha}(\phi,\psi,\zeta)\right\rVert_{L_x^2}^{\frac{1}{2}}\left\lVert\partial^{\alpha}\nabla_x(\phi,\psi,\zeta)\right\rVert_{L_x^2}^{\frac{1}{2}}\left\lVert\partial_{\beta-\beta_1} g\right\rVert_{\sigma,\omega} \left\lVert h\right\rVert_{\sigma,\omega}+\left\lVert\partial^{\alpha}\nabla_x^2(\phi,\psi,\zeta)\right\rVert_{L_x^2} \left\lVert\partial_{\beta-\beta_1} g\right\rVert_{\sigma,\omega}\left\lVert h\right\rVert_{\sigma,\omega} \notag\\ &\leq C\chi(1+t)^{-1}\left\lVert\partial_{\beta-\beta_1}g\right\rVert_{\sigma,\omega}^2+C\chi\left\lVert\partial_x^2\mathbf{v}\right\rVert_{H^1}^2+ C\chi \sum_{1\leq \left\lvert\gamma\right\rvert\leq 3} \left\lVert\partial^{\gamma}g\right\rVert_{\sigma}^2 + C\chi \left\lVert h\right\rVert_{\sigma,\omega}^2. \end{align}\tag{267}\] Combining 265 , 266 and 267 , one has \[\begin{align} \notag I_1 \leq C\check{\delta}(1+t)^{-1}\mathcal{D}_{2,\omega}+C\check{\delta}\mathcal{D}_{3,\omega} + C\check{\delta}\left\lVert h\right\rVert_{\sigma,\omega}^2. \end{align}\] When \(|\alpha_1|=\left\lvert\alpha\right\rvert=2\), using Lemma 20, Lemma 26 and the a priori assumption, we can obtain \[\begin{align} \label{higer-order-N2-1} I_{1}\leq& C\left(\|\partial^{\alpha}(\phi,\psi,\zeta)\|_{L_x^2}+\sum_{|\alpha'|=1}\|[\partial^{\alpha'}(\phi,\psi,\zeta)]^2\|_{L_x^{2}}\right) \Big\||\omega(\beta)\partial_{\beta-\beta_1}g|_{\sigma}\Big\|_{L_{x}^{\infty}}\| \omega(\beta) h\|_{\sigma} \notag\\ &+ C \left(\left\lVert\partial^{\alpha}(\tilde{\rho},\tilde{u},\tilde{\theta})\right\rVert_{L_x^\infty}+\sum_{|\alpha'|=1}\left\lVert[\partial^{\alpha'}(\tilde{\rho},\tilde{u},\tilde{\theta})]^2\right\rVert_{L_x^\infty} \right)\Big\|\omega(\beta)\partial_{\beta-\beta_1}g\Big\|_{\sigma}\| \omega(\beta) h\|_{\sigma} \notag\\ \leq& C\check{\delta}\| \omega(\beta) h\|^{2}_{\sigma}+ C\check{\delta}\mathcal{D}_{3,\omega}(t) + C\check{\delta}(1+t)^{-1} \mathcal{D}_{2,\omega}. \end{align}\tag{268}\] For \(|\alpha_1|\geq2\), using the method similar to that of 268 , we can obtain the same conclusion as that of 268 . Hence, for \(\left\lvert\alpha\right\rvert\geq1\), we deduce from the above estimates that \[|(\partial^\alpha_\beta[ \Gamma(\frac{M-\mu}{\sqrt{\mu}},g)],\omega^2(\beta) h)| \leq C\check{\delta}\big(\|\omega(\beta)h\|_{\sigma}^2+(1+t)^{-1}\mathcal{D}_{2,\omega}(t)+\mathcal{D}_{3,\omega}(t)\big).\] Similar arguments as the above give \[|(\partial^\alpha_\beta [\Gamma(g,\frac{M-\mu}{\sqrt{\mu}})], \omega^2(\beta) h)| \leq C\check{\delta}\big(\|\omega(\beta)h\|_{\sigma}^2+(1+t)^{-1}\mathcal{D}_{2,\omega}(t)+\mathcal{D}_{3,\omega}(t)\big).\] Then we have proved ?? . By ?? and the similar calculations as ?? , we can prove that ?? holds and we omit the details for brevity. This completes the proof of Lemma 41. ◻
The following estimates are concerned with the nonlinear term \(\Gamma(\frac{G}{\sqrt{\mu}},\frac{G}{\sqrt{\mu}})\).
Lemma 42. Under the same assumptions of Theorem 8, let \(|\alpha|+|\beta|\leq 3\) and \(\omega=\omega(\beta)\) be defined by 31 . For \(\alpha=0\) one has \[\label{54633} |(\partial_\beta[\Gamma(\frac{G}{\sqrt{\mu}},\frac{G}{\sqrt{\mu}})], \omega^2(\beta) h)| \leq C\check{\delta}\big(\|\omega(\beta)h\|_{\sigma}^2+\mathcal{D}_{2,\omega}(t)\big)+C\bar{\delta}(1+t)^{-\frac{3}{2}},\qquad{(16)}\] and \[\label{54634} |( \Gamma(\frac{G}{\sqrt{\mu}},\frac{G}{\sqrt{\mu}}),h)| \leq C\check{\delta}\big(\|h\|_{\sigma}^2+\mathcal{D}_{2}(t)\big)+C\bar{\delta}(1+t)^{-\frac{3}{2}}.\qquad{(17)}\] For \(\left\lvert\alpha\right\rvert\geq1\), one has \[\begin{align} \label{54633-1} &|(\partial^\alpha_\beta[\Gamma(\frac{G}{\sqrt{\mu}},\frac{G}{\sqrt{\mu}})], \omega^2(\beta) h)| \leq C\check{\delta}\Big(\|\omega(\beta)h\|_{\sigma}^2+(1+t)^{-1}\mathcal{D}_{2,\omega}(t)+\mathcal{D}_{3,\omega}(t)\Big)+C\bar{\delta}(1+t)^{-\frac{3}{2}-\left\lvert\alpha\right\rvert}, \end{align}\qquad{(18)}\] and \[\begin{align} \label{54634-1} |(\partial^\alpha [\Gamma(\frac{G}{\sqrt{\mu}},\frac{G}{\sqrt{\mu}})],h)| \leq C\check{\delta}\Big(\|h\|_{\sigma}^2+(1+t)^{-1}\mathcal{D}_{2}(t)+\mathcal{D}_{3}(t)\Big)+C\bar{\delta}(1+t)^{-\frac{3}{2}-\left\lvert\alpha\right\rvert}. \end{align}\qquad{(19)}\]
Proof. Recalling that \(G=\bar{G}_0+\sqrt{\mu}g\), we see \[\label{54635} \Gamma(\frac{G}{\sqrt{\mu}},\frac{G}{\sqrt{\mu}})=\Gamma(\frac{\bar{G}_0}{\sqrt{\mu}},\frac{\bar{G}_0}{\sqrt{\mu}}) +\Gamma(\frac{\bar{G}_0}{\sqrt{\mu}},g)+\Gamma(g,\frac{\bar{G}_0}{\sqrt{\mu}})+\Gamma(g,g).\tag{269}\] For the first term in 269 , we have from the similar arguments as 261 that \[\begin{align} \label{54636} &|(\partial^\alpha_\beta [\Gamma(\frac{\bar{G}_0}{\sqrt{\mu}},\frac{\bar{G}_0}{\sqrt{\mu}})], \omega^2(\beta)h)|\leq C\sum_{|\alpha_1|\leq|\alpha|}\sum_{ |\bar{\beta}|\leq|\beta_1|\leq|\beta|} \underbrace{\int_{\Omega}|\mu^{\epsilon}\partial^{\alpha_1}_{\bar{\beta}}(\frac{\bar{G}_0}{\sqrt{\mu}})|_2| \omega(\beta) \partial^{\alpha-\alpha_1}_{\beta-\beta_1}(\frac{\bar{G}_0}{\sqrt{\mu}})|_{\sigma}| \omega(\beta)h|_{\sigma}\,dx}_{I_{2}}. \end{align}\tag{270}\] Note that \(|\alpha_1|\leq|\alpha|\leq 3\) in 270 due to the fact that \(|\alpha|+|\beta|\leq 3\). For \(\left\lvert\alpha-\alpha_1\right\rvert\le\left\lvert\alpha_1\right\rvert\), one has from 270 that \[\begin{align} \label{5-4-1} I_{2}&\leq C \bar{\delta}^{-1} \left\lVert\mu^{\epsilon}\partial_{\bar\beta}^{\alpha_1}(\frac{\bar G_0}{\sqrt{\mu}})\right\rVert_{L^2}^2\left\lVert\omega(\beta)\partial_{\beta-\beta_1}^{\alpha-\alpha_1}(\frac{\bar G_0}{\sqrt{\mu}})\right\rVert_{L^\infty}^2 +C \bar{\delta}\left\lVert\omega(\beta) h\right\rVert_{\sigma}^2. \end{align}\tag{271}\] By Lemma 40 and 271 , it holds \[\begin{align} &I_2 \leq C\bar{\delta}(1+t)^{-\frac{3}{2}}+C\check{\delta}(\left\lVert\omega(\beta)h\right\rVert_{\sigma}^2+\mathcal{D}_{2,\omega}(t)), \quad \text{for}\quad \alpha=0,\\ &I_2 \leq C\bar{\delta}(1+t)^{-\frac{3}{2}-\left\lvert\alpha\right\rvert}+C\check{\delta}\Big(\left\lVert\omega(\beta)h\right\rVert_{\sigma}^2+(1+t)^{-1}\mathcal{D}_{2,\omega}(t)+\mathcal{D}_{3,\omega}(t)\Big),\quad \text{for} \quad \alpha\ge1. \end{align}\] For \(\left\lvert\alpha-\alpha_1\right\rvert\ge\left\lvert\alpha_1\right\rvert\), it can be treated in the same way. By the above estimate and 270 , we have \[\begin{align} &|(\partial^\alpha_\beta [\Gamma(\frac{\bar{G}_0}{\sqrt{\mu}},\frac{\bar{G}_0}{\sqrt{\mu}})], \omega^2(\beta)h)| \leq C\bar{\delta}(1+t)^{-\frac{3}{2}}+C\check{\delta}(\left\lVert\omega(\beta)h\right\rVert_{\sigma}^2+\mathcal{D}_{2,\omega}(t)), \quad \text{for}\quad \alpha=0,\\ &|(\partial^\alpha_\beta [\Gamma(\frac{\bar{G}_0}{\sqrt{\mu}},\frac{\bar{G}_0}{\sqrt{\mu}})], \omega^2(\beta)h)|\leq C\bar{\delta}(1+t)^{-\frac{3}{2}-\left\lvert\alpha\right\rvert}+C\check{\delta}\Big(\left\lVert\omega(\beta)h\right\rVert_{\sigma}^2+(1+t)^{-1}\mathcal{D}_{2,\omega}(t)+\mathcal{D}_{3,\omega}(t)\Big),\quad\text{for} \;\; \alpha\ge1.\notag \end{align}\]
For the second term in 269 , by ?? , we can obtain \[\begin{align} \label{54640c} &|(\partial^\alpha_\beta [\Gamma(\frac{\bar{G}_0}{\sqrt{\mu}},g)], \omega^2(\beta)h)| \leq C\sum_{ |\alpha_1|\leq|\alpha|}\sum_{ |\bar{\beta}|\leq|\beta_1|\leq|\beta|} \underbrace{\int_{\Omega}|\mu^{\epsilon}\partial^{\alpha_1}_{\bar{\beta}}(\frac{\bar{G}_0}{\sqrt{\mu}})|_2| \omega(\beta) \partial^{\alpha-\alpha_1}_{\beta-\beta_1}g|_{\sigma}| \omega(\beta)h|_{\sigma}\,dx}_{I_{3}}. \end{align}\tag{272}\] From the estimate of \(\bar G_0\) in Lemma 40, one has \[\begin{align} \left\lvert\mu^{\epsilon} \partial_{\bar\beta}^{\alpha_1}(\frac{\bar G_0}{\sqrt{\mu}})\right\rvert_2\approx C\bar{\delta}\tilde{D}_{-\frac{1}{2}}\left\lvert\mu^{\epsilon} \partial_{\bar\beta}^{\alpha_1}(\frac{M-\mu}{\sqrt{\mu}})\right\rvert_2. \end{align}\] It follows that the estimate of \(I_3\) is similar to that of \(I_1\) ?? , and using the same method, we obtain from 272 that \[\begin{align} \label{54641c} |(\partial^\alpha_\beta [\Gamma(\frac{\bar{G}_0}{\sqrt{\mu}},g)], \omega^2(\beta)h)| \leq C\bar{\delta}\left(\|\omega(\beta)h\|_{\sigma}^2+(1+t)^{-1}\mathcal{D}_{2,\omega}(t)\right). \end{align}\tag{273}\] Similar arguments as for obtaining 273 imply \[\begin{align} |(\partial^\alpha_\beta [\Gamma(g,\frac{\bar{G}_0}{\sqrt{\mu}})], \omega^2(\beta)h)| \leq C \bar{\delta}\big(\|\omega(\beta)h\|_{\sigma}^2+(1+t)^{-1}\mathcal{D}_{2,\omega}(t)\big). \end{align}\] By ?? , for \(\alpha=0\), we can arrive at \[\begin{align} |(\partial_\beta [\Gamma(g,g)],\omega^2(\beta)h)| &\leq C\sum_{ |\bar{\beta}|\leq|\beta_1|\leq|\beta|} \int_{\mathbb{R}}|\mu^{\epsilon}\partial_{\bar{\beta}}g|_2| \omega(\beta) \partial_{\beta-\beta_1}g|_{\sigma}| \omega(\beta)h|_{\sigma}\,dx \leq C\check{\delta}\|\omega(\beta)h\|_{\sigma}^2+C\check{\delta}\mathcal{D}_{2,\omega}(t). \end{align}\] For \(\left\lvert\alpha\right\rvert\geq1\), one has \[\begin{align} |(\partial^\alpha_\beta [\Gamma(g,g)],\omega^2(\beta)h)| &\leq C\sum_{ |\alpha_1|\leq|\alpha|}\sum_{ |\bar{\beta}|\leq|\beta_1|\leq|\beta|} \int_{\mathbb{R}}|\mu^{\epsilon}\partial^{\alpha_1}_{\bar{\beta}}g|_2| \omega(\beta) \partial^{\alpha-\alpha_1}_{\beta-\beta_1}g|_{\sigma}| \omega(\beta)h|_{\sigma}\,dx \nonumber\\ &\leq C\check{\delta}\|\omega(\beta)h\|_{\sigma}^2+C\check{\delta}\mathcal{D}_{3,\omega}(t). \end{align}\] Plugging all the above estimates back to 270 , one has ?? and ?? . We can use the similar calculations for obtaining ?? and ?? . Therefore, the proof of Lemma 42 is completed. ◻
The research of Renjun Duan was partially supported by the grant from the National Natural Science Foundation of China (Project No. 12425109). The research of Feimin Huang was partially supported by the Strategic Priority Research Program of the Chinese Academy of Sciences (No. XDB0510201) and National Natural Sciences Foundation of China (Project No. 12288201).
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The authors declare that they have no conflict of interest.