Well-posedness and Blow-up Criterion for
Strong Solutions of the compressible Navier-Stokes/Allen-Cahn System with Vacuum
May 23, 2026
This paper is devoted to the study of strong solutions for the compressible Navier-Stokes/Allen-Cahn system in bounded domain \(\Omega\subset\mathbb{R}^3\), allowing for the presence of initial vacuum. A characteristic of this system is the strong coupling between density and the Allen-Cahn equation, which leads to strong degeneracy in vacuum regions. Under a compatibility condition on the initial phase-field variable, we establish the local existence and uniqueness of strong solutions for \(0\le\rho_0\in W^{1,q}\) with \(q\in(3,6)\), \(u_0\in H_0^1\) and \(\chi_0\in H^2\). Owing to time-weighted estimates, no compatibility condition is required for the velocity, but these estimates introduce a singularity in proving uniqueness. We then establish a criterion for the possible breakdown of such a local strong solution at finite time in terms of blow-up of the quantities \(\|\nabla u\|_{L_t^{1} L_x^{\infty}}\), \(\|u\|_{L_t^{2} L_x^{\infty}}\) and \(\|\nabla \chi\|_{L_t^{2} L_x^{\infty}}\).
The dynamic behavior of a binary mixture of two immiscible fluids subject to phase separation is usually described by diffuse interface models. These models consist of the Navier-Stokes equations governing the evolution of the mixture and a phase-field system that introduces diffuse effects, such as the Cahn-Hilliard, Allen-Cahn, or other types of phase-field equations. This paper is concerned with the compressible Navier-Stokes/Allen-Cahn (NSAC) system proposed by Blesgen [1] \[\begin{align} \begin{cases} \partial_t\rho+{\rm div}(\rho u)=0, \\ \partial_t(\rho u)+{\rm div}(\rho u\otimes u)={\rm div}{\mathbb{T}}, \\ \partial_t(\rho\chi)+{\rm div}(\rho\chi u)=-\mu, \\ \rho\mu=-\Delta\chi+\rho\dfrac{\partial \tilde{F}(\rho, \chi)}{\partial\chi}, \end{cases} \end{align}\] where \((x, t)\in\Omega\times(0, T)\), \(\Omega\subset\mathbb{R}^3\) is a bounded domain with smooth boundary \(\partial\Omega\) and \(T>0\) is a given time. \(\rho=\rho(x,t)\) denotes the total density, \(u=u(x,t)\) is the mean velocity, phase-field variable \(\chi=\chi(x,t)\) represents the concentration difference of the two fluids, \(\mu=\mu(x,t)\) is the chemical potential, the Cauchy stress-tensor is given by \[\begin{align} {\mathbb{T}}=&{\mathbb{S}}-\left(\nabla\chi\otimes\nabla\chi-\frac{|\nabla\chi|^2}{2}{\mathbb{I}}\right)-p(\rho, \chi){\mathbb{I}}, \\ {\mathbb{S}}=&\nu\left(\nabla u+\nabla u^T-\frac{2}{3}{\rm div}u{\mathbb{I}}\right)+\eta\,{\rm div}u\,{\mathbb{I}}, \end{align}\] where \(\nu>0\), \(\eta\ge0\) are viscosity coefficients, and \(p(\rho,\chi)=\rho^2\dfrac{\partial \tilde{F}(\rho, \chi)}{\partial\rho}\) denotes the pressure. In this paper we take the specific free energy \(\tilde{F}\) as follows \[\tilde{F}(\rho,\chi)=\frac{A\rho^{\gamma-1}}{\gamma-1}+\frac{(\chi^2-1)^2}{4}.\] For simplicity, we use the Lamé operator instead of \({\rm div}\,{\mathbb{S}}\), i.e. \[\mathcal{L}u=\nu\Delta u + (\nu+\lambda)\nabla (\mathop{\mathrm{div}}u),\] where \(\nu>0\) and \(\lambda=\eta-\frac{2}{3}\nu\) satisfy \(2\nu+3\lambda \geq 0\). Then the compressible NSAC system can be rewritten as follows \[\label{NSAC} \begin{cases} \rho_t + \mathop{\mathrm{div}}(\rho u)=0,\\ \rho u_t + \rho (u\cdot\nabla) u + \nabla (P(\rho)) = \mathcal{L}u - \mathop{\mathrm{div}}( \nabla\chi\otimes\nabla\chi - \frac{|\nabla \chi|^2}{2} \mathbb{I}), \\ \rho \chi_t + \rho ( u \cdot \nabla ) \chi = - \mu, \\ \rho \mu = - \Delta\chi + \rho f(\chi), \end{cases}\tag{1}\] where \(P(\rho)=A\rho^\gamma\) with \(A>0\) and \(\gamma>1\), the free energy density \(f(s)=F'(s)\), and \(F(s)=\frac{1}{4}(s^2-1)^2\) for \(s\in\mathbb{R}\) is the Landau type potential.
There have been many theoretical results concerning the 1D compressible NSAC system. When the initial density is away from vacuum, Ding-Li-Luo [2] proved the existence and uniqueness of global classical solutions, the existence of weak solutions, and the uniqueness of strong solutions for the initial boundary value problem. Later, Zhang [3] improved the regularity to \(H^i (i=2,4)\). Then Ding-Li-Tang [4] considered the free boundary problem and obtained the existence and uniqueness of strong solutions. In the case where the viscosity coefficient depends on the density, Ding-Li-Wang [5] obtained the \(H^{2m}\)-solution \((m\in\mathbb{N})\) of the free boundary problem. Moreover, the existence of time-periodic solutions have been obtained by Song-Zhang-Wang [6]. For the Cauchy problem, Chen-Huang-Shi [7] investigated the global well-posedness of the solutions and their sharp interface limit, while Luo-Yin-Zhu [8], [9] proved the stability of the rarefaction wave and the composite wave. For the non-isentropic NSAC system without vacuum, we refer the readers to [10]–[12] and the references therein. When vacuum is allowed, Chen-Guo [13] established the existence and uniqueness of strong solutions for the initial boundary value problem under the compatibility condition \[\nu u_{0xx}-(A\rho_0^\gamma)_x-\frac{1}{2}(\chi_{0x}^2)_x=\rho_0 f(x), \quad f\in L^2,\] and the existence of unique classical solution under the compatibility conditions \[\begin{align} \begin{cases} \nu u_{0xx}-(A\rho_0^\gamma)_x-\frac{1}{2}(\chi_{0x}^2)_x=\rho_0 g(x), ~ & g\in H^1, \\ \mu_0=\rho_0h(x), & h_{xx}\in L^2 \end{cases} \end{align}\] for \(x\in(0,1)\). Chen-Zhu [14] assumed that the viscosity coefficient depends both on the density and the phase variable, and they established the local existence of strong solutions for the initial boundary value problem along with the corresponding blow-up criteria, imposing the compatibility condition \[[\nu(\rho_0,\chi_0)u_{0x}]_x-(A\rho_0^\gamma)_x-\frac{1}{2}(\chi_{0x}^2)_x=\rho_0f(x), \quad f\in L^2\] for \(x\in(0,1)\). When the viscosity coefficient depends only on the density, local strong solutions can be improved to classical solutions under more complicated compatibility conditions. Meanwhile, Su [15] established the existence of unique strong solutions using time-weighted estimates without any compatibility condition, and these strong solutions become classical if additional compatibility conditions are imposed. It should be pointed out that all results concerning initial vacuum assume the phase-field variable satisfies a Dirichlet boundary condition, i.e. \(\chi\big|_{x=0,1}=0\).
For the 3D compressible NSAC system, Feireisl et al. [16] first established the global existence of weak solutions to the initial boundary boundary problem under the assumption \(\gamma>6\) and without initial vacuum. Later, Chen-Wen-Zhu [17] relaxed the range to \(\gamma>2\), allowing for initial vacuum, and obtained the existence of weak solutions for system with Landau potential. Li-Xie [18] proved that if the Mach number and initial data are small and away from vacuum, global strong spatial-periodic solutions exist. Under small initial perturbations away from vacuum, Zhao [19] proved the global existence of classical solutions to the 3D Cauchy problem. For the strong solutions to the initial boundary value problem of the NSAC system, only Kotschote [20] derived the local well-posedness result for the non-isentropic case. Whether the strong solutions to the initial boundary value problem of the multi-dimensional compressible NSAC system exist is still an open problem.
Noticing that if the phase variable is constant, then the NSAC system reduces to the compressible Navier-Stokes equations, for which the mathematical theory is considerably more developed. For the compressible Navier-Stokes equations with vacuum, the first result was given by Salvi and Straškraba [21], where the compatibility condition on the initial data was introduced to guarantee the uniqueness of local strong solutions, and this analytical framework was subsequently refined and extended in [22]–[24]. Later, Huang-Li-Xin [25] established a global-in-time existence result under a small initial energy assumption. In 1D, Jiu-Li-Ye [26] proved the global existence of classical solutions for arbitrarily large initial data with vacuum. Recently, the compatibility condition above was removed in [27], [28], where the local existence and uniqueness of strong solutions were established. While extensive literature exists on this topic, we do not enumerate all contributions here. Comparing with the compressible Navier-Stokes equations, results on the multi-dimensional compressible NSAC system are relatively rare. In this paper, we establish the local existence and uniqueness of strong solutions allowing for initial vacuum, with a compatibility condition imposed on the phase-field variable but none required for the velocity.
When deriving the existence of local strong solutions, it is natural to consider global well-posedness. However, as is known, the existence of global strong solutions remains an open problem even for the Navier-Stokes equations, so an alternative approach is to consider either smallness restrictions or blow-up criteria, and in this paper, we adopt the latter. As we mentioned before, there is no local existence result for strong solutions to the compressible multi-dimensional NSAC system. But some results have been established for the nonhomogeneous incompressible case, including both the NSAC and Navier-Stokes/Cahn-Hilliard (NSCH) system. The local well-posedness without vacuum was established in [29]. Subsequently, Li-Ding-Huang [30] derived the blow-up criterion for the 3D compressible NSAC system: for the maximum existence time \(T^*\), it holds that \[\lim_{T\to T^*}\left(\|Du\|_{L^1(0,T; L^\infty)}+\|u\|_{L^2(0,T; L^\infty)} +\|\nabla\chi\|_{L^2(0,T; L^\infty)}\right)=\infty.\] Then Zhang [31] and Fan-Li [32] improved the criterion by relaxing the restriction \(\|\nabla\chi\|_{L^2(0,T; L^\infty)}\) to \(\|\nabla\chi\|_{L^2(0,T; BMO)}\) and removing it, respectively. Recently, Fang-Nei-Guo [33] gave a blow-up criterion for local strong solutions to the initial boundary value problem of the NSCH system in the case of initial density away from zero. Moreover, there have been global existence and uniqueness results for both nonhomogeneous NSAC and nonhomogeneous NSCH systems, such as [34]–[36]. For NSCH systems with a non-Newtonian term and initial vacuum, Fang-Duan-Guo [37] proved the global existence of weak solutions for the initial-boundary value problem and the global existence of spatial-periodic strong solutions. In this paper, we establish a blow-up criterion for our local strong solutions analogous to that in [30].
This paper aims to establish the mathematical theory of compressible NSAC systems with vacuum in 3D. Throughout this paper, we consider the following initial value conditions: \[\begin{align} \label{I} (\rho,\, \rho u , \, \chi)\Big|_{t=0}=(\rho_0,\,\rho_0 u_0, \, \chi_0) \quad \text{in }\Omega, \end{align}\tag{2}\] and boundary value conditions \[\begin{align} \label{B1} (u, \partial_{\boldsymbol{n}}\chi)\Big|_{\partial\Omega} = (0, 0) \quad \text{on }\partial\Omega, \end{align}\tag{3}\] where \(\boldsymbol{n}\) is the unit outward normal vector of \(\partial\Omega\). We emphasize that the initial density \(\rho_0\) is allowed to vanish on an open subset of \(\Omega\); in other words, the presence of an initial vacuum is permitted. Throughout this paper, we always assume that \(\int_\Omega\rho_0 \,dx = 1\), thus \(\eqref{NSAC}_1\) implies that \[\int_{\Omega} \rho dx = \int_{\Omega} \rho_0 dx = 1.\]
Before stating our main result, we first explain the notations used throughout this paper. For \(1 \leq p \leq \infty\) and integer \(k \geq 0\), the standard Sobolev spaces are denoted by \[L^p=L^p(\Omega),~~ W^{k, p}=W^{k, p}(\Omega), ~~H^k=H^{k, 2}(\Omega),~~ H_0^1=\left\{u \in H^1; \left.u\right|_{\partial \Omega}=0\right\}.\] Now we define precisely what we mean by strong solutions to the system 1 –\(\eqref{B1}\).
Definition 1 (Strong solutions). For \(T>0\), \((\rho, u, \chi, \mu)\) is called a strong solution to the system \(\eqref{NSAC}\)–\(\eqref{B1}\) in \(\Omega\times(0,T)\) if for some \(q\in(3,6)\), it has the regularities \[\begin{cases} 0 \le \rho \in L^{\infty}(0, T; W^{1, q}) \cap C([0, T]; C(\overline{\Omega})), \quad \rho_t \in L^\infty(0, T; L^2), \\ u \in L^\infty(0, T; H_0^1) \cap L^2(0, T; H^2) \cap L^1(0, T; W^{2,q}), \quad \sqrt{\rho}u \in C([0, T]; L^2), \\ \sqrt{t} \nabla^2 u \in L^{\infty}(0, T; L^2) \cap L^{2}(0, T; L^q), \\ \sqrt{\rho} u_t\in L^2(0, T; L^2), \quad \sqrt{t} u_t \in L^2(0, T; H_0^1), \\ \chi \in C([0, T]; H^1) \cap L^{\infty}(0, T; H^2) \cap L^2(0, T; H^{3}), \quad \sqrt{t} \nabla^3 \chi \in L^{\infty}(0, T; L^2), \\ \chi_t \in L^2(0, T; H^1), \quad \rho \chi_t \in L^\infty(0, T; L^2), \quad \sqrt{t} \nabla\chi_t \in L^\infty(0, T; L^2), \\ \mu \in L^{\infty}(0, T; L^{2}). \end{cases}\] Besides, \((\rho, u, \chi, \mu)\) satisfies 1 almost everywhere in \(\Omega\times(0,T)\) and fulfills the initial condition 2 . The boundary condition \(\partial_{\boldsymbol{n}} \chi = 0\) holds almost everywhere on \(\partial{\Omega}\times(0, T)\).
Our first result is the local existence and uniqueness of strong solutions.
Theorem 1. Let \(\Omega\) be a smooth bounded domain in \({\mathbb{R}}^3\) and the initial data \((\rho_0, u_0, \chi_0)\) satisfies \(0 \le \rho_0 \in W^{1,q}\), with \(q \in (3, 6)\), \(u_0 \in H_0^1\) and \(\chi_0 \in H^2\). If, in addition, the following compatibility conditions hold: \[\label{com1} \left\{ \begin{align} \Delta \chi_0 &= \rho_0 {h}, &&\text{for some }{h} \in L^2, \\ \partial_{\boldsymbol{n}} \chi_0 &= 0, && \text{on } \partial\Omega, \end{align} \right.\qquad{(1)}\] then, there exists a positive time \(T_0>0\), depending only on \(A\), \(\gamma\), \(\nu\), \(\lambda\), \(q\), \(\Omega\) and \(\Phi_0\), such that the system 1 , subject to 2 –3 , admits a unique strong solution \((\rho, u, \chi, \mu)\) in the sense of Definition 1, where \[\begin{align} \Phi_0=\|\rho_0\|_{W^{1,q}} + \|\nabla u_0\|_{L^{2}} + \| \chi_0 \|_{H^2} + \| h \|_{L^2}. \end{align}\]
Remark 1. The main difficulty of this problem arises from the strong coupling between density and the Allen-Cahn equation, which leads to strong degeneracy within the vacuum region. A compatibility condition is imposed on the initial phase-field variable \(\chi_0\) (see ?? ). This is necessary for two reasons. Mathematically, the equation for \(\chi\) involves \(\rho^2\chi_t\), the stronger degeneracy in vacuum prevents the time-weighted estimates used for the velocity from controlling the initial singularity of \(\chi_t\). The compatibility condition ensures that the initial chemical potential \(\mu_0\) is well defined in \(L^2\). Physically, in the vacuum region where \(\rho_0=0\), there is no matter and hence no driving force for phase separation. This forces \(\Delta\chi_0=0\) and thus \(\mu_0=0\), which is exactly encoded in the compatibility condition.
Remark 2. The compatibility condition ?? is also necessary. Let \((\rho, u, \chi, \mu)\) be a strong solution as in Definition 1. From the regularity, \(\mu \rightharpoonup g\) in \(L^2\) as \(t\to 0^+\) for some \(g\in L^2\). Using \(\eqref{NSAC}_4\) and passing to the limit, we obtain \[\Delta \chi_0 = \rho_0 (f(\chi_0) - g) \quad \text{a.e. in } \Omega.\] Hence \(\Delta\chi_0=\rho_0 h\) with \(h=f(\chi_0) - g \in L^2\). Moreover, from the regularity we can deduce \(\chi\in C([0,T]; H^s)\) for some \(3/2<s<2\). Then by the continuity of the trace operator, the boundary condition \(\partial_{\boldsymbol{n}}\chi=0\) on \(\partial\Omega\) for \(t>0\) implies \(\partial_{\boldsymbol{n}}\chi_0=0\). Thus ?? is necessary.
Remark 3. Under the assumptions of Theorem 1, if the compatibility conditions \[\begin{cases} \mathcal{L} u_0 - \nabla P(\rho_0) - \Delta \chi_0 \nabla \chi_0 = \sqrt\rho_0 {g}, & {g} \in L^2, \\ \mu_0 = \rho_0 {h}, & {h} \in H^1 \end{cases}\] are imposed and the initial data are more regular, namely \(u_0 \in H_0^1 \cap H^2\) and \(\chi_0 \in H^3\), then a local strong solution \((\rho, u, \chi, \mu)\) can be obtained in the sense of Definition 1, with all time-weighted norms in Definition 1 replaced by their corresponding non-weighted counterparts. The corresponding a priori estimates are analogous to those in Section 2, and the proof, with only minor modifications, follows arguments similar to those in Theorem 1.2 of [38].
Our second result is a criterion for the possible breakdown of such a local strong solution at finite time. To the best of our knowledge, this is the first blow-up criterion for the compressible NSAC system in 3D bounded domain with vacuum.
Theorem 2. Let \((\rho, u, \chi, \mu)\) be the strong solution of 1 with the initial boundary conditions 2 and 3 . Assume that the initial data \((\rho_0, u_0, \chi_0)\) have the same regularity as in Theorem 1 and satisfy ?? . If \(0 < T^* < \infty\) is the maximum time of existence, then \[\label{Blowup95Criteria} \lim\limits_{T\to T^*} \left(\|\nabla u\|_{L^1(0,T; L^\infty)}+\| u\|_{L^2(0,T; L^\infty)}+\|\nabla \chi\|_{L^2(0,T; L^\infty)}\right) = \infty.\qquad{(2)}\]
The main contributions of this paper are as follows.
\(\bullet\) To the best of our knowledge, this is the first result establishing the local existence of strong solutions to the initial-boundary value problem of the 3D compressible NSAC system allowing initial vacuum. The proof relies on time-weighted estimates that remove the compatibility condition on the velocity, while a mild compatibility condition on the phase-field variable \(\chi_0\) is retained due to the stronger degeneracy \(\rho^2\chi_t\). Moreover, in contrast to existing 1D results that impose Dirichlet conditions, we handle the physically relevant Neumann condition \(\partial_{\boldsymbol{n}}\chi\big|_{\partial\Omega}=0\).
\(\bullet\) Unlike the heat-conductive compressible Navier-Stokes equations where the initial data only prescribe \(\rho_0\theta_0\) and strong thermal nonlinearities force a Lagrangian approach [28], our NSAC system prescribes \(\chi_0\) directly with sufficient regularity and involves no thermal effect. Consequently, we are able to prove uniqueness entirely in Eulerian coordinates using singular time-weighted estimates. This simplifies the argument and highlights the structural difference introduced by the Allen-Cahn coupling.
The remainder of this paper is organized as follows. In Section 2, we derive the a priori estimates, which play a fundamental role in the subsequent analysis. Section 3 is devoted to the construction of approximate solutions via a Galerkin scheme. In Section 4, we present the proof of Theorem 1. Finally, Section 5 is devoted to the proof of Theorem 2. Moreover, one can find the proof of Proposition 2 in Appendix.
Our main purpose of this section is to derive some a priori estimates for the strong or smooth solutions \((\rho, u, \chi, \mu)\) to the system 1 , associated with the initial conditions 2 and the boundary conditions 3 , provided that the initial density function has a positive lower bound, \(\rho_0\geq\delta>0\). All these a priori estimates we will obtain are independent of \(\delta>0\).
Throughout the paper, we denote by \(C\) the generic positive constants that depending only on \(A\), \(\gamma\), \(\nu\), \(\lambda\), \(q\), \({\Omega}\) and \(\Phi_0\), but independent of \(\delta>0\). With out loss of generality, we always assume \(0 < T \le 1\). For each \(t \in (0, T)\) and \(q \in (3,6)\), denote \[\begin{align} \varPhi(t) &:= \sup_{0 \le s \le t} \left(\|\rho\|_{W^{1, q}} + \|\nabla u\|^2_{L^2}+ \| \sqrt{s} \sqrt \rho u_t \|^2_{L^2} + \|\chi\|^2_{H^2} + \|\rho \chi_t\|^2_{L^2} + \| \sqrt{s} \nabla\chi_t \|^2_{L^2} \right) \\ & + \int_0^t {\left( \|\nabla^2 u\|^2_{L^2}+ \| \sqrt{\rho} u_t \|^2_{L^2} + \|\sqrt{s} \nabla u_t \|^2_{L^2} + \| \nabla^3 \chi \|^2_{L^2}+\|\nabla\chi_t\|^2_{L^2} \right)} ds + 1. \end{align}\]
The following lemma in [39] provides a weighted Poincaré-type inequality that will be frequently used throughout the paper.
Lemma 1. If \(\varphi \in H^1(\Omega)\), \(\Omega\subset\subset \mathbb{R}^3\), and \(\int_{\Omega} \rho^{\gamma} \,dx \le E_0\) for some positive number \(E_0\), then \[\| \varphi\|_{L^2}^2 \le C(E_0) \left( \| \nabla \varphi \|_{L^2}^2 + \left(\int_{\Omega} \rho |\varphi| dx\right)^2 \right),\] and \[\| \varphi\|_{L^2}^2 \le C(\Omega,E_0)\left(\| \nabla \varphi \|_{L^2}^2 + \|\rho^{\alpha} \varphi\|_{L^2}^2 \right), \quad\alpha=\frac{1}{2}, 1.\]
We begin with the standard energy estimates.
Lemma 2 (Energy Equality). Under the conditions of Theorem 1, it holds for any \(t \in (0,T)\) that \[\label{E0} \begin{align} &\frac{1}{2}\|\sqrt{\rho} u \|_{L^2}^2 + \frac{1}{2}\|\nabla \chi\|_{L^2}^2 + \frac{A}{\gamma-1} \int_{\Omega} \rho^{\gamma} \,dx + \int_{\Omega} \rho F(\chi) \,dx \\ &\quad + \int_0^t ( \|\mu\|_{L^2}^2 + \nu\|\nabla u\|_{L^2}^2 + ( \lambda + \nu ) \|\mathop{\mathrm{div}}u\|_{L^2}^2 ) \,ds = E_0, \end{align}\qquad{(3)}\] where \[E_0 = \frac{1}{2} \| \sqrt{\rho_0} u_0 \|_{L^2}^2 + \frac{A}{\gamma-1} \int_{\Omega} \rho_0^{\gamma} \,dx + \frac{1}{2} \| \nabla\chi_0 \|_{L^2}^2 + \int_{\Omega} \rho_0 F(\chi_0) \,dx.\]
Proof. Multiplying \(\eqref{NSAC}_2\) by \(u\) and \(\eqref{NSAC}_3\) by \(\mu\), integrating over \(\Omega\times(0,t)\), after integration by parts and utilizing the boundary conditions \(\eqref{B1}\), it is easy to arrive at ?? . ◻
The following result is a direct consequence of Lemma 1 and Lemma 2.
Corollary 1. Let \((\rho, u, \chi, \mu)\) be a smooth solution to the system 1 , associated with the initial boundary conditions 2 3 . It holds that \[\begin{align} \sup_{0 \le t \le T} \left( \|\chi\|_{H^1} + \|f(\chi)\|_{L^2} + \|f'(\chi)\|_{L^3} \right) \le C . \end{align}\]
Before establishing the a priori estimates, we present two preparatory lemmas.
Lemma 3. Let \((\rho, u, \chi, \mu)\) be a smooth solution to the system 1 , associated with the initial boundary conditions 2 3 . It holds that \[\begin{align} \int_{0}^{T} \big( \|\nabla u\|_{L^\infty} + \|\nabla^2 u\|_{L^q} \big) \, dt \le C T^{\frac{6-q}{4q}} \varPhi^{\gamma + 1}(T). \end{align}\]
Proof. By the \(W^{2,q}\)-estimate of Lamé equation (cf. [23], [40]), one has \[\begin{align} \|\nabla^{2}u\|_{L^q}& \le C ( \|\rho u_{t}\|_{L^q} + \| \rho ( u \cdot \nabla ) u \|_{L^q} + \| \nabla (P(\rho))\|_{L^q} + \| \Delta \chi \nabla \chi \|_{L^q} ). \end{align}\] It follows from the Hölder, Sobolev, Poincaré, Gagliardo-Nirenberg inequalities that \[\begin{align} &\|\rho u_{t}\|_{L^q} \le C \|\rho u_{t}\|_{L^2}^{\frac{6-q}{2q}} \|\rho u_{t}\|_{L^6}^{\frac{3q-6}{2q}} \le C \|\rho\|_{L^\infty}^{\frac{5q-6}{4q}} \| \sqrt\rho u_{t}\|_{L^2}^{\frac{6-q}{2q}} \| \nabla u_{t}\|_{L^2}^{\frac{3q-6}{2q}}, \\ &\| \rho ( u \cdot \nabla ) u \|_{L^q} \le \|\rho\|_{L^\infty} \|u\|_{L^\infty} {\|\nabla u\|_{L^q}} \le C \|\rho\|_{L^\infty} \|\nabla u\|_{L^2}^{\frac{1}{2}} \|\nabla^2 u\|_{L^2}^{\frac{3}{2}}, \\ &\| \nabla (P(\rho))\|_{L^q} \le C \|\rho\|_{L^\infty}^{\gamma-1} \|\nabla\rho \|_{L^q}, \\ &\| \Delta\chi \nabla\chi \|_{L^q} \le \|\nabla\chi\|_{L^\infty} \|\nabla^2 \chi\|_{L^q} \le C \|\nabla^2 \chi\|_{L^2}^{\frac{1}{2}} \| \nabla^3 \chi \|_{L^2}^{\frac{3}{2}}. \end{align}\] Integrating the above estimates over \((0,T)\), one has \[\begin{align} \int_0^T \|\rho u_{t}\|_{L^q} \,dt &\le \int_0^T \|\rho\|_{L^\infty}^{\frac{5q-6}{4q}} \| \sqrt\rho u_{t}\|_{L^2}^{\frac{6-q}{2q}} \|\sqrt{t} \nabla u_{t}\|_{L^2}^{\frac{3q-6}{2q}} t^{-\frac{3q-6}{4q}} \,dt \\ & \le C {\varPhi^{\frac{5q-6}{4q}}(T) \left( \int_0^T \|\sqrt\rho u_{t}\|^2_{L^2} \,dt \right)^{\frac{6-q}{4q}} \left( \int_0^T \|\sqrt{t} \nabla u_{t}\|^2_{L^2} \,dt \right)^{\frac{3q-6}{4q}} T^{\frac{6-q}{4q}}} \\ & \le C T^{\frac{6-q}{4q}} \varPhi^{\frac{7q-6}{4q}}(T), \\ \int_0^T \| \rho ( u \cdot \nabla ) u \|_{L^q} \,dt &\le \int_0^T \|\rho\|_{L^\infty} \|\nabla u\|_{L^2}^{\frac{1}{2}} \|\nabla^2 u\|_{L^2}^{\frac{3}{2}} \,dt \\ &\le C \varPhi^{\frac{5}{4}}(T) \left( \int_0^T \|\nabla^2 u \|_{L^2}^2 \,dt \right)^{\frac{3}{4}} T^{\frac{1}{4}} \le CT^{\frac{1}{4}}\varPhi^2(T), \\ \int_0^T \| \nabla (P(\rho))\|_{L^q} \,dt &\le \int_0^T \|\rho\|_{L^\infty}^{\gamma-1} \|\nabla\rho \|_{L^q} \,dt \le CT\varPhi^{\gamma}(T),\\ \int_0^T \| \Delta\chi \nabla\chi \|_{L^q} \,dt &\le C \int_0^T \|\nabla^2 \chi\|_{L^2}^{\frac{1}{2}} \| \nabla^3 \chi \|_{L^2}^{\frac{3}{2}} \,dt \\ &\le C \varPhi^{\frac{1}{4}}(T) \left( \int_0^T \|\nabla^3 \chi\|_{L^2}^2 \,dt \right)^{\frac{3}{4}} T^{\frac{1}{4}} \le CT^{\frac{1}{4}}\varPhi^2(T). \end{align}\] Combining the above estimates, we obtain \[\begin{align} \int_0^T \|\nabla^2 u\|_{L^q}^2 \,dt \le C \left(T^{\frac{6-q}{4q}} \varPhi^{\frac{7q-6}{4q}}(T) + T^{\frac{1}{4}}\varPhi^{2}(T) + T \varPhi^{\gamma}(T) + T^{\frac{1}{4}}\varPhi^2(T)\right) \le C T^{\frac{6-q}{4q}} \varPhi^{1 + \gamma}, \end{align}\] where we have used \(\frac{6-q}{4q} \le \frac{1}{4}\) for each \(q \in (3, 6)\), \(T \le 1\), \(\gamma > 1\) and \(\varPhi(T) \ge 1\). It follows from the Sobolev and Poincaré inequalities that \[\begin{align} \int_{0}^{T} \|\nabla u\|_{L^\infty} \,dt \le C \int_0^T \|\nabla^2 u\|_{L^q} \, dt \le C T^{\frac{6-q}{4q}} \varPhi^{\gamma + 1}(T). \end{align}\] This completes the proof. ◻
Lemma 4. Let \((\rho, u, \chi, \mu)\) be a smooth solution to the system 1 , associated with the initial boundary conditions 2 3 . There exists a constant \(\varepsilon_0 \in ( 0, 1 )\), depending only on \(A\), \(\gamma\), \(\nu\), \(\lambda\), \(q\), \(\Omega\) and \(\Phi_0\), such that, if \[\label{Assump1} T^{\frac{6-q}{4q}}\varPhi^{\gamma + 1}(T) \le \varepsilon_0,\qquad{(4)}\] then it holds that \[\begin{align} \sup_{0 \le t \le T} \left( \|\rho\|_{L^\infty} + \|\rho\|_{W^{1,q}} \right) \le C. \end{align}\]
Proof. Multiplying \(\eqref{NSAC}_1\) by \(q\rho^{q-1}\) and integrating over \(\Omega\), it follows from integration by parts that \[\begin{align} \frac{d}{dt}\|\rho\|_{L^q}^q & = (1 - q ) \int_{\Omega} \rho^{q} \mathop{\mathrm{div}}u \,dx \\ &\le C\|\nabla u\|_{L^\infty}\|\rho\|_{L^q}^q . \end{align}\] By the Grönwall inequality, we obtain \[\begin{align} \sup_{0 \le t \le T}\|\rho\|_{L^q} \le \| \rho_0\|_{L^q} \mathop{\mathrm{exp}} \left\{C\int_{0}^{T}\|\nabla u\|_{L^\infty} \, dt \right\}, \end{align}\] and \[\begin{align} \label{sup95rho95infty} \sup_{0 \le t \le T} \|\rho\|_{L^\infty} \le \|\rho_0\|_{L^\infty} \mathop{\mathrm{exp}} \left\{C \int_{0}^{T} \|\nabla u\|_{L^\infty} \,dt \right\}. \end{align}\tag{4}\] Combining 4 with Lemma 3 to get \[\begin{align} \sup_{0 \le t \le T} \|\rho\|_{L^\infty} \le \|\rho_0\|_{L^\infty} \mathop{\mathrm{exp}}\left\{ C T^{\frac{6-q}{4q}}\varPhi^{\gamma + 1}(T) \right\}. \end{align}\] Choosing \(\varepsilon_0\) sufficiently small, then it follows from ?? that \[\begin{align} \sup_{0 \le t \le T} \|\rho\|_{L^\infty} \le 2 \|\rho_0\|_{L^\infty}. \end{align}\]
Differentiating \(\eqref{NSAC}_1\) with respect to \(x\), multiplying the resultant with \(q|\nabla\rho|^{q-2}\nabla\rho\) and integrating over \(\Omega\), it follows from integration by parts that \[\begin{align} \frac{d}{dt} \|\nabla\rho\|_{L^q}^q &= -q \int_{\Omega} ( \nabla\rho \cdot \nabla ) u \cdot \nabla\rho | \nabla\rho |^{q - 2} \,dx + (1 - q ) \int_{\Omega} \mathop{\mathrm{div}}u |\nabla\rho|^q \,dx \\ &\quad - q \int_{\Omega} \rho \nabla \mathop{\mathrm{div}}u \cdot \nabla\rho |\nabla\rho|^{q-2} \,dx \\ & \le C\|\nabla u\|_{L^\infty}\|\nabla \rho\|_{L^q}^q + C\|\rho\|_{L^\infty} \|\nabla \rho\|_{L^q}^{q-1} \|\nabla^2 u \|_{L^q}, \end{align}\] which indicates that \[\begin{align} \label{nabla95rho95q} \frac{d}{dt} \|\nabla\rho\|_{L^q} \le C\|\nabla u\|_{L^\infty} \|\nabla\rho\|_{L^q} + C\|\rho\|_{L^\infty} \|\nabla^2 u \|_{L^q}. \end{align}\tag{5}\] Then, one obtains \[\begin{align} \frac{d}{dt} \|\rho\|_{W^{1, q}} \le C\|\nabla u\|_{L^\infty} \|\rho\|_{W^{1, q}} + C \|\rho\|_{L^\infty} \|\nabla^2 u \|_{L^q} \le C\|\nabla u\|_{L^\infty} \|\rho\|_{W^{1, q}} + C \|\nabla^2 u \|_{L^q}. \end{align}\] By Lemma 3 and the Grönwall inequality, one has \[\begin{align} \|\rho\|_{W^{1,q}} &\le \left( \| \rho_0\|_{W^{1,q}} + \int_0^T \|\nabla^2 u \|_{L^q} \,dt \right) \mathop{\mathrm{exp}}\left\{C\int_{0}^{T}\|\nabla u\|_{L^\infty} \, dt \right\}\\ &\le C\left(1 + T^{\frac{6-q}{4q}}\varPhi^{\gamma + 1}(T) \right) \mathop{\mathrm{exp}}\left\{ C T^{\frac{6-q}{4q}}\varPhi^{\gamma + 1}(T) \right\} \le C. \end{align}\] This completes the proof. ◻
Remark 4. Under the assumption ?? , it follows from \(\varPhi(T) \ge 1\) that \[\label{A2} T \varPhi^{\alpha}(T) \le \big( T^{\frac{6-q}{4q}} \varPhi^{\gamma+1}(T) \big)^{\frac{4q}{6-q}} (\varPhi (T) \big)^{\alpha - \frac{4q ( \gamma + 1 )}{6-q}} \le C,\qquad{(5)}\] as long as \(\alpha \le 4\gamma +4\).
Now we turn to do some a priori estimates on phase variable \(\chi\).
Lemma 5. Under the assumptions of Lemma 4, and if ?? holds, then the following estimate holds \[\sup_{0 \le t \le T} \| \rho \chi_t \|_{L^2}^2 + \int_{0}^{T} \| \nabla \chi_t \|_{L^2}^2 \,dt \le C .\]
Proof. Rewrite the equations \(\eqref{NSAC}_{3,4}\) as follows \[\label{NSAC9534} \rho^2 \chi_t + \rho^2 u \cdot \nabla \chi = \Delta \chi - \rho f(\chi).\tag{6}\] Differentiating 6 with respect to \(t\) gives \[\label{dt95NSAC9534} 2 \rho \rho_t \chi_t + \rho^2 \chi_{tt} + 2 \rho \rho_t u \cdot \nabla \chi + \rho^2 u_t \cdot \nabla \chi + \rho^2 u \cdot \nabla \chi_t = \Delta \chi_t - \rho_t f(\chi) - \rho f'(\chi) \chi_t.\tag{7}\] Multiplying 7 by \(\chi_t\), and integrating over \(\Omega\), using \(\eqref{NSAC}_1\), applying integration by parts, one has
\[\begin{align} \frac{1}{2} \frac{d}{dt} \|\rho \chi_t\|_{L^2}^{2} + \|\nabla\chi_t\|_{L^2}^{2} =& \frac{1}{2} \int_{\Omega} \rho^2 \mathop{\mathrm{div}}u \chi_t^{2} \,dx - 2 \int_{\Omega} \rho^{2} ( u \cdot \nabla \chi_t) \chi_t\,dx + 2 \int_{\Omega} \rho \rho_t (u \cdot \nabla \chi ) \chi_t \,dx\\ & + \int_{\Omega}\rho ( u_t \cdot \nabla \chi ) \rho \chi_t \,dx - \int_{\Omega} \rho ( u \cdot \nabla\chi_t ) f(\chi) \,dx \\ & - \int_{\Omega} \rho f'(\chi) ( u \cdot \nabla\chi ) \chi_t \,dx - \int_{\Omega} \rho f'(x) \chi_t^{2} \,dx \\ =& \sum_{i=1}^{7}{I}_{i}. \end{align}\] Combining Lemma 1, Lemma 4, and Corollary 1, we deduce that \[\begin{align} I_1 + I_2 &\le C \|\rho\|_{L^\infty} ( \|\nabla u\|_{L^3} \|\chi_t\|_{L^6} + \|u\|_{L^\infty} \|\nabla\chi_t\|_{L^2} ) \|\rho \chi_t\|_{L^2} \\ & \le C \|\nabla u\|_{L^2}^{\frac{1}{2}} \|\nabla^2 u\|_{L^2}^{\frac{1}{2}} ( \| \rho \chi_t\|_{L^2} + \|\nabla\chi_t\|_{L^2}) \|\rho \chi_t\|_{L^2} \\ & \le \frac{1}{4} \| \nabla \chi_t \|_{L^2}^2 + C (1 + \|\nabla u\|_{L^2} \|\nabla^2 u\|_{L^2} ) \|\rho \chi_t\|_{L^2}^2, \\ I_3 + I_6 &\le C ( \|\rho_t\|_{L^3} + \|f'(\chi)\|_{L^3} ) \|u\|_{L^\infty} \|\nabla\chi\|_{L^6} \| \rho\chi_t\|_{L^2} \\ &\le C ( \|\rho\|_{L^\infty} \|\nabla u\|_{L^3} + \|\nabla\rho\|_{L^3} \|u\|_{L^\infty} + 1 ) \|u\|_{L^\infty} \|\nabla^2 \chi\|_{L^2} \|\rho \chi_t\|_{L^2} \\ & \le C (1 + \|\nabla u\|_{L^2} \|\nabla^2 u\|_{L^2} ) \|\nabla^2 \chi\|_{L^2} \| \rho\chi_t \|_{L^2}, \\ I_4 &\le \|\rho\|_{L^\infty}^\frac{1}{2} \| \sqrt\rho u_t \|_{L^2} \|\nabla\chi\|_{L^\infty} \| \rho \chi_t\|_{L^2} \\ &\le C \| \sqrt\rho u_t\|_{L^2} \|\nabla^3\chi\|_{L^2}^{\frac{1}{2}} \|\nabla^2\chi\|_{L^2}^{\frac{1}{2}} \|\rho \chi_t\|_{L^2}, \\ I_5 + I_7 &\le \|\rho\|_{L^\infty} \|u\|_{L^\infty} \|\nabla\chi_t\|_{L^2} \|f(\chi)\|_{L^2} + \| \rho \chi_t\|_{L^2} \|f'(\chi)\|_{L^3} \|\chi_t\|_{L^6} \\ & \le C \|\nabla u\|_{L^2}^{\frac{1}{2}} \|\nabla^2 u\|_{L^2}^{\frac{1}{2}} \|\nabla\chi_t\|_{L^2} + C \|\rho \chi_t\|_{L^2} (\|\rho \chi_t\|_{L^2} + \|\nabla\chi_t\|_{L^2}) \\ & \le \frac{1}{4} \|\nabla\chi_t\|_{L^2}^2 + C \|\nabla u\|_{L^2} \|\nabla^2 u\|_{L^2} + C \| \rho \chi_t\|_{L^2}^2. \end{align}\] Then we obtain \[\begin{align} \frac{d}{dt} \|\rho \chi_t\|_{L^2}^{2} + \|\nabla \chi_t \|_{L^2}^{2} & \le C (1 + \|\nabla u\|_{L^2} \|\nabla^2 u\|_{L^2} ) (1+ \|\nabla^2 \chi\|_{L^2}^2 + \| \rho\chi_t \|_{L^2}^2 )\\ &\quad + C \| \sqrt\rho u_t\|_{L^2} \|\nabla^3\chi\|_{L^2}^{\frac{1}{2}} \|\nabla^2\chi\|_{L^2}^{\frac{1}{2}} \|\rho \chi_t\|_{L^2} . \end{align}\] Integrating it over \((0, T )\), using ?? and ?? , together with the fact that \(\varPhi(T) \geq 1\), we deduce from the Hölder inequality that \[\begin{align} &\sup_{ 0 \le t \le T} \|\rho \chi_t\|_{L^2}^{2} + \int_0^T \| \nabla\chi_t \|_{L^2}^{2} \,dt\\ \le& \lim_{\tau \rightarrow 0}\|\rho \chi_t\|_{L^2}^{2} (\tau) + T + T \varPhi(T)+ T^{\frac{1}{2}} \varPhi^{\frac{3}{2}}(T) \left( \int_0^T \| \nabla^2 u \|_{L^2}^2 \,dt \right)^{\frac{1}{2}}\\ & + T^{\frac{1}{4}} \varPhi^{\frac{3}{4}}(T) \left( \int_0^T \| \sqrt{\rho} u_t \|_{L^2}^2 \,dt \right)^{\frac{1}{2}} \left( \int_0^T \| \nabla^3 \chi \|_{L^2}^2 \,dt \right)^{\frac{1}{4}}\\ \le& C + C T \varPhi^2(T) + T^{\frac{1}{4}} \varPhi^{\frac{3}{2}}(T) \le C, \end{align}\] where we have used \[\begin{align} \lim_{\tau \rightarrow 0}\|\rho \chi_t\|_{L^2}^{2} (\tau) &\le C ( \| \rho_0 ( u_0 \cdot \nabla\chi_0 ) \|_{L^2}^2 + \left\| \frac{\Delta\chi_0}{\rho_0} \right\|_{L^2}^2 + \| f(\chi_0) \|_{L^2}^2 ) \\ & \le C ( \|\rho_0\|_{L^\infty}^2 \|u_0\|_{L^6}^2 \| \nabla \chi_0\|_{L^3}^2 +\| h \|_{L^2}^2 + 1 ) \le C. \end{align}\] ◻
Lemma 6. Under the assumptions of Lemma 4, and if ?? holds, then the following estimate holds \[\int_0^T \|\nabla^3\chi\|_{L^2}^2 \,dt \le C .\]
Proof. Applying the gradient operator to 6 , we obtain \[\begin{align} \|\nabla \Delta \chi\|_{L^2} &\le \big( \| 2 \rho \nabla\rho \chi_t \|_{L^2} + \| \rho^2 \nabla\chi_t \|_{L^2} + \| 2 \rho \nabla\rho ( u \cdot \nabla\chi ) \|_{L^2} + \|\rho^2 \nabla u \nabla\chi\|_{L^2} \\ &\quad + \| \rho^2 \nabla^2\chi u \|_{L^2} + \|\nabla\rho f(\chi)\|_{L^2} + \|\rho f'(\chi) \nabla\chi\|_{L^2} \big). \end{align}\] By Lemma 1, Lemma 4 and Lemma 5, a direct computation shows \[\begin{align} \| 2 \rho \nabla\rho \chi_t \|_{L^2} + \| \rho^2 \nabla\chi_t \|_{L^2} &\le 2 \|\rho\|_{L^\infty} \|\nabla\rho\|_{L^3} \|\chi_{t}\|_{L^6} + \|\rho\|_{L^\infty}^2 \|\nabla\chi_{t}\|_{L^2} \\ &\le C (1 + \|\nabla\chi_{t}\|_{L^2} ), \\ \|2 \rho \nabla\rho ( u \cdot \nabla ) \chi \|_{L^2} + \|\rho^2 \nabla u \nabla\chi\|_{L^2} &\le ( 2 \|\rho\|_{L^\infty} \|\nabla\rho\|_{L^3} \|u \|_{L^6} + \|\rho\|_{L^\infty}^2 \|\nabla u\|_{L^2} ) \|\nabla\chi\|_{L^\infty} \\ &\le C \|\nabla u\|_{L^2} \|\nabla^2\chi\|_{L^2}^{\frac{1}{2}} \| \nabla ^3 \chi\|_{L^2} ^{\frac{1}{2}} \\ &\le C \|\nabla u\|_{L^2} \|\nabla^2\chi\|_{L^2}^{\frac{1}{2}} \| \nabla \Delta \chi\|_{L^2} ^{\frac{1}{2}} \\ &\le \frac{1}{2} \|\nabla \Delta \chi\|_{L^2} + C \|\nabla u\|_{L^2}^2 \|\nabla^2\chi\|_{L^2},\\ \|\nabla\rho f(\chi)\|_{L^2} + \|\rho f'(\chi) \nabla\chi\|_{L^2} &\le C (\|\nabla \rho\|_{L^3} \|f(\chi) \|_{L^6} + \|\rho\|_{L^\infty} \|f'(\chi) \|_{L^3}\|\nabla \chi\|_{L^6}) \\ &\le C(1 + \|\nabla^2\chi\|_{L^2}), \end{align}\] and \[\begin{align} \| \rho^2 \nabla^2\chi u \|_{L^2} &\le \|\rho\|_{L^\infty}^2 \|\nabla^2\chi\|_{L^2} \|u\|_{L^\infty}\\ &\le C \|\nabla^2\chi\|_{L^2} \|\nabla u\|_{L^2}^{\frac{1}{2}} \|\nabla^2 u\|_{L^2}^{\frac{1}{2}}. \end{align}\] Then, using the \(H^3\)-estimates of Neumann-Laplacian and 3 , it follows that \[\label{nabla95cube95chi} \begin{align} \|\nabla^3\chi\|_{L^2}^2 &\le C \|\nabla \Delta \chi\|_{L^2}^2 \\ &\le C (1 + \|\nabla\chi_{t}\|_{L^2}^2 + \|\nabla u\|_{L^2}^4 \|\nabla^2 \chi\|_{L^2}^2 + \|\nabla^2\chi\|_{L^2}^2 + \|\nabla^2\chi\|_{L^2}^2 \|\nabla u\|_{L^2} \|\nabla^2 u\|_{L^2}). \end{align}\tag{8}\] Therefore, one deduces by ?? , 8 , Lemma 5 and the Hölder inequality that \[\begin{align} &\int_0^T \|\nabla^3\chi\|_{L^2}^2 \,dt\\ \le&{ C\int_0^T \left(1 + \|\nabla\chi_{t}\|_{L^2}^2 + \|\nabla u\|_{L^2}^4 \|\nabla^2 \chi\|_{L^2}^2 + \|\nabla^2\chi\|_{L^2}^2 + \|\nabla^2\chi\|_{L^2}^2 \|\nabla u\|_{L^2} \|\nabla^2 u\|_{L^2} \right) \,dt }\\ \le& C + C T \varPhi^3(T) + T \varPhi(T) + T^{\frac{1}{2}} \varPhi^2(T) \le C. \end{align}\] This completes the proof. ◻
The following a priori estimates are derived by treating the convection term \(\| \rho ( u \cdot \nabla ) \chi \|_{L^2}^2\) as an energy, without requiring high regularity of \(u_t\).
Lemma 7. Under the assumptions of Lemma 4 and assuming that ?? holds, we have \[\sup_{0 \le t \le T} \left( \|\nabla u\|_{L^2}^2 + \|\mu\|_{L^2}^2 + \| \rho ( u \cdot \nabla ) \chi \|_{L^2}^2 \right) + \int_0^T ( \| \sqrt{\rho} u_t \|_{L^2}^2 + \| \nabla^2u \|_{L^2}^2 ) \,dt \le C .\]
Proof. By Lemma 4 and \(\eqref{NSAC}_2\), one deduces by the standard \(H^2\)-estimate for the Lamé system that \[\begin{align} \|\nabla^2 u\|_{L^2}^2 &\le C ( \| \rho u_t \|_{L^2}^2 + \| \rho ( u \cdot \nabla ) u \|_{L^2}^2 + \|\nabla P(\rho)\|_{L^2}^2 + \| \Delta\chi \nabla\chi \|_{L^2}^2 ) \\ &\le C ( \|\rho\|_{L^\infty} \| \sqrt{\rho} u_t \| _{L^2}^2 + \|\rho\|_{L^\infty}^2 \|u\|_{L^6}^2 \|\nabla u\|_{L^3}^2 + \|\rho\|_{L^\infty}^{2\gamma-2} \|\nabla\rho\|_{L^2}^2 + \|\nabla\chi\|_{L^\infty}^2 \|\nabla^2\chi\|_{L^2}^2 ) \\ &\le C (1+ \|\sqrt{\rho} u_t\| _{L^2}^2 + \|\nabla u\|_{L^2}^3 \|\nabla^2 u\|_{L^2} + \|\nabla^2\chi\|_{L^2}^3 \| \nabla^3 \chi \|_{L^2} )\\ &\le \frac{1}{2} \|\nabla^2 u\|_{L^2}^2 + C (1+ \| \sqrt{\rho} u_t \| _{L^2}^2 + \|\nabla u\|_{L^2}^6 + \|\nabla^2\chi\|_{L^2}^3 \| \nabla^3 \chi \|_{L^2} ), \end{align}\] thus we have \[\label{lem:u1:1} \|\nabla^2 u \|_{L^2}^2 \le C (1 +\| \sqrt{\rho} u_t \| _{L^2}^2 + \|\nabla u\|_{L^2}^6 + \|\nabla^2\chi\|_{L^2}^3 \|\nabla^3 \chi\|_{L^2}).\tag{9}\] From \(\eqref{NSAC}_1\) and 6 , after integration by parts, one obtains \[\begin{align} &\int_{\Omega} \nabla P(\rho) \cdot u_{t} \,dx = -\frac{d}{dt} \int_\Omega P(\rho) \mathop{\mathrm{div}}u \,dx + \int_\Omega P'(\rho) \rho_t \mathop{\mathrm{div}}u \,dx, \\ &\int_\Omega P'(\rho) \rho_t \mathop{\mathrm{div}}u \,dx = -\int_\Omega( \nabla P(\rho) \cdot u ) \mathop{\mathrm{div}}u \,dx - \gamma \int_\Omega P(\rho) | \mathop{\mathrm{div}}u |^2 \,dx ,\\ &- \int_\Omega\mathop{\mathrm{div}}( \nabla\chi \otimes \nabla\chi - \frac{ | \nabla\chi |^2}{2} \mathbb{I} ) \cdot u_t \,dx = - \frac{d}{dt} \int_\Omega\Delta \chi \nabla\chi \cdot u \,dx - \int_\Omega(\nabla \chi_t \cdot \nabla ) u \cdot \nabla \chi \,dx\\ &\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad- \int_\Omega(\nabla \chi \cdot \nabla ) u \cdot \nabla \chi_t \,dx + \int_\Omega\nabla \chi \cdot \nabla \chi_t \mathop{\mathrm{div}}u \,dx, \end{align}\] and \[\begin{align} - \frac{d}{dt} \int_\Omega \Delta \chi \nabla\chi \cdot u \,dx &= - \frac{d}{dt} \int_\Omega ( \rho^2\chi_t + \rho^2 ( u \cdot \nabla\chi ) + \rho F'(\chi) ) ( u\cdot \nabla\chi ) \,dx \\ &= - \frac{1}{2} \frac{d}{dt} \left( \| \rho ( u \cdot \nabla\chi ) \|_{L^2}^2 + \|\mu\|_{L^2}^2 \right) \\ &\quad+ \frac{d}{dt} \left( \frac{1}{2} \|\rho \chi_t\|_{L^2}^2 - \int_\Omega\rho F'(\chi) (u \cdot \nabla\chi )\,dx \right), \end{align}\] where we have used the identity \[\begin{align} \|\mu\|_{L^2}^2 = \|\rho \chi_t\|_{L^2}^2 + 2 \int_\Omega\rho^2 \chi_t ( u \cdot \nabla\chi ) \,dx + \| \rho ( u \cdot \nabla\chi ) \|_{L^2}^2. \end{align}\] Multiplying \(\eqref{NSAC}_2\) by \(u_{t}\), one deduces by the above identities that \[\begin{align}\label{lem:u1:2} &\frac{1}{2} \frac{d}{dt} \left( \nu \|\nabla u\|_{L^2}^2 + (\lambda + \nu ) \|\mathop{\mathrm{div}}u\|_{L^2}^2 + \|\mu\|_{L^2}^2 + \| \rho ( u \cdot \nabla\chi ) \|_{L^2}^2 \right) + \|\sqrt{\rho} u_t\|_{L^2}^2\\ =& - \int_{\Omega} \rho ( u \cdot \nabla ) u \cdot u_t \,dx + \gamma \int_{\Omega} P(\rho) | \mathop{\mathrm{div}}u|^2 \,dx + \int_{\Omega} \nabla P(\rho) \cdot u \mathop{\mathrm{div}}u \,dx \\ &- \int_\Omega( \nabla\chi_t \cdot \nabla ) u \cdot \nabla\chi \,dx - \int_\Omega( \nabla\chi \cdot \nabla ) u \cdot \nabla\chi_t \,dx + \int_\Omega\nabla\chi \cdot \nabla\chi_t \mathop{\mathrm{div}}u \,dx\\ & + \frac{d}{dt} \left( \int_{\Omega} P(\rho) \mathop{\mathrm{div}}u \,dx + \frac{1}{2} \|\rho \chi_t\|_{L^2}^2 - \int_\Omega\rho F'(\chi) (u \cdot \nabla\chi )\,dx \right)\\ =& \sum_{i=1}^6 J_i + G'(t), \end{align}\tag{10}\] where \[G(t) := \int_{\Omega} P(\rho) \mathop{\mathrm{div}}u \,dx + \frac{1}{2} \|\rho \chi_t\|_{L^2}^2 - \int_\Omega\rho F'(\chi) (u \cdot \nabla\chi )\,dx.\] For each \(\eta > 0\), it follows from Lemma 4 that, \[\begin{align} J_1 &\le \|\rho\|_{L^\infty}^{\frac{1}{2}} \| \sqrt{\rho}u_t \|_{L^2} \|u\|_{L^6} \|\nabla u\|_{L^3} \le C \| \sqrt{\rho} u_t \|_{L^2} \|\nabla u\|_{L^2}^{\frac{3}{2}} \|\nabla^2 u\|_{L^2}^{\frac{1}{2}} \\ &\le \frac{1}{4} \| \sqrt{\rho} u_t \|_{L^2}^2 + \eta \|\nabla^2 u\|_{L^2}^2 + C_\eta \|\nabla u\|_{L^2}^6, \\ J_2 + J_3 &\le C \|\rho\|_{L^\infty}^{\gamma} \|\nabla u\|_{L^2}^2 + C \|\rho\|_{L^\infty}^{\gamma-1} \|\nabla\rho\|_{L^3} \|u\|_{L^6} \|\nabla u\|_{L^2} \le C \|\nabla u\|_{L^2}^2, \\ \sum_{i=4}^6 {J_i} &\le C \|\nabla\chi_t\|_{L^2} \|\nabla\chi\|_{L^\infty} \|\nabla u\|_{L^2} \le C \|\nabla\chi_t\|_{L^2} \|\nabla^2\chi\|_{L^2}^{\frac{1}{2}} \| \nabla^3 \chi\|_{L^2}^{\frac{1}{2}}\|\nabla u\|_{L^2}. \end{align}\] Substituting the above estimates and 9 into 10 , we obtain \[\begin{align} &\frac{1}{2} \frac{d}{dt} \left( \nu \|\nabla u\|_{L^2}^2 + (\lambda + \nu ) \|\mathop{\mathrm{div}}u\|_{L^2}^2 + \|\mu\|_{L^2}^2 + \| \rho ( u \cdot \nabla\chi ) \|_{L^2}^2 \right) + \frac{3}{4} \|\sqrt{\rho} u_t\|_{L^2}^2 + \eta \|\nabla^2 u\|_{L^2} \\ \le& 2 \eta \|\nabla^2 u\|_{L^2}^2 + C_\eta \|\nabla u\|_{L^2}^6 + C ( \|\nabla\chi_t\|_{L^2} \|\nabla^2\chi\|_{L^2}^{\frac{1}{2}} \|\nabla^3\chi\|_{L^2}^{\frac{1}{2}} \|\nabla u\|_{L^2} + \|\nabla u\|_{L^2}^2 ) + G'(t)\\ \le& C \eta \| \sqrt{\rho} u_t \| _{L^2}^2 + \tilde{C}_{\eta} \|\nabla u\|_{L^2}^6 + C (\|\nabla^2\chi\|_{L^2}^3 \|\nabla^3 \chi\|_{L^2} + \|\nabla\chi_t\|_{L^2} \|\nabla^2\chi\|_{L^2}^{\frac{1}{2}} \|\nabla^3\chi\|_{L^2}^{\frac{1}{2}} \|\nabla u\|_{L^2} \\ &+ \|\nabla u\|_{L^2}^2 + 1 ) + G'(t). \end{align}\] Choosing \(\eta\) sufficiently small, one deduces by the Young inequality that \[\label{lem:u1:3} \begin{align} &\frac{d}{dt} \left( \nu \|\nabla u\|_{L^2}^2 + (\lambda + \nu ) \|\mathop{\mathrm{div}}u\|_{L^2}^2 + \|\mu\|_{L^2}^2 + \| \rho ( u \cdot \nabla\chi ) \|_{L^2}^2 \right) + \|\sqrt{\rho} u_t\|_{L^2}^2 + 2\eta \|\nabla^2 u\|_{L^2}\\ \le&C ( \|\nabla u\|_{L^2}^6 + \|\nabla^2\chi\|_{L^2}^6 + \|\nabla^3 \chi\|_{L^2}^2 + \|\nabla \chi_t\|_{L^2}^2 + \|\nabla u\|_{L^2}^8 + \|\nabla^2\chi\|_{L^2}^4 + 1 ) + 2 G'(t). \end{align}\tag{11}\] It follows from ?? , Lemma 4 and Lemma 5 that \[\begin{align} G(t) &\le \|P(\rho)\|_{L^2} \|\nabla u\|_{L^2} + \frac{1}{2} \|\rho \chi_t\|_{L^2} + \| \rho ( u \cdot \nabla ) \chi \|_{L^2} \|f(\chi)\|_{L^2}\\ &\le \frac{\nu}{4} \|\nabla u\|_{L^2}^2 + \frac{1}{4} \| \rho ( u \cdot \nabla ) \chi \|_{L^2}^2 + C, \\ |G(0)| &\le \|P(\rho_0)\|_{L^2} \|\nabla u_0\|_{L^2} + \frac{1}{2} \|\rho \chi_t\|_{L^2}(0) + \| \rho_0 ( u_0 \cdot \nabla ) \chi_0 \|_{L^2} \| f(\chi_0) \|_{L^2}\\ &\le C + \|\rho_0\|_{L^3} \|u_0\|_{L^6} \|\nabla \chi_0\|_{L^6}\le C. \end{align}\] Plugging the above estimates into 11 and integrating over \((0,T)\), it follows from ?? , ?? , Lemma 5 and Lemma 6 that \[\begin{align} &\sup_{0 \le t \le T} \left( \|\nabla u\|_{L^2}^2 +\|\mu\|_{L^2}^2 + \| \rho ( u \cdot \nabla ) \chi \|_{L^2}^2 \right) + \int_0^T ( \| \sqrt{\rho} u_t \|_{L^2}^2 + \| \nabla^2u \|_{L^2}^2 ) \,dt\\ \le& C \int_0^T ( \|\nabla u\|_{L^2}^6 + \|\nabla^2\chi\|_{L^2}^6 + \| \nabla^3\chi\|_{L^2}^2 + \|\nabla\chi_t\|_{L^2}^2 + \|\nabla u\|_{L^2}^8 + \|\nabla^2\chi\|_{L^2}^4) \,dt + C \\ \le& C T \varPhi^{4}(T) + C T \varPhi^3(T) + C T \varPhi^2(T) + C \\ \le& C. \end{align}\] This completes the proof. ◻
Corollary 2. Under the assumptions of Lemma 4 and assuming that ?? holds, we have \[\sup_{0 \le t \le T} \left( \|\rho_t\|_{L^2} + \|\nabla^2\chi\|_{L^2} + \|\chi\|_{L^\infty} \right) \le C .\]
Proof. It follows from \(\eqref{NSAC}_1\), \(\eqref{NSAC}_4\), Lemma 1, Lemma 4, the Sobolev and Poincaré inequalities that \[\begin{align} &\|\rho_t\|_{L^2} \le \|\nabla\rho\|_{L^3}\|u\|_{L^6} + \|\rho\|_{L^\infty} \|\nabla u\|_{L^2} \le C \|\nabla u\|_{L^2} ,\\ &\|\nabla^2\chi\|_{L^2} \le \|\rho\|_{\infty} (\|\mu\|_{L^2} + \|f(\chi)\|_{L^2} ) \le C (1 + \|\mu\|_{L^2} ), \\ &\|\chi\|_{L^\infty} \le C \|\chi\|_{H^2} \le C (1 + \|\nabla^2\chi\|_{L^2} ), \end{align}\] By using Lemma 7, the proof is complete. ◻
In what follows, we do some time-weighted a priori estimates.
Lemma 8. Under the assumptions of Lemma 4 and assuming that ?? holds, there holds \[\sup_{0 \le t \le T} \| \sqrt{t}\sqrt{\rho}u_t \|_{L^2}^2 + \int_0^T \|\sqrt{t} \nabla u_t \|_{L^2}^2 \,dt \le C .\]
Proof. Differentiating \(\eqref{NSAC}_2\) with respect to \(t\), we have \[\begin{align}\label{lem:u2:1} &\rho_t u_t + \rho u_{tt} + \rho_t ( u \cdot \nabla ) u + \rho ( u_t \cdot \nabla ) u + \rho( u \cdot \nabla ) u_t + \nabla \Big(P'(\rho) \rho_t \Big) \\ &\quad = \mathcal{L} u_t - \mathop{\mathrm{div}}( \nabla\chi_t \otimes \nabla\chi + \nabla\chi \otimes \nabla\chi_t - \nabla\chi \cdot \nabla\chi_t \mathbb{I} ). \end{align}\tag{12}\] It follows from integration by parts and \(\eqref{NSAC}_1\) that \[\begin{align} &- \int_{\Omega} \rho_t | u_t |^2 \,dx = \int_{\Omega} \mathop{\mathrm{div}}( \rho u ) | u_t |^2 \,dx = - \int_{\Omega} \rho u \cdot \nabla|u_t|^2 \,dx, \\ &- \int_{\Omega} u_{t} \cdot\nabla \Big(P'(\rho) \rho_t \Big) \,dx = - A \gamma \int_{\Omega} \rho^{\gamma-1} \rho_t \mathop{\mathrm{div}}u_{t} \,dx. \end{align}\] Testing 12 with \(u_{t}\), and using the above identities, one deduces that \[\begin{align} &\frac{1}{2} \frac{d}{dt} \| \sqrt\rho u_{t} \|_{L^2}^{2} + \nu \|\nabla u_{t} \|_{L^2}^{2} + ( \lambda + \nu ) \| \mathop{\mathrm{div}}u_{t} \|_{L^2}^{2} \\ =& - \int_{\Omega} \rho u \cdot \nabla|u_t|^2 \,dx - \int_{\Omega} \rho_t ( u\cdot\nabla ) u \cdot u_t \,dx - \int_{\Omega} \rho ( u_t \cdot \nabla ) u \cdot u_t \,dx \\ & + A \gamma \int_{\Omega} \rho^{\gamma-1} \rho_t \mathop{\mathrm{div}}u_t \,dx + \int_{\Omega} ( \nabla\chi_t \cdot \nabla) u_t \cdot \nabla\chi \,dx + \int_{\Omega} ( \nabla\chi \cdot \nabla) u_t \cdot \nabla\chi_t \,dx \\ & - \int_{\Omega} \nabla\chi \cdot \nabla\chi_t \mathop{\mathrm{div}}u_t \,dx. \end{align}\] Multiplying the above equation by \(t\), then we have \[\begin{align} \label{lem:u2:2} &\frac{1}{2} \frac{d}{dt} \| \sqrt{t} \sqrt\rho u_{t} \|_{L^2}^{2} + \nu \|\sqrt{t} \nabla u_{t}\|_{L^2}^{2} + ( \lambda + \nu ) \| \sqrt{t} \mathop{\mathrm{div}}u_{t} \|_{L^2}^{2} \\ =& \frac{1}{2}\|\sqrt\rho u_{t} \|_{L^2}^{2} - t \int_{\Omega} \rho u \cdot \nabla|u_t|^2 \,dx - t \int_{\Omega} \rho_t ( u \cdot \nabla ) u \cdot u_t \,dx \\ &- t \int_{\Omega} \rho ( u_t \cdot \nabla ) u \cdot u_t \,dx + A \gamma t \int_{\Omega} \rho^{\gamma-1} \rho_t \mathop{\mathrm{div}}u_t \,dx + t \int_{\Omega} ( \nabla\chi_t \cdot \nabla) u_t \cdot \nabla\chi \,dx \\ & + t \int_{\Omega} ( \nabla\chi \cdot \nabla) u_t \cdot \nabla\chi_t \,dx - t \int_{\Omega} \nabla\chi \cdot \nabla\chi_t \mathop{\mathrm{div}}u_t\,dx\\ =&\sum_{i=1}^{8}{K}_i. \end{align}\tag{13}\] Using the Hölder, Sobolev, Young, Poincaré and Gagliardo-Nirenberg inequalities, it follows from Lemma 4, Lemma 7 and Corollary 2 that \[\begin{align} K_2+ K_4 &\le C \|\rho\|_{L^\infty}^{\frac{1}{2}} \| \sqrt{t} \sqrt\rho u_t\|_{L^2} ( \|u\|_{L^\infty} \|\sqrt{t} \nabla u_t\|_{L^2} + \|\nabla u\|_{L^3}\| \sqrt{t} u_t\|_{L^6} ) \\ &\le C \|\nabla u\|_{L^2}^{\frac{1}{2}} \|\nabla^2 u\|_{L^2}^{\frac{1}{2}} \| \sqrt{t} \sqrt\rho u_t \|_{L^2}\|\sqrt{t} \nabla u_t\|_{L^2} \\ &\le \frac{\nu}{6} \| \sqrt{t} \nabla u_t\|_{L^2}^2 + C \|\nabla^2 u\|_{L^2} \|\sqrt{t} \sqrt\rho u_t \|_{L^2}^2, \\ K_3 + K_5 &\le C t \| \rho_t\|_{L^2}( \|u\|_{L^6} \|\nabla u\|_{L^6} \| u_t \|_{L^6} + \|\rho\|_{L^\infty}^{\gamma-1} \| \nabla u_t \|_{L^2} ) \\ &\le C\sqrt{t} ( \|\nabla u\|_{L^2} \|\nabla^2 u\|_{L^2} + 1 ) \| \sqrt{t}\nabla u_t\|_{L^2} \\ & \le \frac{\nu}{6} \|\sqrt{t} \nabla u_t\|_{L^2}^2 + C(1 + \|\sqrt{t} \nabla^2 u\|_{L^2}^2 ),\\ K_6 + K_7 + K_8 &\le C \|\sqrt{t} \nabla\chi_t\|_{L^2} \|\nabla\chi\|_{L^\infty} \| \sqrt{t} \nabla u_t\|_{L^2} \\ &\le C \|\sqrt{t} \nabla\chi_t\|_{L^2} \|\nabla^2\chi\|_{L^2}^{\frac{1}{2}} \|\nabla^3\chi\|_{L^2}^{\frac{1}{2}} \| \sqrt{t} \nabla u_t \|_{L^2} \\ &\le \frac{\nu}{6} \| \sqrt{t} \nabla u_t\|_{L^2}^2 + C \| \nabla^3 \chi\|_{L^2} \|\sqrt{t} \nabla\chi_t\|_{L^2}^2. \end{align}\] Substituting the above estimates into 13 , we obtain \[\begin{align} &\frac{d}{dt}\|\sqrt{t}\sqrt\rho u_{t} \|_{L^2}^{2} + \nu \|\sqrt{t}\nabla u_{t} \|_{L^2}^{2} + (\lambda+\nu) \|\sqrt{t} \mathop{\mathrm{div}}u_{t} \|_{L^2}^{2} \\ \le& \| \sqrt\rho u_{t} \|_{L^2}^{2} + C (1 + \| \sqrt{t} \nabla^2 u \|_{L^2}^2 ) + C \|\nabla^2 u\|_{L^2} \| \sqrt{t} \sqrt\rho u_t \|_{L^2}^2 + C \| \nabla^3 \chi\|_{L^2} \|\sqrt{t} \nabla\chi_t\|_{L^2}^2. \end{align}\] Integrating it over \((0,T)\), one deduces by Lemma 6, Lemma 7 and ?? that \[\begin{align} &\sup_{0 \le t \le T} \| \sqrt{t} \sqrt{\rho}u_t \|_{L^2}^2 + \nu \int_0^T \|\sqrt{t} \nabla u_t \|_{L^2}^2 \,dt\\ \le& C \int_0^T (1 + \| \sqrt\rho u_{t} \|_{L^2}^{2} + \|\nabla^2 u\|_{L^2}^2 ) \,dt + T^{\frac{1}{2}} \varPhi(T) \left( \big( \int_0^T \|\nabla^2 u\|_{L^2}^2 \,dt \big)^{\frac{1}{2}} + \big( \int_0^T\| \nabla^3 \chi\|_{L^2} \,dt \big)^{\frac{1}{2}} \right) \\ \le& C (1 + T^{\frac{1}{2}} \varPhi(T)) \le C. \end{align}\] The proof is complete. ◻
Lemma 9. Under the assumptions of Lemma 4 and assuming that ?? holds, there holds \[\sup_{0 \le t \le T} \|\sqrt{t}\nabla\chi_t\|_{L^2}^2 + \int_0^T \|\sqrt{t} \rho \chi_{tt} \|_{L^2}^2 \,dt \le C .\]
Proof. Differentiating 6 with respect to \(t\), one obtains \[\label{lem:chi3:1} 2 \rho \rho_t \chi_t + \rho^2 \chi_{tt} + 2 \rho \rho_t u \cdot \nabla \chi + \rho^2 u_t \cdot \nabla \chi + \rho^2 u \cdot \nabla \chi_t = \Delta \chi_t - \rho_t f(\chi) - \rho f'(\chi) \chi_t.\tag{14}\] Note that \[\begin{align} - \int_{\Omega} \rho_{t} f(\chi) \chi_{tt} dx \, &= - \frac{d}{dt} \int_{\Omega} \rho_t f(\chi)\chi_t \,dx + \int_{\Omega} \rho_{tt} f(\chi) \chi_t \,dx + \int_{\Omega} \rho_t ( f'(\chi) \chi_t ) \chi_t \,dx \\ &= - \frac{d}{dt}\int_{\Omega} \rho_t f(\chi)\chi_t \,dx - \int_{\Omega} \mathop{\mathrm{div}}(\rho_t u + \rho u_t) f(\chi) \chi_t \,dx + \int_{\Omega} \rho_t f'(\chi) \chi_t^2 \,dx \\ &= - \frac{d}{dt}\int_{\Omega} \rho_t f(\chi)\chi_t \,dx + \int_{\Omega} (\rho_t u + \rho u_t) ( f'(\chi) \nabla\chi \chi_t + f(\chi) \nabla\chi_t ) \,dx \\ &\quad+ \int_{\Omega} \rho_t f'(\chi) \chi_t^2 \,dx. \end{align}\] Multiplying 14 by \(\chi_{tt}\) and utilizing the above identities, it follows from integration by parts that \[\begin{align}\label{lem:chi3:2} &\frac{d}{dt} \left( \frac{1}{2}\|\nabla\chi_t\|_{L^2}^2 + \int_{\Omega} \rho_t f(\chi) \chi_t \,dx \right) + \| \rho \chi_{tt} \|_{L^2}^2 \\ =& -2 \int_{\Omega} \rho \rho_t \chi_t \chi_{tt} \,dx - 2 \int_{\Omega} \rho \rho_t ( u \cdot \nabla\chi ) \chi_{tt} \,dx - \int_{\Omega} \rho^2 ( u_t \cdot \nabla \chi ) \chi_{tt} \,dx \\ &- \int_{\Omega} \rho^2 ( u \cdot \nabla\chi_t ) \chi_{tt} \,dx + \int_{\Omega} \rho_t f'(\chi) ( u \cdot \nabla\chi ) \chi_t \,dx + \int_{\Omega} \rho_t f(\chi) (u \cdot \nabla\chi_t ) \,dx \\ & + \int_{\Omega} \rho f'(\chi) ( u_t \cdot \nabla\chi ) \chi_t \,dx + \int_{\Omega} \rho f(\chi) ( u_t \cdot \nabla\chi_t) \,dx + \int_{\Omega} \rho_t f'(\chi) \chi_t^2 \,dx \\ & - \int_{\Omega} \rho f'(\chi) \chi_t \chi_{tt} \,dx. \end{align}\tag{15}\] Multiplying the above identity with \(t\) to obtain \[\label{lem:chi3:3} \begin{align} &\frac{1}{2} \frac{d}{dt} \|\sqrt{t} \nabla\chi_t\|_{L^2}^2 + \| \sqrt{t} \rho\chi_{tt}\|_{L^2}^2 \\ =& \frac{d}{dt} [ t H(t) ] - H(t) + \frac{1}{2}\|\nabla\chi_t\|_{L^2} - 2 t \int_{\Omega} \rho \rho_t \chi_t \chi_{tt} \,dx - 2 t \int_{\Omega} \rho \rho_t ( u \cdot \nabla\chi ) \chi_{tt} \,dx \\ & - t \int_{\Omega} \rho^2 ( u_t \cdot \nabla \chi ) \chi_{tt} \,dx - t \int_{\Omega} \rho^2 ( u \cdot \nabla\chi_t ) \chi_{tt} \,dx + t \int_{\Omega} \rho_t f'(\chi) ( u \cdot \nabla\chi ) \chi_t \,dx \\ & + t \int_{\Omega} \rho_t f(\chi) (u \cdot \nabla\chi_t ) \,dx + t \int_{\Omega} \rho f'(\chi) ( u_t \cdot \nabla\chi ) \chi_t \,dx + t \int_{\Omega} \rho f(\chi) ( u_t \cdot \nabla\chi_t) \,dx \\ & + t \int_{\Omega} \rho_t f'(\chi) \chi_t^2 \,dx - t \int_{\Omega} \rho f'(\chi) \chi_t \chi_{tt} \,dx \\ =& \frac{d}{dt} [ t H(t) ] - H(t) + \frac{1}{2}\|\nabla\chi_t\|_{L^2} + \sum_{i=1}^{10}{L_i}, \end{align}\tag{16}\] where \(H(t) := - \int_{\Omega} \rho_t f(\chi)\chi_t \,dx\). It follows from Lemma 1, Lemma 4–8, the Hölder, Poincaré, Young and Gagliardo-Nirenberg inequalities that \[\begin{align} | H(t) | &\le \|\rho_t\|_{L^2} \|f(\chi)\|_{L^\infty} \|\chi_t\|_{L^2}\\ &\le C (\|\rho\chi_{t}\|_{L^2} + \|\nabla\chi_{t}\|_{L^2}) \\ &\le C (1 + \|\nabla\chi_{t}\|_{L^2} ), \\ L_1 + L_2 &\le \sqrt{t} \|\sqrt{t} \rho \chi_{tt}\|_{L^2} \|\rho_t\|_{L^3} ( \| \chi_{t} \|_{L^6} + \|u\|_{L^\infty} \|\nabla\chi\|_{L^6} ) \\ &\le \sqrt{t} \|\sqrt{t} \rho \chi_{tt}\|_{L^2} \|\nabla^2 u\|_{L^2}^{\frac{1}{2}} (\|\rho\chi_{t}\|_{L^2} + \|\nabla\chi_{t}\|_{L^2} + \|u\|_{L^\infty} \|\nabla^2\chi\|_{L^2} ) \\ &\le \sqrt{t} \|\sqrt{t} \rho \chi_{tt}\|_{L^2} \|\nabla^2 u\|_{L^2}^{\frac{1}{2}} (1 + \|\nabla\chi_t\|_{L^2} + \|\nabla^2 u\|_{L^2}^{\frac{1}{2}} ) \\ &\le \frac{1}{8} \|\sqrt{t} \rho \chi_{tt}\|_{L^2}^2 + C \|\nabla^2 u\|_{L^2} (1 + \|\sqrt{t}\nabla\chi_t\|_{L^2}^2) + C \|\nabla^2 u\|_{L^2}^2, \\ L_3 &\le \|\rho\|_{L^\infty}^{\frac{1}{2}} \|\sqrt{t} \sqrt\rho u_{t}\|_{L^2} \|\nabla\chi\|_{L^\infty} \|\sqrt{t} \rho \chi_{tt}\|_{L^2} \\ &\le C \|\nabla^2\chi\|_{L^2}^{\frac{1}{2}} \| \nabla^3 \chi \|_{L^2}^{\frac{1}{2}} \|\sqrt{t} \rho \chi_{tt}\|_{L^2} \\ & \le \frac{1}{8} \|\sqrt{t} \rho \chi_{tt}\|_{L^2}^2 + C \|\nabla^3\chi\|_{L^2}, \\ L_4 &\le \|\sqrt{t} \rho\chi_{tt}\|_{L^2} \|\rho\|_{L^\infty} \|u\|_{L^\infty} \|\sqrt{t} \nabla\chi_t\|_{L^2} \\ &\le C \| \rho \chi_{tt} \|_{L^2} \|\nabla u\|_{L^2}^{\frac{1}{2}} \|\nabla^2 u\|_{L^2}^{\frac{1}{2}} \|\sqrt{t} \nabla\chi_t\|_{L^2} \\ &\le \frac{1}{8} \|\sqrt{t} \rho\chi_{tt}\|_{L^2}^2 + C \|\nabla^2 u\|_{L^2} \|\sqrt{t} \nabla\chi_t\|_{L^2}^2, \\ L_5 + L_6 &\le C \sqrt{t} \|\rho_t\|_{L^2} \|u\|_{L^\infty} ( \|f'(\chi)\|_{L^6} \|\nabla\chi\|_{L^6} \|\sqrt{t} \chi_t\|_{L^6} + \|f(\chi)\|_{L^\infty} \|\sqrt{t} \nabla\chi_t\|_{L^2}) \\ &\le C \|\nabla u\|_{L^2}^{\frac{1}{2}} \|\nabla^2 u\|_{L^2}^{\frac{1}{2}} (\|\rho\chi_{t}\|_{L^2} + \|\sqrt{t} \nabla\chi_t\|_{L^2} ) \\ &\le C(1 + \|\nabla^2 u\|_{L^2} + \|\sqrt{t} \nabla\chi_t\|_{L^2}^2 ), \\ L_7 +L_8&\le C \|\rho\|_{L^\infty}^{\frac{1}{2}} \|\sqrt{t} \sqrt\rho u_{t}\|_{L^2} ( \|f'(\chi)\|_{L^6} \|\nabla\chi\|_{L^6} \|\sqrt{t} \chi_t\|_{L^6} + \|f(\chi)\|_{L^\infty} \|\sqrt{t} \nabla\chi_t\|_{L^2}) \\ &\le C (\|\rho\chi_{t}\|_{L^2}+ \|\sqrt{t} \nabla\chi_t\|_{L^2} ) \\ &\le C (1 + \|\sqrt{t} \nabla\chi_t\|_{L^2}^2 ), \end{align}\] and \[\begin{align} L_9 + L_{10}&\le C (\|f'(\chi)\|_{L^6} \|\rho_t\|_{L^2} \|\sqrt{t} \chi_t\|_{L^6}^2 + \|f'(\chi)\|_{L^3} \|\sqrt{t}\rho\chi_{tt}\|_{L^2} \|\sqrt{t}\chi_{t}\|_{L^6}) \\ &\le \frac{1}{8} \|\sqrt{t} \rho\chi_{tt}\|_{L^2}^2 + C (\|\rho\chi_t\|_{L^2}^2 + \|\sqrt{t} \nabla\chi_t\|_{L^2}^2) \\ & \le \frac{1}{8} \|\sqrt{t} \rho\chi_{tt}\|_{L^2}^2 + C (1 + \|\sqrt{t} \nabla\chi_t\|_{L^2}^2). \end{align}\] Plugging the above estimates into 16 and using the Young inequality, one obtains \[\begin{align} \label{lem:chi3:4} & \frac{d}{dt} \|\sqrt{t} \nabla\chi_t\|_{L^2}^2 + \|\sqrt{t} \rho \chi_{tt}\|_{L^2}^2 \\ \le& 2 \frac{d}{dt} [ t H(t) ] + C (1 + \|\nabla\chi_t\|_{L^2}^2 + \|\nabla^2 u\|_{L^2}^2 + \| \nabla^3 \chi \|_{L^2}^2 ) + C \|\nabla^2 u\|_{L^2} \|\sqrt{t}\nabla\chi_t\|_{L^2}^2. \end{align}\tag{17}\] By Lemma 1, Lemma 5, Corollary 2, and Sobolev inequality, we have \[\begin{align}\label{lem:chi3:5} 2 t H(t) &\le 2 \sqrt{t} \|\rho_t\|_{L^2} \|f(\chi)\|_{L^3} \|\sqrt{t} \chi_t\|_{L^6}\\ &\le \frac{1}{2} \|\sqrt{t} \nabla \chi_t\|_{L^2}^2 + C. \end{align}\tag{18}\] Integrating 17 over \((0, T)\) and using the Hölder inequality, one deduces by ?? , 18 , and Lemma 7 that \[\begin{align} &\sup_{0 \le t \le T} \|\sqrt{t} \nabla\chi_t\|_{L^2}^2 + \int_0^T \|\sqrt{t} \rho \chi_{tt}\|_{L^2}^2 \,dt\\ \le& 2 \sup_{0 \le t \le T} \left( t H(t) \right) + C(T+\varPhi(T)) + T^{\frac{1}{2}} \varPhi(T) \left( \int_0^T \|\nabla^2 u\|_{L^2}^2 \,dt \right)^{\frac{1}{2}} \\ \le& \frac{1}{2} \sup_{0 \le t \le T} \|\sqrt{t} \nabla \chi_t\|_{L^2}^2 + C. \end{align}\] Therefore, one has \[\sup_{0 \le t \le T} \|\sqrt{t}\nabla\chi_t\|_{L^2}^2 + \int_0^T \|\sqrt{t} \rho \chi_{tt} \|_{L^2}^2 \,dt \le C .\] The proof is complete. ◻
Finally, we summarize the prior estimates obtained previously and derive some additional estimates essential for proving uniqueness.
Proposition 1. Under the assumptions of Lemma 4 and assuming that ?? holds, there holds \[\begin{align} &\varPhi(T) + \sup_{0 \le t \le T} \left( \|\rho_t\|_{L^2}^2 + \| \sqrt{t} \nabla^2 u \|_{L^2}^2 + \| \sqrt{t} \nabla^3\chi \|_{L^2}^2 + \|\mu\|_{L^2}^2 \right) \\ &\quad+ \int_0^T \left( \|\nabla^2 u\|_{L^q} + \|\sqrt{t} \nabla^2 u\|_{L^q}^2 + \|\chi_t\|_{H^1}^2 + \| \sqrt{t} \rho \chi_{tt} \|_{L^2}^2\right) \,dt \le C. \end{align}\]
Proof. It follows from Lemma 3–9 and Corollary 2 that \[\varPhi(T) + \sup_{0 \le t \le T} (\|\rho_t\|_{L^2}^2 + \|\mu\|_{L^2}^2) + \int_0^T \left( \|\nabla^2 u\|_{L^q} + \|\sqrt{t} \rho \chi_{tt}\|_{L^2}^2 \right)\,dt \le C.\] Recalling 8 and 9 , one deduces by the Young inequality that \[\begin{align} \|\nabla^3\chi\|_{L^2}^2 \leq C(1 + \| \nabla\chi_{t}\|_{L^2}^2 + \| \sqrt{\rho} u_t \| _{L^2}^2), \end{align}\] which further indicates that \[\begin{align} \sup_{0 \le t \le T} \| \sqrt{t} \nabla^3\chi \|_{L^2}^2 &\le C \sup_{0 \le t \le T} \left(1 + \| \sqrt{t} \nabla\chi_{t}\|_{L^2}^2 + \| \sqrt{t} \sqrt{\rho} u_t \| _{L^2}^2 \right)\\ &\le C\left(1 + \varPhi(T) \right) \le C. \end{align}\] Moreover, one has \[\begin{align} \sup_{0 \le t \le T} \| \sqrt{t} \nabla^2 u \|_{L^2}^2 &\le C \sup_{0 \le t \le T} \left( \| \sqrt{t} \sqrt{\rho} u_t \| _{L^2}^2 + \|\nabla u\|_{L^2}^6 + \|\nabla^2\chi\|_{L^2}^3 \| \sqrt{t} \nabla^3 \chi\|_{L^2} + 1 \right) \\ &\le C \varPhi(T) + C \varPhi^3(T) + C \varPhi^{\frac{3}{2}} \sup_{0 \le t \le T} \| \sqrt{t} \nabla^3\chi \|_{L^2}^2 \le C. \end{align}\] One also deduces by Lemma 1 and \(\varPhi(T) \le C\) that \[\int_0^T \|\chi_t\|_{H^1}^2 \,dt \le C.\] Besides, it follows from the \(W^{2, q}-\)estimates of \(\eqref{NSAC}_2\), the Hölder, Poincaré, Sobolev inequalities that \[\label{nabla2uq} \begin{align} \|\nabla^{2}u\|_{L^q}&\le C ( \|\rho u_{t}\|_{L^q} + \|\rho ( u \cdot \nabla ) u\|_{L^q} + \|\nabla (P(\rho))\|_{L^q} + \|\Delta \chi \nabla \chi\|_{L^q}) \\ &\le C (\|\rho\|_{L^\infty} \|u_t\|_{L^6} + \|\rho\|_{L^\infty} \|u\|_{L^\infty} \|\nabla u\|_{L^6} \\ &\quad+ \|\nabla \rho\|_{L^q} \|\rho\|_{L^\infty}^{\gamma-1} + \|\nabla \chi\|_{L^\infty} \|\nabla^2\chi\|_{L^q}) \\ &\le C (1 + \|\nabla u_t\|_{L^2} + \|\nabla^2 u\|_{L^2}^{\frac{3}{2}} + \|\nabla^3 \chi\|_{L^2}^{\frac{3}{2}}). \end{align}\tag{19}\] Consequently, one has \[\begin{align} \int_0^T \|\sqrt{t} \nabla^2 u\|_{L^q}^2 \,dt &\le C \int_0^T (1 + \|\sqrt{t} \nabla u_t\|_{L^2}^2 + \|\sqrt{t} \nabla^2 u\|_{L^2} \|\nabla^2 u\|_{L^2}^2 + \|\sqrt{t}\nabla^3 \chi\|_{L^2} \|\nabla^3 \chi\|_{L^2}^2) \,dt \\ &\le C (1 + \varPhi(T)) \le C. \end{align}\] The conclusion follows directly from the estimates above. ◻
Let \(\{w_i\}_{i=1}^\infty\) be the eigenfunctions associated with the eigenvalues \(\{\lambda_i\}_{i=1}^\infty\) of the operator \(\mathcal{L}\), solving \[\begin{align} \begin{cases} - \mathcal{L} w_i = \lambda_i w_i \quad &\text{in } \Omega,\\ w_i = 0 \quad &\text{on } \partial \Omega. \end{cases} \end{align}\] Here \(0<\lambda_1\le\lambda_2\le\cdots\) and \(\lambda_i\to\infty\) as \(i\to\infty\). The family \(\{w_i\}\) can be chosen as an orthonormal basis of \(L^2(\Omega)\) and an orthogonal basis of \(H_0^1(\Omega)\). For any \(N\in\mathbb{N}_+\), we define \[X_N := \mathrm{span}\{w_1,w_2,\ldots,w_N\},\] and denote by \(\|\cdot\|_{X_N}\) the induced norm on \(X_N\). Let \[\rho_{0N} = \rho_0 * \eta_{\frac{1}{N}} + \frac{1}{N}, \qquad h_N = h * \eta_{\frac{1}{N}} - C_N,\] where \(\eta_{\frac{1}{N}}\) is a standard mollifier and \(C_N = \frac{\overline{\rho_{0N} h * \eta_{\frac{1}{N}}}}{\overline{\rho_{0N}}}\). Here \(\overline{f}:= \frac{1}{|\Omega|}\int_\Omega f\,dx\) denotes the spatial average over \(\Omega\) of an integrable function \(f\). Then we have \[\begin{align} \rho_{0N} \to \rho_0 \quad &\text{in } W^{1,q}, \label{rho0n}\\ h_N \to h \quad &\text{in } L^2, \notag \\ \overline{\rho_{0N} h_N} = 0. \notag \end{align}\tag{20}\]
By classical elliptic theory (see, e.g., [41], [42]), there exists a \(\chi_{0N}\in H^2\) satisfying \[\begin{align} \begin{cases} \Delta \chi_{0N} = \rho_{0N} h_N - \rho_0 h \quad &\text{in } \Omega,\\ \partial_{\boldsymbol{n}}\chi_{0N} = 0 \quad &\text{on } \partial \Omega, \end{cases} \end{align}\] and \[\begin{align} \label{chi0n} \chi_{0N} \to \chi_0 \quad \text{in } H^2. \end{align}\tag{21}\] Let \(u_{0N}=P_Nu_0\) be the \(L^2\)-projection of \(u_0\) onto \(X_N\), where \(P_N\) is defined by \[P_N g := \sum_{i=1}^N \langle g,w_i\rangle w_i, \qquad g\in L^2.\] It is straightforward to verify that \[\begin{align} \label{u0n} u_{0N} \to u_0 \quad \text{in } H^1. \end{align}\tag{22}\] We are now ready to formulate the Galerkin approximate system: \[\label{appro95pro} \begin{cases} \partial_t\rho_N + \mathop{\mathrm{div}}(\rho_N u_N) = 0,\\ \rho_N\partial_t u_N + P_N\!\left[\rho_N(u_N\!\cdot\!\nabla)u_N + \nabla P(\rho_N)\right] = \mathcal{L}u_N - P_N\!\left[\mathop{\mathrm{div}}\!\left(\nabla\chi_N\otimes\nabla\chi_N -\frac{|\nabla\chi_N|^2}{2}\mathbb{I}\right)\right],\\ \rho_N\partial_t\chi_N + \rho_N u_N\cdot\nabla\chi_N = -\mu_N,\\ \rho_N\mu_N = -\Delta\chi_N + \rho_N F'(\chi_N),\\ (\rho_N,u_N,\chi_N)|_{t=0} = (\rho_{0N},u_{0N},\chi_{0N}),\\ (u_N,\partial_{\boldsymbol{n}}\chi_N)|_{\partial\Omega} = (0,0). \end{cases}\tag{23}\]
By Schaefer’s fixed point theorem (cf. [43]), the well-posedness of system 23 can be established in a standard manner.
Proposition 2. For any \(T>0\) and \(N\in \mathbb{N}_+\), there exists some \((\rho_N, u_N, \chi_N, \mu_N)\) solving 23 , such that:
\(\rho_N \in C^1(\overline{\Omega} \times[0,T])\) and satisfies \[\partial_t \rho_N + \mathop{\mathrm{div}}(\rho_N u_N) = 0 \quad \text{\rm in } \Omega\times(0,T); \\None\]
\(u_N \in C([0, T]; X_N)\) and, for every \(\varphi_N \in X_N\) and almost every \(t \in (0, T)\), satisfies \[\label{Galerkin95Equation} \begin{align} &\langle \rho_N \partial_t u_N + \rho_N (u_N\cdot\nabla) u_N + \nabla (P(\rho_N)), \varphi_N \rangle + \nu \langle \nabla u_N, \nabla \varphi_N \rangle + (\lambda + \nu) \langle \mathop{\mathrm{div}}u_N, \mathop{\mathrm{div}}\varphi_N \rangle \\ =&\langle\mathop{\mathrm{div}}(\nabla \chi_N \otimes \nabla \chi_N - \frac{|\nabla \chi_N|^2}{2}\mathbb{I}), \varphi_N \rangle; \end{align}\qquad{(6)}\]
\(\chi_N \in C([0, T]; H^1) \cap L^{\infty} (0, T; H^2) \cap L^2(0,T;H^3)\), with \(\partial_t \chi_N \in L^\infty ( 0, T; H^1)\), and satisfies \[\rho_N^2 \partial_t \chi_N + \rho_N^2 u_N \cdot \nabla \chi_N - \Delta \chi_N + \rho_N F'(\chi_N) = 0 \quad \text{\rm a.e. in } \Omega\times (0, T);\]
the initial and boundary conditions hold \[(\rho_N(0), u_N(0), \chi_N(0))= (\rho_{0N}, u_{0N}, \chi_{0N}) \quad \text{a.e. in } \Omega,\] and \[\partial_{\boldsymbol{n}} \chi_N = 0 \quad \text{a.e. on } \partial \Omega\times (0, T);\]
the following energy equality holds for almost every \(t \in (0, T)\) \[\begin{align} E_N(t) + \int_0^t ( \| \mu_N \|_{L^2}^2 + \nu \| \nabla u_N \|_{L^2}^2 + ( \lambda + \nu ) \| \mathop{\mathrm{div}}u_N \|_{L^2}^2 ) \, ds = E_N(0), \end{align}\] where \[\label{EN} E_N(t) = \frac{1}{2} \| \sqrt{\rho_N} u_N \|_{L^2}^2 + \frac{A}{\gamma-1} \int_{\Omega} \rho_N^{\gamma} \,dx + \frac{1}{2} \| \nabla\chi_N \|_{L^2}^2 + \int_{\Omega} \rho_N F(\chi_N)\,dx.\qquad{(7)}\]
For the sake of completeness, the proof of Proposition 2 is given in the Appendix.
Let \((\rho_N, u_N, \chi_N, \mu_N)\) be the strong solutions in Proposition 2. Set \[\begin{align} \varPhi_N(t) &= \sup_{0 \le s \le t} \left(\|\rho_N\|_{W^{1, q}} + \|\big(\nabla u_N, \sqrt{s} \sqrt{\rho_N} \partial_t u_N, \nabla^2 \chi_N, \rho_N \partial_t \chi_N, \sqrt{s} \nabla \partial_t \chi_N\big) \|^2_{L^2}\right) \\ &\quad+ \int_0^t \left(\|\big(\nabla^2 u_N, \sqrt{\rho_N} \partial_t u_N, \sqrt{s} \nabla \partial_t u_N, \nabla^3 \chi_N, \nabla \partial_t \chi_N \big)\|^2_{L^2}\right) \,ds + 1. \end{align}\]
Proposition 3. Let \(\varepsilon_0\) be the constant given in Lemma 4. Then, there exists a time \(T_0>0\) and a constant \(C>0\), depending only on \(A\), \(\gamma\), \(\nu\), \(\lambda\), \(q\), \(\Omega\) and \(\Phi_0\), such that \[T_0^{\frac{6-q}{4q}} \varPhi_N^{\gamma + 1}(T_0) \le \varepsilon_0.\] Moreover, we have \[\begin{align} &\varPhi_N(T_0) + \sup_{0 \le t \le T_0} \left(\|\partial_t \rho_N\|_{L^2}^2 + \|\sqrt{t} \nabla^2 u_N\|_{L^2}^2 + \|\sqrt{t} \nabla^3 \chi_N\|_{L^2}^2 + \|\mu_N\|_{L^2}^2\right) \\ \quad& + \int_0^{T_0} \left(\|\nabla^2 u_N\|_{L^q} + \|\sqrt{t} \nabla^2 u_N\|_{L^q}^2 + \|\partial_t \chi_N\|_{H^1}^2 + \|\sqrt{t} \rho_N \partial_{tt}^2 \chi_N\|^2_{L^2}\right) \,dt \le C \end{align}\] for all \(N \in \mathbb{N}\).
Proof. Taking \(T = 1\) in Proposition 2 and denoting \[\Psi_N(t) = t^{\frac{6-q}{4q}} \varPhi_N^{\gamma + 1 }(t).\] Since \(\varepsilon_0 \in (0,1)\) and \(\Psi_N\) is continuous on \([0,1]\), there exists some \(T_N \in (0,1)\) such that \(\Psi_N(T_N) = \varepsilon_0\). In a manner similar to that in Section 2, one can show that \(\varPhi_N(T_N) \le C\) uniformly in \(N\). By the definition of \(\Psi_N\), we have \[\begin{align} T_N &= \left( \varPhi_N(T_N)^{ -\gamma - 1 } \varepsilon_0 \right)^{\frac{4q}{6-q}} \\ &\ge \left( C^{ -\gamma - 1 } \varepsilon_0 \right)^{\frac{4q}{6-q}}, \end{align}\] which implies that \({T_N}\) admits a uniform positive lower bound, independent of \(N\).
Now set \(T_0 = \inf\limits_{N \in \mathbb{N}}{T_N}\), then \(\varPhi_N(T_0) \le C\). Moreover, as in Section 2, we can show that \[\begin{align} \int_0^{T_0} \| \nabla^2 u_N \|_{L^q} \,dt + \sup_{0 \le t \le T_0} \left( \| \partial_t \rho_N \|_{L^2}^2 + \| \sqrt{t} \partial_t \nabla \chi_N \|_{L^2}^2 + \| \sqrt{t} \nabla^3 \chi_N \|_{L^2}^2 + \| \sqrt{t} \nabla^2 u_N \|_{L^2}^2 \right) \le C. \end{align}\] However, the upper bound of \(\int_0^{T_0} \|\nabla^2 u\|_{L^q} \,dt\) is estimated slightly different to that in Section 2. To be specific, taking \(\varphi_N = P_N( | \mathcal{L} u_N|^{q - 2} \mathcal{L} u_N )\) in ?? and using integration by parts, we have \[\begin{align} &\| \nabla^2 u_N \|_{L^q}^q \le C \| \mathcal{L}u_N \|_{L^q}^q = C \langle \mathcal{L} u_N, \varphi_N \rangle \\ &\quad= C \langle \rho_N \partial_t u_N + \rho_N (u_N\cdot\nabla) u_N + \nabla (P(\rho_N)) + \Delta \chi_N \nabla \chi_N , \varphi_N \rangle\\ &\quad\le C (\|\rho_N \partial_t u_N \|_{L^q} + \| \rho_N (u_N\cdot\nabla) u_N \|_{L^q} + \| \nabla (P(\rho_N)) \|_{L^q} + \| \Delta \chi_N \nabla \chi_N \|_{L^q} ) \| \varphi_N \|_{L^{\frac{q}{q - 1}}}\\ &\quad\le C (\|\rho_N \partial_t u_N \|_{L^q} + \| \rho_N (u_N\cdot\nabla) u_N \|_{L^q} + \| \nabla (P(\rho_N)) \|_{L^q} + \| \Delta \chi_N \nabla \chi_N \|_{L^q} ) \| \nabla^2 u_N \|_{L^q}^{q - 1}. \end{align}\] That is, \[\begin{align} \| \nabla^2 u_N \|_{L^q} &\le C (\|\rho_N \partial_t u_N \|_{L^q} + \| \rho_N (u_N\cdot\nabla) u_N \|_{L^q} + \| \nabla (P(\rho_N)) \|_{L^q} + \| \Delta \chi_N \nabla \chi_N \|_{L^q} ). \end{align}\] The rest are analogous to that in Section 2. ◻
In this section, we prove the convergence of the approximate solutions in Proposition 2 and derive the regularity of the limit, which will be used in the proof of Theorem 1.
Theorem 3. Assume that all the conditions of Theorem 1 are satisfied. Then, there exists a positive time \(T_0>0\), depending only on \(A\), \(\gamma\), \(\nu\), \(\lambda\), \(q\), \(\Omega\) and \(\Phi_0\), such that the system 1 , subject to 2 3 , in \(\Omega\times (0, T_0)\), admits a solution \((\rho, u, \chi, \mu)\), satisfying all the properties listed in Definition 1, except that the property \(\sqrt{\rho}u \in C([0, T_0];L^2)\) is replaced by \[\begin{align} \sqrt{\rho}u \in C((0, T_0];L^2),\quad \rho u \in C([0, T_0];L^2), \end{align}\] and \[\label{normcon} \lim_{t \rightarrow 0^+} \| \sqrt{\rho} u \|_{L^2}(t) = \| \sqrt{\rho_0} u_0 \|_{L^2}.\qquad{(8)}\]
Proof. Step 1. Limit passage. Let \((\rho_{0N}, u_{0N}, \chi_{0N})\) be chosen as in Section 3, then we have \[\begin{align} &\rho_{0N} \rightarrow \rho_0 \qquad \mathrm{in}\;W^{1,q}, \\ &u_{0N} \rightarrow u_0 \qquad \mathrm{in}\;H^1, \\ &\chi_{0N} \rightarrow \chi_0 \qquad \mathrm{in}\; H^2. \end{align}\] It follows from Proposition 2 and Proposition 3 that, there exists a time \(T_0>0\) and a constant \(C>0\), depending only on \(A\), \(\gamma\), \(\nu\), \(\lambda\), \(q\), \(\Omega\) and \(\Phi_0\), such that \[\begin{align} &\sup\limits_{0 \le t \le T_0} \left(\|\partial_t \rho_N\|_{L^2}^2 + \|\rho_N\|_{W^{1,q}} \right) \le C, \\ &\sup\limits_{0 \le t \le T_0} \left(\|\nabla u_N\|_{L^2}^2 + \|\sqrt{t} \nabla^2 u_N\|_{L^2}^2 \right) + \int_0^{T_0} \left(\| \sqrt{t} \nabla \partial_t u_N \|_{L^2}^2 + \| \nabla^2 u_N \|_{L^2}^2 \right) \,dt \le C,\\ &\sup\limits_{0 \le t \le T_0} \left( \| \chi_N \|_{H^2}^2 + \| \sqrt{t} \nabla^3 \chi_N \|_{L^2}^2 \right) + \int_0^{T_0} \| \partial_t \chi_N \|_{H^1}^2 \,dt\le C,\\ &\sup\limits_{0 \le t \le T_0} \left( \| \rho_N \partial_t \chi_N\|_{L^2}^2 + \|\mu_N\|_{L^2}^2 + \| \sqrt{t} \partial_t \nabla\chi_N\|_{L^2}^2 + \| \sqrt{t} \sqrt{\rho_N} \partial_t u_N \|_{L^2}^2 \right) \le C, \\ &\int_0^{T_0} \left( \|\sqrt{\rho_N} \partial_t u_N\|_{L^2}^2 + \| \nabla^2 u_N \|_{L^q} + \|\sqrt{t} \nabla^2 u\|_{L^q}^2 + \|\nabla^3 \chi_N \|_{L^2}^2 + \| \sqrt{t} \rho_N \partial_{tt}^2 \chi_N \|_{L^2}^2 \right) \,dt \le C, \end{align}\] for any \(N \in \mathbb{N}\). Using the standard diagonal argument, we can extract a subsequence of \((\rho_N, u_N, \chi_N, \mu_N)\), still denoted by \((\rho_N, u_N, \chi_N, \mu_N)\), such that \[\label{wc1} \begin{align} \begin{aligned} &\rho_N \rightharpoonup\rho &&\quad\text{weakly-* in }L^{\infty}(0, T_0; W^{1,q}), \\ &\partial_t \rho_N \rightharpoonup\rho_t &&\quad\text{weakly-* in }L^{\infty}(0, T_0; L^2), \\ &u_N \rightharpoonup u &&\quad\text{weakly-* in }L^{\infty}(0, T_0; H_0^1),\\ &u_N \rightharpoonup u &&\quad\text{weakly in }L^2(0, T_0; H^2),\\ &\partial_t u_N \rightharpoonup u_t &&\quad\text{weakly in }L^2(\delta, T_0; H^1_0),\\ \end{aligned} \qquad \begin{align} &\chi_N \rightharpoonup\chi &&\quad\text{weakly-* in }L^{\infty}(0, T_0; H^2),\\ &\chi_N \rightharpoonup\chi &&\quad\text{weakly in }L^2(0, T_0; H^3),\\ &\partial_t \chi_N \rightharpoonup\chi_t &&\quad\text{weakly in }L^2(0, T_0; H^1),\\ &\partial_t \chi_N \rightharpoonup\chi_t &&\quad\text{weakly-* in }L^\infty(\delta, T_0; H^1),\\ &\mu_N \rightharpoonup\mu && \quad\text{weakly-* in }L^\infty(0, T_0; L^2), \end{align} \end{align}\tag{24}\] for any \(\delta \in (0, T_0)\). Here \((\rho, u, \chi, \mu)\) satisfies \[\label{reg} \begin{cases} 0 \le \rho \in L^{\infty}(0, T_0; W^{1, q}), \quad \rho_t \in L^\infty(0, T_0; L^2), \\ u \in L^\infty(0, T_0; H_0^1) \cap L^2(0, T_0; H^2) \cap L^1(0, T_0; W^{2,q}), \quad \sqrt{t} u_t \in L^2(0, T_0; H_0^1), \\ \sqrt{t} \nabla^2 u \in L^\infty(0, T_0;L^2) \cap L^2(0, T_0; L^q), \\ \chi \in L^{\infty}(0, T_0; H^2) \cap L^2(0, T_0; H^{3}), \quad \sqrt{t} \nabla^3 \chi \in L^{\infty}(0, T_0; L^2), \\ \chi_t \in L^2(0, T_0; H^1), \quad \sqrt{t} \nabla\chi_t \in L^\infty(0, T_0; L^2), \\ \mu \in L^\infty(0, T_0; L^2). \end{cases}\tag{25}\] Therefore, by the Aubin-Lions lemma, together with the embeddings \(H^3 \hookrightarrow \hookrightarrow H^2 \hookrightarrow\hookrightarrow H^1\hookrightarrow \hookrightarrow L^2\), and \(W^{1, q} \hookrightarrow \hookrightarrow C(\overline{\Omega}) \hookrightarrow L^2\), for \(q \in (3, 6)\), we infer that \[\begin{align} \rho_N \rightarrow \rho &\quad\text{strongly in }C([0, T_0]; C(\overline{\Omega})), \tag{26}\\ u_N \rightarrow u &\quad\text{strongly in }C([\delta, T_0]; L^2) \cap L^2(\delta, T_0; H_0^1), \tag{27} \\ \chi_N \rightarrow \chi &\quad\text{strongly in }C([0, T_0]; H^1) \cap L^2(0, T_0; H^2) \tag{28} \end{align}\] for any \(\delta \in (0, T_0)\). By the convergence of the nonlinear terms 24 and 26 –28 , we have the following convergence of the nonlinear terms \[\label{cnl} \begin{align} (\rho_N u_N, \sqrt{\rho_N} u_N) &\rightarrow (\rho u,\sqrt{\rho} u) &&\quad\text{strongly in }C([\delta, T_0]; L^2), \\ (\rho_N\partial_t u_N ,\sqrt{\rho_N} \partial_t u_N) &\rightharpoonup(\rho u_t ,\sqrt{\rho} u_t) &&\quad\text{weakly in }L^2(\delta, T_0; L^2), \\ \rho_N ( u_N \cdot \nabla ) u_N &\rightarrow \rho (u \cdot \nabla) u && \quad\text{strongly in }L^1(\Omega\times (\delta, T_0)), \\ P(\rho_N) &\rightarrow P(\rho) &&\quad\text{strongly in }C([0, T_0]; C(\overline{\Omega})), \\ \Delta\chi_N \nabla\chi_N &\rightarrow \Delta\chi \nabla\chi &&\quad\text{strongly in } L^1(\Omega\times (0, T_0)). \\ (\rho_N^2 \partial_t\chi_N, \rho_N \partial_t\chi_N) &\rightharpoonup (\rho^2 \partial_t\chi, \rho \partial_t\chi) &&\quad\text{weakly in }L^2(0, T_0; L^2), \\ \rho_N^2 ( u_N \cdot \nabla ) \chi_N &\rightharpoonup\rho^2 ( u \cdot \nabla ) \chi &&\quad\text{weakly in }L^1(\Omega\times (\delta, T_0)), \\ \rho_N F'(\chi_N) &\rightarrow \rho F'(\chi) &&\quad\text{strongly in }C([0, T_0]; L^2) \end{align}\tag{29}\] for any \(\delta \in (0, T_0)\). From the weakly lower semi-continuity of the norms and 29 , one concludes that \[\begin{align} \int_{\delta}^{T_0} \| \sqrt \rho u_t \|_{L^2}^2 \,dt \le \liminf_{N \rightarrow \infty} \int_{\delta}^{T_0} \| \sqrt{\rho_N} \partial_t u_N \|_{L^2}^2 \,dt \le C, \end{align}\] for any \(\delta \in (0,T_0)\). Thus, we have \[\begin{align} \sqrt{\rho} u_t \in L^2(0, T; L^2). \end{align}\]
Step 2. The existence. It follows from 24 , 27 , 28 , 29 , and Proposition 2 that \((\rho, u, \chi, \mu)\) satisfies 1 in the sense of distributions and also satisfies 3 almost everywhere on \(\partial \Omega \times (0, T_0)\). Furthermore, in view of the regularity property 25 and an integration by parts argument, we conclude that \((\rho, u, \chi, \mu)\) satisfies 1 almost everywhere in \(\Omega \times (0, T_0)\). Thus, we have shown that \((\rho, u, \chi, \mu)\) satisfies all properties in Definition 1, except \(\sqrt{\rho}u \in C([0, T_0];L^2)\) and the initial conditions 2 .
It follows from 20 , 21 , 26 and 28 that \[\begin{cases}\label{CRC} \rho \rightarrow \rho_0 \quad \text{in } C(\overline{\Omega}) \quad\text{as } t \rightarrow 0^+, \\ \chi \rightarrow \chi_0 \quad \text{in } H^1 \quad\text{as } t \rightarrow 0^+. \\ \end{cases}\tag{30}\] We also deduce by 29 that \[\begin{align} &(\sqrt{\rho}u, \rho u) \in C((0, T];L^2). \end{align}\] By letting \(N \rightarrow \infty\) in ?? and 26 –28 , one has \[\begin{align} &\frac{1}{2}\|\sqrt{\rho} u \|_{L^2}^2(t) + \frac{1}{2} \|\nabla \chi\|_{L^2}^2 (t) + \frac{A}{\gamma-1} \int_{\Omega} \rho^{\gamma}(x, t) \,dx + \int_{\Omega} \rho(x, t) F(\chi(x, t)) \,dx \\ &\quad + \int_0^t ( \|\mu\|_{L^2}^2 + \nu \|\nabla u\|_{L^2}^2 + ( \lambda + \nu ) \|\mathop{\mathrm{div}}u\|_{L^2}^2 ) \, ds \\ &= \frac{1}{2} \|\sqrt{\rho_0} u_0 \|_{L^2}^2 + \frac{1}{2} \|\nabla \chi_0\|_{L^2}^2 + \frac{A}{\gamma-1} \int_{\Omega} \rho_0^{\gamma} \,dx + \int_{\Omega} \rho_0 F(\chi_0) \,dx \end{align}\] for any \(t \in (0, T_0)\). It follows from 30 that \[\begin{align} &\lim_{t \rightarrow 0^+} \|\sqrt{\rho} u \|_{L^2}^2(t) = \|\sqrt{\rho_0} u_0 \|_{L^2}^2. \end{align}\]
It remains to show that \[\label{CRU} \lim_{t \rightarrow 0^+} \| \rho u - \rho_0 u_0 \|_{L^2}(t) = 0.\tag{31}\] One deduces by the Gagliardo-Nirenberg and Hölder inequalities that \(\int_{0}^{{T_0}} \| \partial_{t} (\rho_N u_N) \|_{L^2}^{2} \,dt \le C\). It follows from the Hölder inequality that \[\begin{align} &\| \rho u - \rho_0 u_0 \|_{L^2}(t) \\ \le& \| \rho u - \rho_N u_N \|_{L^2}(t) + \| \rho_N u_N -\rho_{0N}u_{0N} \|_{L^2}(t) + \| \rho_{0N} u_{0N} - \rho_{0N} u_0 \|_{L^2} + \| \rho_{0N} u_0 - \rho_{0} u_0 \|_{L^2} \\ \le& \| \rho u - \rho_N u_N \|_{L^2}(t) + \int_0^{t}\|\partial_{t}(\rho_N u_N) \|_{L^2} \,ds + \| \rho_{0N} \|_{L^\infty} \| u_{0N} - u_0 \|_{L^2} +\frac{\|\rho_0\|_{L^\infty}}{N} \|u_0\|_{L^2} \\ \le& \| \rho u - \rho_N u_N \|_{L^2}(t) + C \sqrt{t} + C \| u_{0N} - u_0 \|_{L^2} +\frac{C}{N} \|u_0\|_{L^2}. \end{align}\] Recalling 22 , one gets \(\| \rho u - \rho_0 u_0 \|_{L^2}(t) \le C \sqrt{t}\). This completes the proof. ◻
To prove Theorem 1, it suffices to show that the solution \((\rho, u, \chi, \mu)\) obtained in Theorem 3 satisfies \(\sqrt{\rho}u \rightarrow \sqrt{\rho_0} u_0\) in \(L^2\) as \(t\rightarrow 0\). Thanks to ?? , it is sufficient to show \(\sqrt{\rho}u \rightharpoonup\sqrt{\rho_0} u_0\) in \(L^2\) as \(t\rightarrow 0\). Let \(\{t_k\}\) be any decreasing sequence such that \(t_k \downarrow 0\), It remains to show that there exists a subsequence \(\{t_{k_j}\}\) of \(\{t_k\}\) such that \[\label{CRU2} \sqrt{\rho} u (t_{k_j}) \rightharpoonup\sqrt{\rho_0} u_0 \quad \text{in } L^2 \quad \text{as } j\rightarrow \infty.\tag{32}\] Owing to the uniform bound of \(\|\sqrt{\rho} u\|_{L^2}(t_k)\), one can extract a subsequence \(\{t_{k_j}\}\) of \(\{t_k\}\) such that \[\sqrt{\rho}u (t_{k_j}) \rightharpoonup\psi \quad \text{in } L^2 \quad \text{as } j\rightarrow \infty,\] for some \(\psi\in L^2\). Consequently, \[\sqrt{\rho_0}\sqrt{\rho}u (t_{k_j}) \rightharpoonup\sqrt{\rho_0}\psi \quad \text{in } L^2 \quad \text{as } j\rightarrow \infty.\] On the other hand, by using 30 , the Hölder inequality and the uniform bounds of \(u\) in \(L^6\) and \(\sqrt{\rho}\) in \(L^\infty\), we have \(\|\sqrt{\rho_0}\sqrt{\rho}u-\rho u\|_{L^2}\le \|\sqrt{\rho_0}-\sqrt{\rho}\|_{L^3}\|\sqrt{\rho}\|_{L^\infty}\|u\|_{L^6}\rightarrow 0\), as \(t\rightarrow 0^+\). Hence, from 31 we can deduce that \[\sqrt{\rho_0} \sqrt{\rho} u (t) \rightarrow \rho_0 u_0 \quad \text{in } L^2 \quad \text{as } t\rightarrow 0^+.\] By the uniqueness of weak limits, it follows that \(\sqrt{\rho_0} \psi = \rho_0 u_0\) a.e. in \(\Omega\). We decompose the domain into the non-vacuum and the vacuum regions: \[\Omega= \Omega_+ \cup \Omega_0, \quad \Omega_+ = \{x\in\Omega:\rho_0(x) > 0\}, \quad\Omega_0 = \{x\in\Omega:\rho_0(x) = 0\}.\] Hence, one deduces that \(\psi = \sqrt{\rho_0} u_0\) a.e. in \(\Omega_+\). Let \(\phi\in L^2\) be arbitrary, then we have \[\label{CRU3} \int_{\Omega_+} (\sqrt{\rho} u - \sqrt{\rho_0} u_0)\phi \,dx (t_{k_j}) =\int_{\Omega_+} (\sqrt{\rho} u - \psi)\phi \,dx (t_{k_j}) \rightarrow 0 \quad \text{as } j\rightarrow \infty\tag{33}\] and \[\begin{align}\label{CRU4} \int_{\Omega_0} (\sqrt{\rho} u - \sqrt{\rho_0} u_0)\phi \,dx (t_{k_j}) &=\int_{\Omega_0} \sqrt{\rho} u \phi \,dx (t_{k_j}) \\ &= \int_{\Omega_0} (\sqrt{\rho} - \sqrt{\rho_0}) u \phi \,dx (t_{k_j}) \rightarrow 0 \quad \text{as } j\rightarrow \infty. \end{align}\tag{34}\] Combining 33 and 34 yields 32 . Since the sequence \(\{t_k\}\) is arbitrary, it follows that \[\sqrt{\rho}u(t) \rightharpoonup\sqrt{\rho_0} u_0 \quad \text{in } L^2 \quad \text{as } t\rightarrow 0^+.\] This completes the proof. 0◻
This section is devoted to proving the uniqueness of the strong solutions obtained in the previous section.
Let \((\rho_i,u_i,\chi_i)\) \((i=1,2)\) be two strong solutions to system 1 in \(\Omega\times(0,T_0)\), subject to the initial condition 2 and the boundary condition 3 . Denote \(\rho=\rho_1-\rho_2\), \(u=u_1-u_2\), \(\chi=\chi_1-\chi_2\). It is straightforward to verify that \((\rho, u, \chi)\) satisfies that \[\begin{align}\label{NSAC95diff} \begin{cases} \rho_{t} + u_1 \cdot \nabla \rho + \rho \mathop{\mathrm{div}}u_1 + u \cdot \nabla \rho_2 + \rho_2 \mathop{\mathrm{div}}u=0, \\ \rho_1 u_{t}-\mathcal{L} u = -\rho \partial_t u_2 - \rho_1 ( u_1 \cdot \nabla ) u - \rho_1 ( u \cdot \nabla ) u_2 - \rho ( u_2 \cdot \nabla) u_2 \\\quad- \nabla\bigl( P(\rho_1) - P(\rho_2) \bigr) - \mathop{\mathrm{div}}\bigl( \nabla \chi_1 \otimes \nabla \chi + \nabla \chi \otimes \nabla \chi_2 \bigr) - \frac{1}{2} \nabla \bigl( \nabla \chi \cdot ( \nabla \chi_1 + \nabla \chi_2) \bigr), \\ \rho_1^2 \chi_{t} - \Delta \chi = - \rho ( \rho_1 + \rho_2 ) \partial_t \chi_2 - \rho_1^2 u_1 \cdot \nabla \chi - \rho_1^2 u \cdot \nabla \chi_2 - \rho (\rho_1 + \rho_2) u_2 \cdot \nabla \chi_2 \\ \quad - \rho_1 \chi ( \chi_1^2 + \chi_1 \chi_2 + \chi_2^2 - 1) - \rho ( \chi_2^3 - \chi_2) \end{cases} \end{align}\tag{35}\] a.e. in \(\Omega\times(0, T_0)\) and \(( u, \partial_{\boldsymbol{n}} \chi ) \big|_{\partial \Omega} = 0\). It follows from the Hölder and Poincaré inequalities that \[\begin{align} \|\sqrt{\rho_1} u\|_{L^2}^2(\tau) &= \|\rho_1 |u|^2\|_{L^1}(\tau) \le \|\rho_1 u\|_{L^2}(\tau) \|u\|_{L^2}(\tau) \\ &\le C \left(\|\rho_1 u_1 - \rho_2 u_2\|_{L^2}(\tau) + \|(\rho_1 - \rho_2) u_2 \|_{L^2}(\tau)\right) \|\nabla u\|_{L^2}(\tau) \\ &\le C \left(\|\rho_1 u_1 - \rho_2 u_2\|_{L^2}(\tau) + \|\rho_1 - \rho_2\|_{L^3}(\tau) \| u_2 \|_{L^6}(\tau)\right) \\ &\le C \left(\|\rho_1 u_1 - \rho_2 u_2\|_{L^2}(\tau) + \|\rho_1 - \rho_2\|_{L^3}(\tau)\right) \end{align}\] for every \(\tau>0\). Hence, one deduces by the regularities of \((\rho_i, u_i, \chi_i)\) that \[\label{IU} \lim_{\tau \rightarrow 0} \|(\rho, \sqrt{\rho_1}u, \chi )\|_{L^2}(\tau) = 0,\tag{36}\]
The proof of uniqueness proceeds in three steps: energy estimates for \((\rho, u, \chi)\), growth estimates for \((\rho, \chi)\), and a singular-weighted Grönwall argument.
Step 1. Energy inequalities. Multiplying \(\eqref{NSAC95diff}_1\) by \(\rho\) and integrating over \(\Omega\), it follows from integration by parts, the Hölder, Sobolev and Poincaré inequalities that \[\label{rho95diff} \begin{align} \frac{d}{dt} \|\rho\|_{L^2}^{2} &= - \int_{\Omega} \rho^{2} \mathop{\mathrm{div}}u_{1} \,dx - 2 \int_{\Omega} \rho u \cdot \nabla \rho_{2} \,dx - 2 \int_{\Omega} \rho_{2} \mathop{\mathrm{div}}u \rho \,dx \\ &\le C \| \nabla u_1 \|_{L^\infty} \|\rho\|_{L^2}^2 + C \|u\|_{L^6} \|\rho\|_{L^2} \| \nabla \rho_2 \|_{L^3} + C \| \rho_2\|_{L^\infty} \|\nabla u\|_{L^2} \|\rho\|_{L^2}\\ &\le C \|\nabla^2 u_1 \|_{L^q} \|\rho\|_{L^2}^2 + C \|\nabla u\|_{L^2} \|\rho\|_{L^2}. \end{align}\tag{37}\]
Testing \(\eqref{NSAC95diff}_2\) with \(u\), it follows from integration by parts that \[\begin{align} &\frac{1}{2} \frac{d}{dt} \|\sqrt{\rho_1} u\|_{L^2}^2 + \nu \|\nabla u\|_{L^2}^2 + ( \lambda + \nu ) \|\mathop{\mathrm{div}}u\|_{L^2}^2 \\ =& \int_\Omega\rho_1 ( u \cdot \nabla ) u \cdot u_1 \,dx - \int_\Omega\rho_1 ( u_1 \cdot \nabla ) u \cdot u \,dx - \int_\Omega\rho_1 ( u \cdot \nabla ) u_2 \cdot u \,dx - \int_\Omega\rho ( u_2 \cdot \nabla ) u_2 \cdot u \,dx\\ & - \int_\Omega\rho \partial_t u_2 \cdot u \,dx + \int_\Omega[ P( \rho_1 ) - P( \rho_2 )] \mathop{\mathrm{div}}u \,dx + \int_\Omega( \nabla \chi_1 \otimes \nabla \chi + \nabla \chi \otimes \nabla \chi_2 ) : \nabla u \,dx \\ & + \frac{1}{2} \int_\Omega\nabla \chi \cdot ( \nabla \chi_1 + \nabla \chi_2 ) \mathop{\mathrm{div}}u \,dx \\ =& \sum_{i=1}^8 M_i. \end{align}\] Using the Hölder, Young, Sobolev and Gagliardo-Nirenberg inequalities, one obtains \[\begin{align} M_1+M_2 &\le 2 \|\rho_1\|_{L^\infty}^{\frac{1}{2}} \|\sqrt{\rho_1} u\|_{L^2} \|\nabla u\|_{L^2} \| u_1 \|_{L^\infty}\\ &\le C \|\sqrt{\rho_1}u\|_{L^2}\|\nabla u\|_{L^2} \|\nabla u_1\|_{L^2}^{\frac{1}{2}} \|\nabla^2 u_1\|_{L^2}^{\frac{1}{2}}\\ &\le \frac{\nu}{10}\|\nabla u\|_{L^2}^2 + C \| \nabla^2 u_1 \|_{L^2} \|\sqrt{\rho_1}u\|_{L^2}^2, \\ M_3 &\le \| \nabla u_2 \|_{L^\infty} \|\sqrt{\rho_1} u\|_{L^2}^2 \le C \|\nabla^2 u_2\|_{L^q} \|\sqrt{\rho_1} u\|_{L^2}^2, \\ M_4 &\le \|\rho\|_{L^2} \| u_2 \|_{L^6} \| \nabla u_2 \|_{L^6} \| u \|_{L^6}\\ &\le \|\rho\|_{L^2} \| \nabla u_2 \|_{L^2} \|\nabla^2 u_2\|_{L^2} \|\nabla u\|_{L^2}\\ &\le \frac{\nu}{10}\|\nabla u\|_{L^2}^2 + C \|\nabla^2 u_2\|_{L^2}^2 \|\rho\|_{L^2}^2. \\ M_5 &\le \|\rho\|_{L^2}\|\partial_t u_2\|_{L^3}\|u\|_{L^6}\\ &\le \frac{\nu}{10}\|\nabla u\|_{L^2}^2 + C\| \nabla \partial_t u_2\|_{L^2}^2 \|\rho\|_{L^2}^2. \\ M_6 &\le C (\|\rho_1\|_{L^\infty}^{\gamma-1} + \|\rho_2\|_{L^\infty}^{\gamma-1} )\int_\Omega|\rho \mathop{\mathrm{div}}u | \,dx \\ &\le \frac{\nu}{10} \|\nabla u\|_{L^2}^2 + C \|\rho\|_{L^2}^2. \\ M_7 + M_8 &\le C \|\nabla \chi\|_{L^2} \big( \| \nabla \chi_1 \|_{L^\infty} + \| \nabla \chi_2 \|_{L^\infty} \big) \|\nabla u\|_{L^2}\\ &\le C \|\nabla \chi\|_{L^2} \big( \| \nabla^2 \chi_1 \|_{L^2}^\frac{1}{2} \|\nabla^3 \chi_1 \|_{L^2}^\frac{1}{2} + \|\nabla^2 \chi_2\|_{L^2}^\frac{1}{2} \|\nabla^3 \chi_2 \|_{L^2}^\frac{1}{2} \big) \|\nabla u\|_{L^2}\\ &\le \frac{\nu}{10}\|\nabla u\|_{L^2}^2 + C (\|\nabla^3 \chi_1 \|_{L^2} + \|\nabla^3 \chi_2 \|_{L^2}) \|\nabla\chi\|_{L^2}^2. \end{align}\] Thus, we have \[\label{rho1u} \begin{align} &\frac{d}{dt} \|\sqrt{ \rho_1} u \|_{L^2}^2 + \nu \|\nabla u\|_{L^2}^2 \le C \big(\| \nabla^2 u_1 \|_{L^2} + \|\nabla^2 u_2\|_{L^q} \big) \|\sqrt{\rho_1} u\|_{L^2}^2 \\ &\quad+ C \big(\|\nabla^3 \chi_1 \|_{L^2} + \|\nabla^3 \chi_2 \|_{L^2}\big) \|\nabla\chi\|_{L^2}^2 + \big(\|\nabla \partial_t u_2\|_{L^2}^2 + 1 \big) \|\rho\|_{L^2}^2. \end{align}\tag{38}\]
Multiplying \(\eqref{NSAC95diff}_3\) by \(\chi\) and integrating over \(\Omega\),
it follows from integration by parts and the identity \(\rho (\rho_1 + \rho_2) = - \rho^2 + 2 \rho \rho_1\) that \[\begin{align} &\frac{1}{2} \frac{d}{dt} \|\rho_{1} \chi\|_{L^2}^2 + \|\nabla \chi\|_{L^2}^2 \\ =& -\frac{1}{2} \int_{\Omega} \rho_1^{2} \mathop{\mathrm{div}}u_1 \chi^2 \,dx - \int_{\Omega} \rho (\rho_1 + \rho_2) \partial_t \chi_{2} \chi \,dx - \int_{\Omega} \rho_1^2 ( u \cdot \nabla) \chi_2 \chi \,dx \\ & + \int_{\Omega} \rho^2 ( u_2 \cdot \nabla ) \chi_2 \chi \,dx - 2 \int_{\Omega} \rho_{1} \rho ( u_2 \cdot \nabla ) \chi_2 \chi \,dx - \int_{\Omega} \rho_{1} \chi^2 (\chi _{1}^{2}+\chi _{1}\chi _{2}+\chi _{2}^{2}-1) \,dx \\ & - \int_{\Omega} \rho(\chi_{2}^3 - \chi_2 ) \chi \,dx \\ =& \sum_{i=1}^{7} M_{i}. \end{align}\] Using Lemma 1, the Hölder, Young and Sobolev inequalities, one arrives at \[\begin{align} M_1 &\le \frac{1}{2} \| \nabla u_1 \|_{L^\infty} \|\rho_1 \chi\|_{L^2}^2 \le C \| \nabla^2 u_1 \|_{L^q} \|\rho_1 \chi\|_{L^2}^2, \\ M_2 &\le ( \|\rho_1\|_{L^\infty} + \|\rho_2\|_{L^\infty} ) \|\rho\|_{L^2} \|\chi\|_{L^3} \| \partial_t \chi_2 \|_{L^6}\\ &\le C \|\rho\|_{L^2} (\|\rho_1\chi\|_{L^2} + \|\nabla \chi\|_{L^2}) (\|\rho_2 \partial_t \chi_2\|_{L^2} + \|\nabla \partial_t \chi_2\|_{L^2}) \\ &\le \frac{1}{6} \|\nabla \chi\|_{L^2}^2 + C (1 + \|\nabla \partial_t \chi_2\|_{L^2}^2) \|\rho\|_{L^2}^2 + C \|\rho_1 \chi\|_{L^2}^2, \\ M_3 &\le \|\rho_1\|_{L^\infty}^{\frac{1}{2}} \|\sqrt{\rho_1} u\|_{L^2} \|\rho_1 \chi\|_{L^2} \|\nabla \chi_2\|_{L^\infty} \\ &\le C \|\sqrt{\rho_1} u\|_{L^2} \|\rho_1 \chi\|_{L^2} \|\nabla^2 \chi_2\|_{L^2}^\frac{1}{2} \|\nabla^3 \chi_2\|_{L^2}^\frac{1}{2} \\ &\le \frac{\eta}{2} \|\nabla^3 \chi_2\|_{L^2} \|\rho_1 \chi\|_{L^2}^2 + \frac{C}{\eta} \|\sqrt{\rho_1} u\|_{L^2}^2, \\ M_4 &\le \|\rho\|_{L^2}^2 \| u_2 \|_{L^\infty} \| \nabla \chi_2 \|_{L^\infty} \|\chi\|_{L^\infty}\\ &\le C \| \nabla u_2 \|_{L^2}^{\frac{1}{2}} \|\nabla^2 u_2\|_{L^2}^{\frac{1}{2}} \| \nabla^2 \chi_2 \|_{L^2}^{\frac{1}{2}} \|\nabla^3 \chi_2\|_{L^2}^{\frac{1}{2}} \|\rho\|_{L^2}^2 \\ &\le C (1 + \|\nabla^2 u_2\|_{L^2} \|\nabla^3 \chi_2\|_{L^2}) \|\rho\|_{L^2}^2, \\ M_5 &\le 2 \|\rho\|_{L^2} \|\rho_1 \chi\|_{L^2} \| u_2 \|_{L^\infty} \| \nabla \chi_2 \|_{L^\infty} \\ &\le C \|\nabla^2 u_2\|_{L^2}^{\frac{1}{2}} \|\nabla^3 \chi_2\|_{L^2}^{\frac{1}{2}} \|\rho\|_{L^2} \|\rho_1 \chi\|_{L^2} \\ &\le C \|\rho_1 \chi\|_{L^2}^2 + C \|\nabla^2 u_2\|_{L^2} \|\nabla^3 \chi_2\|_{L^2} \|\rho\|_{L^2}^2, \\ M_6 &\le \|\rho_1 \chi\|_{L^2} \|\chi\|_{L^6} \| \chi_1^2 +\chi_2^2 + \chi_1 \chi_2 -1 \|_{L^3} \\ &\le C \|\rho_1 \chi\|_{L^2} (\|\rho_1 \chi\|_{L^2} + \|\nabla \chi\|_{L^2}) ( \|\chi_1\|_{L^6}^2 + \|\chi_1\|_{L^6} \|\chi_2\|_{L^6} + \|\chi_2\|_{L^6}^2 + 1) \\ &\le \frac{1}{6} \|\nabla \chi\|_{L^2}^2 + C \|\rho_1 \chi\|_{L^2}^2, \\ M_7 &\le \|\rho\|_{L^2} \|\chi\|_{L^6} \|\chi_2^3 - \chi_2\|_{L^3} \\ &\le C \|\rho\|_{L^2} (\|\rho_1 \chi\|_{L^2} + \|\nabla \chi\|_{L^2}) \|\chi_2\|_{L^3} (\|\chi_2\|_{L^\infty}^2 + 1) \\ &\le \frac{1}{6} \|\nabla \chi\|_{L^2}^2 + C \|\rho\|_{L^2}^2 + C \|\rho_1 \chi\|_{L^2}^2 \end{align}\] for every \(\eta>0\). Hence, we have \[\label{rho1chi} \begin{align} &\frac{d}{dt} \|\rho_1 \chi\|_{L^2}^2 + \|\nabla \chi\|_{L^2}^2 \le \eta \|\nabla^3 \chi\|_{L^2} \|\rho_1 \chi\|_{L^2}^2 + C (1 + \|\nabla^2 u_1\|_{L^q}) \|\rho_1 \chi\|_{L^2}^2 \\ &\quad+ C (1 + \|\nabla \partial_t \chi_2\|_{L^2}^2 + \|\nabla^2 u_2\|_{L^2} \|\nabla^3 \chi_2\|_{L^2}) \|\rho\|_{L^2}^2 + \frac{C}{\eta} \| \sqrt{\rho_1}u\|_{L^2} \end{align}\tag{39}\] for every \(\eta>0\).
Step 2. Growth estimates. It follows from 37 that \[\frac{d}{dt}\|\rho\|_{L^2}(t) \le C \|\nabla^2 u_1 \|_{L^q} \|\rho\|_{L^2} + C.\] By applying the Grönwall inequality, 36 yields that \[\label{GE1} \|\rho\|_{L^2}(t) \le C t, \quad \forall t \in (0, T_0).\tag{40}\]
Using \(\eqref{GE1}\), one deduces by 39 and the Poincaré inequality that \[\begin{align} &\frac{d}{dt} \|\rho_1 \chi\|_{L^2}^2 + \|\nabla \chi\|_{L^2}^2 \le C (1 + \|\nabla^3 \chi\|_{L^2} + \|\nabla^2 u_1\|_{L^q}) \|\rho_1 \chi\|_{L^2}^2 \\ &\qquad+ C (1 + \|\sqrt{t}\nabla \partial_t \chi_2\|_{L^2}^2 + \|\sqrt{t}\nabla^2 u_2\|_{L^2} \|\sqrt{t}\nabla^3 \chi_2\|_{L^2}) t + C \|\sqrt{\rho_1}\|_{L^\infty} \|\nabla u\|_{L^2} \\ &\quad\le C (1 + \|\nabla^3 \chi\|_{L^2} + \|\nabla^2 u_1\|_{L^q}) \|\rho_1 \chi\|_{L^2}^2 + C. \end{align}\] Hence, it follows from the Grönwall inequality and 36 that \[\label{GE2} \|\rho_1 \chi\|_{L^2}^2(t) + \int_0^t \|\nabla \chi\|_{L^2}^2 (s) \, ds \le C t, \quad \forall t \in (0, T_0).\tag{41}\]
Step 3. Singular \(t\)-weighted energy inequalities. Multiplying 37 by \(\frac{1}{t}\), one deduces by the Young inequality that \[\begin{align} \frac{d}{dt} \frac{\|\rho\|_{L^2}^2}{t} + \frac{\|\rho\|_{L^2}^2}{t^2} \le C \| \nabla^2 u_1 \|_{L^q} \frac{\|\rho\|_{L^2}^2}{t} + \frac{1}{2}\frac{\|\rho\|_{L^2}^2}{t^2} + C_1 \|\nabla u\|_{L^2}^2 . \end{align}\] Therefore, we have \[\label{wrho} \frac{d}{dt} \frac{\|\rho\|_{L^2}^2}{t} + \frac{\|\rho\|_{L^2}^2}{2 t^2} \le C \| \nabla^2 u_1 \|_{L^q} \frac{\|\rho\|_{L^2}^2}{t} + C_1 \|\nabla u\|_{L^2}^2.\tag{42}\]
Multiplying 39 by \(\frac{1}{\sqrt{t}}\) yields \[\begin{align} &\frac{d}{dt} \frac{\|\rho_{1} \chi\|_{L^2}^2}{\sqrt{t}} + \frac{\|\rho_{1} \chi\|_{L^2}^2}{2 t^{\frac{3}{2}}} + \frac{\|\nabla \chi\|_{L^2}^2}{ \sqrt{t}} \\ \le& \frac{C}{\eta\sqrt{t}} \| \sqrt{\rho_1}u\|_{L^2} + \eta \sqrt{t} \|\sqrt{t} \nabla^3 \chi\|_{L^2} \frac{\|\rho_1 \chi\|_{L^2}^2}{t^{\frac{3}{2}}} + C (1 + \|\nabla^2 u_1\|_{L^q}) \frac{\|\rho_{1} \chi\|_{L^2}^2}{\sqrt{t}} \\ &+ C (1 + \sqrt{t}\|\nabla \partial_t \chi_2\|_{L^2}^2 + \|\nabla^2 u_2\|_{L^2} \|\sqrt{t}\nabla^3 \chi_2\|_{L^2}) \frac{\|\rho\|_{L^2}^2}{t} \\ \le& \frac{C}{\eta\sqrt{t}} \| \sqrt{\rho_1}u\|_{L^2} + \hat{C} \eta \sqrt{t} \frac{\|\rho_1 \chi\|_{L^2}^2}{t^{\frac{3}{2}}} + (1 + \|\nabla^2 u_1\|_{L^q}) \frac{\|\rho_{1} \chi\|_{L^2}^2}{\sqrt{t}} \\ &\qquad+ C (1 + \|\nabla \partial_t \chi_2\|_{L^2} + \|\nabla^2 u_2\|_{L^2}) \frac{\|\rho\|_{L^2}^2}{t} \\ \end{align}\] for each \(\eta>0\). Let \(\eta = \frac{1}{4\sqrt{t}\hat{C}}\), one gets \[\begin{align}\label{wchi} &\frac{d}{dt} \frac{\|\rho_{1} \chi\|_{L^2}^2}{\sqrt{t}} + \frac{\|\rho_{1} \chi\|_{L^2}^2}{4 t^{\frac{3}{2}}} + \frac{\|\nabla \chi\|_{L^2}^2}{ \sqrt{t}} \le C \| \sqrt{\rho_1}u\|_{L^2} + (1 + \|\nabla^2 u_1\|_{L^q}) \frac{\|\rho_{1} \chi\|_{L^2}^2}{\sqrt{t}} \\ &\quad + C (1 + \|\nabla \partial_t \chi_2\|_{L^2} + \|\nabla^2 u_2\|_{L^2}) \frac{\|\rho\|_{L^2}^2}{t}. \\ \end{align}\tag{43}\]
Recalling 38 , we have \[\label{wu} \begin{align} &\frac{d}{dt} \|\sqrt{ \rho_1} u \|_{L^2}^2 + \nu \|\nabla u\|_{L^2}^2 \le C \big(\| \nabla^2 u_1 \|_{L^2} + \|\nabla^2 u_2\|_{L^q} \big) \|\sqrt{\rho_1} u\|_{L^2}^2 \\ & + C \big(\|\sqrt{t}\nabla^3 \chi_1 \|_{L^2} + \|\sqrt{t}\nabla^3 \chi_2 \|_{L^2}\big) \frac{\|\nabla\chi\|_{L^2}^2}{\sqrt{t}} + C \big(\|\sqrt{t} \nabla \partial_t u_2\|_{L^2}^2 + 1 \big) \frac{\|\rho\|_{L^2}^2}{t} \\ \le& C_2 \frac{\|\nabla\chi\|_{L^2}^2}{\sqrt{t}} + C \big(\| \nabla^2 u_1 \|_{L^2} + \|\nabla^2 u_2\|_{L^q} \big) \|\sqrt{\rho_1} u\|_{L^2}^2 + C \big(\|\sqrt{t} \nabla \partial_t u_2\|_{L^2}^2 + 1 \big) \frac{\|\rho\|_{L^2}^2}{t} \end{align}\tag{44}\] Let \(C_3\) and \(C_4\) be positive constants such that \(\nu C_3 \ge 2C_1\) and \(C_4 \ge 2C_2C_3\). Multiplying 43 by \(C_4\) and 44 by \(C_3\), and combining the resulting with 42 , one obtains by the Hölder inequality that \[\begin{align} &\frac{d}{dt} \left( \frac{\|\rho\|_{L^2}^2}{t} + C_3 \|\sqrt{ \rho_1} u \|_{L^2}^2 + C_4 \frac{\|\rho_{1} \chi\|_{L^2}^2}{\sqrt{t}} \right) + \frac{\|\rho\|_{L^2}^2}{2 t^2} + C_1 \|\nabla u\|_{L^2}^2 + C_4 \frac{\|\rho_{1} \chi\|_{L^2}^2}{4 t^{\frac{3}{2}}} + C_2 C_3 \frac{\|\nabla \chi\|_{L^2}^2}{\sqrt{t}} \\ &\quad \le C (1 + \| \nabla^2 u_1 \|_{L^q} + \|\nabla^2 u_2\|_{L^q} + \| \nabla \partial_t \chi_2 \|_{L^2} +\| \sqrt{t}\nabla \partial_t u_2 \|_{L^2}^2 + 1) \left( \frac{\|\rho\|_{L^2}^2}{t} + \|\sqrt{ \rho_1} u \|_{L^2}^2\right) \\ &\qquad+ C (1 + \|\nabla^2 u_1\|_{L^q}) \frac{\|\rho_{1} \chi\|_{L^2}^2}{\sqrt{t}}. \end{align}\] Therefore, noting 36 , 40 and 41 , one concludes from the Grönwall inequality that \[\left(\frac{\|\rho\|_{L^2}^2}{t}+ \|\sqrt{\rho_1} u\|_{L^2}^2 + \frac{\|\rho_{1} \chi\|_{L^2}^2}{\sqrt{t}} \right)(t) + \int_0^t \left(\|\nabla u\|_{L^2}^2 + \frac{\|\nabla \chi\|_{L^2}^2}{\sqrt{t}}\right) \,ds \equiv 0\] for every \(t \in (0, T_0)\). By Lemma 1 and the Poincaré inequality, we have \((\rho, u, \chi) \equiv 0\) in \((0, T_0)\), which completes the proof of uniqueness. 0◻
Finally, we establish a blow-up criterion (Theorem 1.2) which shows that the obtained local strong solution cannot break down unless certain norms blow up. We argue by contradiction. Were ?? false, i.e. \[\label{assump95contra} C_0:= \int_0^{T^*} \left( \|\nabla u\|_{L^\infty} + \|u\|_{L^\infty}^2 + \|\nabla \chi\|_{L^\infty}^2 \right) < \infty,\tag{45}\] we will show that there exists a generic constant \(C\), depending only on that depend on \(C_0\), \(A\), \(\gamma\), \(\nu\), \(\lambda\), \(q\), \(\Omega\) and \(\Phi_0\), such that \[\sup_{0 \le t < T^*} \| ( \nabla \rho, \rho \chi_t, \mu, \nabla u, \mathop{\mathrm{div}}u) \|_{L^2}^2 + \int_0^{T^*} { \| ( \sqrt{\rho} u_t, \nabla\chi_t, \nabla^2 u ) \|_{L^2}^2 } \, dt \le C\] and \[\sup_{0 \le t < T^*} ( \|\nabla \rho\|_{L^q}^2 + \| ( \sqrt{t\rho} u_t, \sqrt{t} \nabla \chi_t, \sqrt{t} \nabla^2u ) \|_{L^2}^2 ) + \int_0^{T^*} { \| ( \sqrt{t} \nabla u_t, \sqrt{t} \mathop{\mathrm{div}}u_t, \sqrt{t} \rho \chi_{tt}) \|_{L^2}^2 } \, dt \le C, \\\] which contradicts the definition of \(T^*\).
Lemma 10. Under the assumption 45 , we have \[\begin{align} \sup_{0 \le t < T_*} \|\rho\|_{L^\infty}^2 + \int_{0}^{T_*} ( \| \rho \chi_t \|_{L^2}^2 + \|\nabla^2 \chi\|_{L^2}^2 ) ds \le C. \end{align}\]
Proof. According to 4 and 45 , we have \[\begin{align} \sup_{0 \le t \le T_*} \|\rho\|_{L^\infty} \le \| \rho_0\|_{L^\infty} \mathop{\mathrm{exp}}\left\{ \int_{0}^{T_*} \|\nabla u\|_{L^\infty} \,dt \right\} \le C. \end{align}\] It follows from \((\ref{NSAC})_3\) that \[\begin{align} \| \rho\chi_t \|_{L^2}^2 & \le C\| \rho u \cdot \nabla \chi \|_{L^2}^2 + C\|\mu\|_{L^2}^2 \\ & \le C( \|\rho\|_{L^\infty} \| \sqrt\rho u \|_{L^2}^2 \| \nabla\chi \|_{L^\infty}^2 + \|\mu\|_{L^2}^2 ). \end{align}\] Integrating the above estimates over \((0,T_*)\), combining 45 with ?? , we have \[\begin{align} \int_{0}^{T_*} \| \rho\chi_t \|_{L^2}^2 \,dt \le C \int_{0}^{T_*} ( \|\rho\|_{L^\infty} \| \sqrt\rho u \|_{L^2}^2 \| \nabla\chi \|_{L^\infty}^2 + \|\mu\|_{L^2}^2 ) \,dt \le C. \end{align}\] By using \(\eqref{NSAC}_4\) and ?? , we obtain \[\begin{align} \int_{0}^{T_*} \left\| \Delta\chi \right\|_{L^2}^2 \, dt & \le C \int_{0}^{T_*} ( \| \rho \mu \|_{L^2}^2 + \| \rho f(\chi) \|_{L^2}^2 ) \, dt \\ & \le C \int_{0}^{T_*} \|\rho\|_{L^\infty}^2 ( \|\mu\|_{L^2}^2 + \|f(\chi)\|_{L^2}^2) \, dt. \end{align}\] ◻
Lemma 11. Under the assumption 45 , we have \[\sup_{0 \le t < T^*} \| ( \nabla \rho, \rho \chi_t, \mu, \nabla u, \mathop{\mathrm{div}}u) \|_{L^2}^2 + \int_0^{T^*} { \| ( \sqrt{\rho} u_t, \nabla\chi_t, \nabla^2 u ) \|_{L^2}^2 } \, dt \le C.\]
Proof. Denote \[\begin{align} \mathcal{E}_1(t) &= \|(\nabla \rho, \rho \chi_t, \mu, \nabla u, \mathop{\mathrm{div}}u)\|_{L^2}^2(t),\\ \mathcal{D}_1( t ) &= \| ( \sqrt{\rho} u_t, \nabla\chi_t, \nabla^2 u ) \|_{L^2}^2 (t), \\ \mathcal{F}_1(t) &= 1 + \|u\|_{L^\infty}^2(t) + \|\nabla \chi \|_{L^\infty}^2(t) + \|\nabla u\|_{L^\infty}(t). \end{align}\] It follows from \(\eqref{NSAC}_1\) that \[\begin{align} \label{be2:1} \frac{d}{dt}\|\nabla\rho\|_{L^2}^2 &\le C\|\nabla u\|_{L^\infty}\|\nabla\rho\|_{L^2}^2 + C \|\rho\|_{L^\infty} \|\nabla^2 u\|_{L^2} \|\nabla\rho\|_{L^2} \notag\\ &\le \eta\|\nabla^2 u\|_{L^2}^2 + C_\eta(\|\nabla u\|_{L^\infty}+1) \|\nabla\rho\|_{L^2}^2 \end{align}\tag{46}\] for every \(\eta > 0\).
Differentiating 6 with respect to \(t\), multiplying by \(\chi_t\), and integrating over \(\Omega\), we can obtain \[\begin{align} &\frac{1}{2} \frac{d}{dt} \|\rho \chi_t\|_{L^2}^{2} + \| \nabla \chi_t \|_{L^2}^{2} \\ =& \frac{1}{2} \int_{\Omega} \rho^2 \mathop{\mathrm{div}}u \chi_t^{2} \,dx - 2 \int_{\Omega} \rho^{2} (u \cdot \nabla\chi_t)\chi_t\,dx + \int_{\Omega} \rho^2 \mathop{\mathrm{div}}u ( u\cdot \nabla \chi) \chi_t \,dx \\ & -\int_{\Omega} \rho^{2} ( u \cdot \nabla ) u \cdot \nabla \chi \chi_t \,dx - \int_\Omega\rho^2 u \otimes u : \nabla^2\chi \chi_t \,dx - \int_\Omega\rho^2 (u \cdot \nabla\chi) (u \cdot \nabla\chi_t) \,dx \\ & - \int_{\Omega} \rho (u_t \cdot\nabla \chi ) \rho \chi_t \,dx - \int_{\Omega} \rho ( u \cdot \nabla \chi_t ) f(\chi) \,dx - \int_{\Omega} \rho ( u\cdot \nabla \chi ) f'(\chi) \chi_t \,dx \\ & - \int_{\Omega} \rho f'(x) \chi_t^{2} \,dx \\ =& \sum_{i=1}^{10} O_i. \end{align}\] Applying the Hölder, Sobolev and Young inequalities, and using \(\eqref{NSAC}_3\), Lemma 1–2, Lemma 10 and Corollary 1, we obtain \[\begin{align} O_1 + O_3 &\le \|\nabla u\|_{L^\infty} (\|\rho \chi_t\|_{L^2}^2 + \|\mu\|_{L^2}^2 ), \\ O_2 + O_6 + O_8 &\le C \|\rho\|_{L^\infty} \| u\|_{L^\infty} \|\nabla \chi_t\|_{L^2} (\|\rho \chi_t\|_{L^2} + \|\mu\|_{L^2} + \|f(\chi)\|_{L^2})\\ &\le \frac{1}{4} \|\nabla \chi_t\|_{L^2}^2 + C \| u\|_{L^\infty}^2 ( \|\rho \chi_t\|_{L^2}^2 + \|\mu\|_{L^2}^2 + 1 ),\\ O_4 &\le C \|\rho\|_{L^\infty} \|u \|_{L^\infty} \|\nabla \chi\|_{L^\infty} \|\nabla u\|_{L^2} \|\rho \chi_t \|_{L^2} \\ & \le C( \|u\|_{L^\infty}^2 + \|\nabla\chi\|_{L^\infty}^2 ) ( \|\nabla u\|_{L^2}^2 + \|\rho \chi_t \|_{L^2}^2 ), \\ O_5 & \le C \|\rho\|_{L^\infty} \|u\|_{L^\infty}^2 \|\nabla^2 \chi \|_{L^2} \|\rho\chi_t\|_{L^2} \\ &\le C\|u\|_{L^\infty}^2 ( \| \Delta \chi \|_{L^2}^2 + \|\rho \chi_t \|_{L^2}^2 ) \\ & \le C \|u\|_{L^\infty}^2 ( \|\mu\|_{L^2}^2 + \|f(\chi)\|_{L^2}^2 + \|\rho \chi_t \|_{L^2}^2 ) \\ &\le C \| u\|_{L^\infty}^2 ( \|\rho \chi_t\|_{L^2}^2 + \|\mu\|_{L^2}^2 + 1 ),\\ O_7 & \le \|\rho\|_{L^\infty}^{\frac{1}{2}} \| \sqrt \rho u_t \|_{L^2} \|\nabla \chi\|_{L^\infty} \|\rho\chi_t \|_{L^2} \le \frac{1}{16} \| \sqrt{\rho} u_t \|_{L^2}^2 + C \|\nabla \chi\|_{L^\infty}^2 \|\rho \chi_t\|_{L^2}^2 , \\ O_9 + O_{10} &\le \| f'(\chi)\|_{L^3} \|\mu\|_{L^2}\|\chi_t\|_{L^6} \le C\|\mu\|_{L^2} ( \| \rho\chi_t \|_{L^2} + \|\nabla\chi_t\|_{L^2} ) \\ &\le \frac{1}{4} \| \nabla \chi_t \|_{L^2}^2 + C \|\mu\|_{L^2}^2. \end{align}\] Thus, one obtains \[\begin{align} \label{be2:2} \frac{d}{dt}\|\rho \chi_t \|_{L^2}^{2} +\|\nabla \chi_t \|_{L^2}^{2} \le \frac{1}{8}\|\sqrt \rho u_t \|_{L^2}^{2} + C \mathcal{F}_1(t) {(\mathcal{E}_1(t) + 1)}. \end{align}\tag{47}\]
Testing \(\eqref{NSAC}_2\) with \(u_{t}\), and integrating over \(\Omega\), one deduces by 10 and inetagration by parts that \[\begin{align} &\frac{1}{2}\frac{d}{dt} \left(\nu \|\nabla u\|_{L^2}^2 + (\lambda + \nu) \|\mathop{\mathrm{div}}u\|_{L^2}^2 + \|\mu\|_{L^2}^2 + \| \rho ( u \cdot \nabla \chi) \|_{L^2}^2\right) + \| \sqrt \rho u_t \|_{L^2}^2 \\ =& - \int_\Omega\rho (u \cdot \nabla )u \cdot u_t \,dx + ( \gamma - 1) \int_\Omega P(\rho) | \mathop{\mathrm{div}}u|^2 \,dx - \int_\Omega P(\rho) u \cdot \nabla \mathop{\mathrm{div}}u \,dx \\ & - \int_\Omega(\nabla \chi_t \cdot \nabla ) u \cdot \nabla \chi \,dx - \int_\Omega(\nabla \chi \cdot \nabla ) u \cdot \nabla \chi_t \,dx + \int_\Omega\nabla \chi \cdot \nabla \chi_t \mathop{\mathrm{div}}u \,dx\\ & + \frac{d}{dt} \left(\int_\Omega P(\rho) \mathop{\mathrm{div}}u \,dx + \frac{1}{2} \|\rho \chi_t\|_{L^2}^2 - \int_\Omega\rho F'(\chi) (u \cdot \nabla \chi) \right) \\ =& \sum_{i=1}^{6} P_i + \hat{G}'(t) + \frac{1}{2} \frac{d}{dt} \|\rho \chi_t\|_{L^2}^2. \end{align}\] where \(\hat{G}(t) := \int_{\Omega} P(\rho) \mathop{\mathrm{div}}u \,dx - \int_\Omega\rho F'(\chi) (u \cdot \nabla\chi )\,dx\). By virtue of the Hölder and Young inequalities, together with Lemma 2 and Lemma 10, we have \[\begin{align} &P_1 \le C \|\rho\|_{L^\infty}^{\frac{1}{2}} \|u\|_{L^\infty} \|\nabla u\|_{L^2} \| \sqrt \rho u_t \|_{L^2} \le \frac{1}{8} \| \sqrt \rho u_t \|_{L^2}^2 + C \|u\|_{L^\infty}^2 \|\nabla u\|_{L^2}^2, \\ &P_2 + P_3 \le C \|\rho\|_{L^\infty}^\gamma \|\nabla u\|_{L^2}^2 + C \|\rho\|_{L^\infty}^{\gamma - \frac{1}{2}} \| \sqrt \rho u \|_{L^2}^2 \|\nabla^2 u\|_{L^2}^2 \le \eta \|\nabla^2 u\|_{L^2}^2 + C, \\ &P_4 + P_5 + P_6 \le 3 \| \nabla \chi_t \|_{L^2} \|\nabla u\|_{L^2} \|\nabla \chi\|_{L^\infty} \le \frac{1}{2} \| \nabla \chi_t \|_{L^2}^2 + C \|\nabla u\|_{L^2}^2 \|\nabla \chi\|_{L^\infty}^2 \end{align}\] for every \(\eta > 0\). Thus, we have \[\begin{align} \label{be2:3} &\frac{1}{2}\frac{d}{dt} \left(\nu \|\nabla u\|_{L^2}^2 + (\lambda + \nu) \|\mathop{\mathrm{div}}u\|_{L^2}^2 + \|\mu\|_{L^2}^2 + \| \rho ( u \cdot \nabla \chi) \|_{L^2}^2\right) + \frac{7}{8} \|\sqrt \rho u_t\|_{L^2}^2 \notag\\ &\quad \le \frac{1}{2} \| \nabla \chi_t \|_{L^2}^2 + \eta\|\nabla^2 u\|_{L^2}^2 + {C \mathcal{F}_1(t) (\mathcal{E}_1(t) + 1) + \frac{1}{2} \frac{d}{dt} \|\rho \chi_t\|_{L^2}^2} + \hat{G}'(t). \end{align}\tag{48}\] We deduce by 46 , 47 and 48 that \[\begin{align}\label{be2:4} &\frac{1}{2}\frac{d}{dt} \left( 2\|\nabla \rho\|_{L^2}^2 + \|\rho \chi_t\|_{L^2}^2 + \|\mu\|_{L^2}^2 + \|\rho (u \cdot \nabla \chi)\|_{L^2}^2 + \nu\|\nabla u\|_{L^2}^2 + (\lambda + \nu)\|\mathop{\mathrm{div}}u\|_{L^2}^2 \right) \\ &\quad + \frac{3}{4}\|\sqrt\rho u_t \|_{L^2}^2 + \|\nabla\chi_t\|_{L^2}^2 \le 2 \eta \|\nabla^2 u\|_{L^2}^2 + C \mathcal{F}_1(t) (\mathcal{E}_1(t) + 1) + \hat{G}'(t). \end{align}\tag{49}\]
Applying the elliptic estimates to \(\eqref{NSAC}_2\), one has \[\begin{align} \| \nabla^2u \|_{L^2}^2 &\le C ( \| \rho u_t \|_{L^2}^2 + \| \rho ( u \cdot \nabla ) u \|_{L^2}^2 + \| \nabla P \|_{L^2}^2 + \| \Delta \chi \nabla \chi \|_{L^2}^2 ) \\ &\le C ( \|\rho\|_{L^\infty}^{\frac{1}{2}} \| \sqrt \rho u_t \|_{L^2}^2 + \|\rho\|_{L^\infty}^2 \|u\|_{L^\infty}^2 \|\nabla u\|_{L^2}^2 + \|\rho\|_{L^\infty}^{2 ( \gamma - 1 ) } \|\nabla \rho\|_{L^2}^2 + \|\nabla \chi\|_{L^\infty}^2 \| \Delta \chi \|_{L^2}^2 ) \\ & \le C ( \| \sqrt \rho u_t \|_{L^2}^2 + \|u\|_{L^\infty}^2 \|\nabla u\|_{L^2}^2 + \|\nabla \rho\|_{L^2}^2 + \|\nabla \chi\|_{L^\infty}^2 (\|\mu\|_{L^2}^2 + 1 ) ), \end{align}\] that is, \[\begin{align} \label{be2:5} \|\nabla^2 u\|_{L^2}^2 \le C \|\sqrt \rho u_t\|_{L^2}^2 + C (\|u\|_{L^\infty}^2 + \|\nabla \chi\|_{L^\infty}^2 + 1) (\mathcal{E}_1(t) + 1). \end{align}\tag{50}\] Plugging 50 into 49 and choosing \(\eta\) sufficiently small, one gets \[\begin{align} &\frac{1}{2}\frac{d}{dt} \left( 2\|\nabla \rho\|_{L^2}^2 + \|\rho \chi_t\|_{L^2}^2 + \|\mu\|_{L^2}^2 + \|\rho (u \cdot \nabla \chi)\|_{L^2}^2 + \nu\|\nabla u\|_{L^2}^2 + (\lambda + \nu)\|\mathop{\mathrm{div}}u\|_{L^2}^2 \right)\\ &\quad+ \frac{1}{2}\|\sqrt\rho u_t\|_{L^2}^2 + \frac{1}{2}\|\nabla\chi_t\|_{L^2}^2 + \eta \|\nabla^2 u\|_{L^2}^2 \le C \mathcal{F}_1(t) (\mathcal{E}_1(t) + 1) + \hat{G}'(t). \end{align}\] Integrating the above estimate over \((0, t)\), we obtain \[\begin{align} \label{be2:6} \mathcal{E}_1(t) + \int_0^t \mathcal{D}_1(s) \, ds \le C \int_0^t \mathcal{F}_1(s) ( \mathcal{E}_1(s) + 1 )\, ds + C_2 |\hat{G}(t)|, \end{align}\tag{51}\] where we have used the fact that \[\begin{align} |\hat{G}(0)| &\le \left| \int_\Omega P(\rho_0) \mathop{\mathrm{div}}u_0 \,dx \right| + \left| \int_\Omega\rho_0 F'(\chi_0)(u_0 \cdot \nabla\chi_0) \,dx \right| \\ & \le C \|P(\rho_0)\|_{L^2} \|\nabla u_0\|_{L^2} + C \|\rho_0\|_{L^\infty} \| F'(\chi_0) \|_{L^2} \|u_0\|_{L^6} \|\nabla \chi_0\|_{L^2}^{1/2}\|\nabla \chi_0\|_{L^6}^{1/2} \\ &\le C, \end{align}\] and \(\mathcal{E}_1(0) \le C.\) Besides, observing that \[\begin{align} C_2 |\hat{G}(t)| &\le C \|P(\rho)\|_{L^2} \|\nabla u\|_{L^2}(t) + C \|F'(\chi)\|_{L^2} (\|\mu\|_{L^2} + \|\rho \chi_t\|_{L^2})(t)\\ &\le C \mathcal{E}_1^{\frac{1}{2}}(t) \le \frac{1}{2} \mathcal{E}_1(t) + C. \end{align}\] Plugging the above estimate into 51 , then we have \[\mathcal{E}_1(t) + \int_0^t \mathcal{D}_1(s) \, ds \le C \int_0^t \mathcal{F}_1(s) ( \mathcal{E}_1(s) + 1 )\, ds.\] It follows from \(\mathcal{F}_1 \in L^1(0, T^*)\) and the Grönwall inequality that \[\sup_{0 \le t < T^*} \mathcal{E}_1(t) + \int_0^{T^*} \mathcal{D}_1(s) \, ds \le C.\] The proof is complete. ◻
The last estimates is the most difficult, as it requires combining the estimates for \(\rho, u, \chi\) and carefully handling the time weights.
Lemma 12. Under the assumption 45 , we have \[\begin{align} &\sup_{0 \le t < T^*} \left(\|\nabla \rho\|_{L^q}^2 + \|(\sqrt{t} \sqrt\rho u_t, \sqrt{t} \nabla \chi_t, \sqrt{t} \nabla^2u , \sqrt{t} \nabla^3\chi)\|_{L^2}^2 \right) \\ &+ \int_0^{T^*} { \big(\|(\sqrt{t} \nabla u_t, \sqrt{t} \mathop{\mathrm{div}}u_t, \sqrt{t} \rho \chi_{tt}, \nabla^3 \chi) \|_{L^2}^2 + \|\nabla^2 u\|_{L^q} + \|\sqrt{t} \nabla^2 u\|_{L^q}^2}\big) \,dt \le C. \end{align}\]
Proof. Denote \[\begin{align} \mathcal{E}_2(t) &= \|\nabla\rho\|_{L^q}^2(t) + \|( \sqrt{t} \sqrt\rho u_t, \sqrt{t} \nabla \chi_t ) \|_{L^2}^2(t), \\ \mathcal{D}_2(t) &= \|( \sqrt{t} \nabla u_t, \sqrt{t} \mathop{\mathrm{div}}u_t, \sqrt{t} \rho \chi_{tt}, \nabla^3 \chi ) \|_{L^2}^2(t), \\ \mathcal{F}_2(t) &= 1 + t^{- \frac{3q-6}{2q}} + \|\nabla u\|_{L^\infty}(t) + \|\nabla \chi\|_{L^\infty}^2(t) + \|( \nabla \chi_t, \nabla^2 u, \sqrt{\rho} u_t ) \|_{L^2}^2(t). \end{align}\]
Step 1. Estimates for the density. It follows from 5 that \[\begin{align} \frac{d}{dt} \|\nabla \rho\|_{L^q}^2 \le C_1 ( \|\nabla u\|_{L^\infty} \|\nabla\rho\|_{L^q}^2 + \|\nabla^2 u\|_{L^q} \|\nabla\rho\|_{L^q} ). \end{align}\] Recall the elliptic estimates \[\begin{align} \|\nabla^2 u\|_{L^q} &\le C_2 ( \|\rho u_{t}\|_{L^q} + \| \rho ( u \cdot \nabla ) u \|_{L^q} + \| \nabla (P(\rho)) \|_{L^q} + \| \Delta \chi \nabla \chi \|_{L^q} ). \end{align}\] Combining the Hölder, Young and Gagliardo-Nirenberg inequalities with Lemma 1 and Lemma 10–11, we have \[\begin{align} \|\rho u_{t}\|_{L^q} &\le C \|\rho\|_{L^\infty}^{\frac{5q - 6}{4q}} \|\sqrt \rho u_{t}\|_{L^2}^{\frac{6 - q}{2q}} \|\nabla u_{t}\|_{L^2}^{\frac{3q - 6}{2q}}\\ &\le \frac{\eta}{C_1 C_2} \|\sqrt{t} \nabla u_{t}\|_{L^2}^2 + C (\|\sqrt \rho u_{t}\|_{L^2}^2 + \big( \frac{1}{\eta t} \big)^{\frac{3q - 6}{2q}}), \\ \| \rho ( u \cdot \nabla ) u \|_{L^q} &\le \|\rho\|_{L^\infty} \|u\|_{L^\infty} \|\nabla u\|_{L^q}\\ &\le C \|\nabla u\|_{L^2}^{\frac{1}{2}} \|\nabla^2 u\|_{L^2}^{\frac{1}{2}} \|\nabla u\|_{L^2}^{\frac{6 - q}{2q}} \|\nabla^2 u\|_{L^2}^{\frac{3q - 6}{2q}} \le C\|\nabla^2 u\|_{L^2}^{2 - \frac{3}{q}},\\ \|\nabla P(\rho)\|_{L^q} &\le C \|\rho\|_{L^\infty}^{\gamma - 1} \|\nabla \rho\|_{L^q} \le C\|\nabla \rho\|_{L^q},\\ \| \Delta \chi \nabla \chi \|_{L^q} &\le \|\nabla \chi\|_{L^\infty} \|\nabla^2 \chi\|_{L^q}\\ &\le C\|\nabla^2 \chi\|_{L^2}^{\frac{1}{2}} \| \nabla^3 \chi \|_{L^2}^{\frac{1}{2}} \|\nabla^2 \chi\|_{L^2}^{\frac{6 - q}{2q}} \| \nabla^3 \chi \|_{L^2} ^{\frac{3q - 6}{2q}} \le C\| \nabla^3 \chi \|_{L^2} ^{2 - \frac{3}{q}} \end{align}\] for every \(\eta > 0\). By using the Young inequality, the above estimates yields \[\begin{align} \label{be3:rho1} \|\nabla^2 u\|_{L^q} &\le C ( \|\sqrt{t} \nabla u_{t}\|_{L^2}^2 + \mathcal{F}_2(t) +\mathcal{E}_2(t) + \| \nabla^3 \chi \|_{L^2} ^{2 - \frac{3}{q}} ), \end{align}\tag{52}\] and \[\begin{align} \label{be3:rho2} \frac{d}{dt} \|\nabla \rho\|_{L^q}^2 \le \eta \|\sqrt{t} \nabla u_{t}\|_{L^2}^2 \|\nabla\rho\|_{L^q} + C \mathcal{F}_2(t) \mathcal{E}_2(t) + C ( \big( \frac{1}{\eta t} \big)^{\frac{3q - 6}{2q}} + \| \nabla^3 \chi \|_{L^2} ^{2 - \frac{3}{q}} ) \|\nabla\rho\|_{L^q} \end{align}\tag{53}\] for every \(\eta > 0\). Taking \(\eta = \frac{\nu}{8 \big( \nabla \|\rho\|_{L^q} + 1\big)}\) in 53 , one has \[\begin{align} \label{be3:rho3} \frac{d}{dt} \|\nabla \rho\|_{L^q}^2 \le \frac{\nu}{8} \|\sqrt{t} \nabla u_{t}\|_{L^2}^2 + C \mathcal{F}_2(t) \big( \mathcal{E}_2(t) + 1 \big) + C \| \nabla^3 \chi \|_{L^2} ^{2 - \frac{3}{q}} \|\nabla\rho\|_{L^q}. \end{align}\tag{54}\]
Applying the operator \(\nabla\) to \(\eqref{NSAC}_4\), then we have \[\begin{align} \|\nabla \Delta \chi\|_{L^2} &\le \big( \| 2 \rho \nabla\rho \chi_t \|_{L^2} + \| \rho^2 \nabla\chi_t \|_{L^2} + \| 2 \rho \nabla\rho ( u \cdot \nabla\chi ) \|_{L^2} + \|\rho^2 \nabla u \nabla\chi\|_{L^2} \\ &\quad + \| \rho^2 \nabla^2\chi u \|_{L^2} + \|\nabla\rho f(\chi)\|_{L^2} + \|\rho f'(\chi) \nabla\chi\|_{L^2} \big). \end{align}\] By Lemma 1–2, Lemma 10–11, the Hölder, Young, Sobolev and Gagliardo-Nirenberg inequalities, terms on the right-hand side are estimated as follows \[\begin{align} \| 2 \rho \nabla\rho \chi_t \|_{L^2} + \| \rho^2 \nabla\chi_t \|_{L^2} &\le 2 \|\rho\|_{L^\infty} \|\nabla\rho\|_{L^3} \|\chi_{t}\|_{L^6} + \|\rho\|_{L^\infty}^2 \|\nabla\chi_{t}\|_{L^2} \\ &\le C \|\nabla\rho\|_{L^2}^{\frac{2q-6}{3q-6}} \|\nabla\rho\|_{L^q}^{\frac{q}{3q-6}} ( \| \rho \chi_{t} \|_{L^2} + \|\nabla\chi_{t}\|_{L^2} ) + C \|\nabla\chi_{t}\|_{L^2}\\ &\le C (1 + \|\nabla\chi_{t}\|_{L^2} ) ( \|\nabla \rho\|_{L^q}^{\frac{q}{3q-6}} + 1 ), \\ \|2 \rho \nabla\rho (u \cdot \nabla) \chi\|_{L^2} + \|\rho^2 \nabla u \nabla\chi\|_{L^2} &\le ( 2 \|\rho\|_{L^\infty} \|\nabla\rho\|_{L^2} \|u\|_{L^\infty} + \|\rho\|_{L^\infty}^2 \|\nabla u\|_{L^2} ) \|\nabla\chi\|_{L^\infty} \\ &\le C ( \|\nabla u\|_{L^2}^{\frac{1}{2}} \|\nabla^2 u\|_{L^2}^{\frac{1}{2}} + 1 ) \|\nabla^2 \chi\|_{L^2}^{\frac{1}{2}} \|\nabla ^3 \chi\|_{L^2} ^{\frac{1}{2}} \\ &\le C ( \|\nabla^2 u\|_{L^2}^{\frac{1}{2}} + 1 ) \| \nabla \Delta \chi\|_{L^2} ^{\frac{1}{2}} \\ &\le \frac{1}{2} \|\nabla \Delta \chi\|_{L^2} + C ( \|\nabla^2 u\|_{L^2} + 1 ),\\ \| \rho^2 \nabla^2\chi u \|_{L^2} &\le C \|\rho\|_{L^\infty}^2 \|\nabla^2 \chi\|_{L^2} \| u\|_{L^\infty} \le C \|\nabla^2 u\|_{L^2}^{\frac{1}{2}} \\ &\le C ( \|\nabla^2 u\|_{L^2} + 1 ),\\ \|\nabla\rho f(\chi)\|_{L^2} + \|\rho f'(\chi) \nabla\chi\|_{L^2} &\le C ( \|\nabla \rho\|_{L^2} \|f(\chi) \|_{L^\infty} + C \|\rho\|_{L^\infty} \|f'(\chi) \|_{L^3}\|\nabla \chi\|_{L^6} ) \le C. \end{align}\] Thus, we have \[\begin{align} \|\nabla \Delta \chi\|_{L^2} &\le C (1 + \|\nabla\chi_{t}\|_{L^2} ) (1 + \|\nabla \rho\|_{L^q}^{\frac{q}{3q-6}} ) + C \|\nabla^2 u\|_{L^2}. \end{align}\] Then the \(H^3-\) estimates of Neumann-Laplacian implies that \[\begin{align} \label{be3:chi1} \| \nabla^3 \chi \|_{L^2} &\le C (1 + \|\nabla\chi_{t}\|_{L^2} ) (1 + \|\nabla \rho\|_{L^q}^{\frac{q}{3q-6}} ) + C \|\nabla^2 u\|_{L^2}. \end{align}\tag{55}\] Noticing that for \(q\in(3,6)\), it holds that \[\begin{align} \| \nabla^3 \chi \|_{L^2} ^{2 - \frac{3}{q}} \|\nabla\rho\|_{L^q} &\le C (1 + \|\nabla\chi_{t}\|_{L^2}^{2 - \frac{3}{q}} ) (1 + \|\nabla \rho\|_{L^q}^{\frac{q}{3q-6}(2 - \frac{3}{q})} ) \|\nabla\rho\|_{L^q} + C \|\nabla^2 u\|_{L^2}^{2 - \frac{3}{q}} \|\nabla\rho\|_{L^q}\\ &\le C (1 + \|\nabla\chi_{t}\|_{L^2}^2 + \|\nabla^2 u\|_{L^2}^2 ) (1 + \|\nabla \rho\|_{L^q}^2 ), \end{align}\] Combining 54 with 55 , and using the Young inequality, one gets \[\begin{align} \label{be3:rho4} \frac{d}{dt} \|\nabla \rho\|_{L^q}^2 \le \frac{\nu}{8} \|\sqrt{t} \nabla u_{t}\|_{L^2}^2 + C \mathcal{F}_2(t) \big( \mathcal{E}_2(t) + 1 \big). \end{align}\tag{56}\]
Step 2. Estimates for the velocity. Recalling 13 , we have \[\begin{align} \label{be3:u1} &\quad \frac{1}{2} \frac{d}{dt} \| \sqrt{t} \sqrt\rho u_{t} \|_{L^2}^{2} + \nu \|\sqrt{t} \nabla u_{t}\|_{L^2}^{2} + ( \lambda + \nu ) \| \sqrt{t} \mathop{\mathrm{div}}u_{t} \|_{L^2}^{2} \\ &= \frac{1}{2}\|\sqrt\rho u_{t}\|_{L^2}^{2} - t \int_{\Omega} \rho u \cdot \nabla|u_t|^2 \,dx - t \int_{\Omega} \rho_t ( u\cdot\nabla ) u \cdot u_t \,dx\\ &\quad - t \int_{\Omega} \rho ( u_t \cdot \nabla ) u \cdot u_t \,dx + A \gamma t \int_{\Omega} \rho^{\gamma-1} \rho_t \mathop{\mathrm{div}}u_t \,dx + 2 t \int_{\Omega} ( \nabla\chi_t \otimes \nabla\chi ) : Du_t \,dx \\ &\quad - t \int_{\Omega} \nabla\chi \cdot \nabla\chi_t \mathop{\mathrm{div}}u_t\,dx = \sum_{i=1}^{7}{Q}_i. \end{align}\tag{57}\] Using Lemma 1–2, Lemma 10–11, and applying the Hölder, Sobolev, Young, Korn, Poincaré and Gagliardo-Nirenberg inequalities, one obtains \[\begin{align} Q_2 &\le C\|\rho\|_{L^\infty}^{\frac{1}{2}} \|u\|_{L^\infty} \| \sqrt{t} \sqrt\rho u_t\|_{L^2} \|\sqrt{t} \nabla u_t\|_{L^2} \\ &\le C \|\nabla u\|_{L^2}^{\frac{1}{2}} \|\nabla^2 u\|_{L^2}^{\frac{1}{2}} \| \sqrt{t} \sqrt\rho u_t \|_{L^2}\|\sqrt{t} \nabla u_t\|_{L^2} \\ &\le \frac{\nu}{8} \| \sqrt{t} \nabla u_t\|_{L^2}^2 + C \|\sqrt{t} \sqrt\rho u_t \|_{L^2}^2 \|\nabla^2 u\|_{L^2}, \\ Q_3 + Q_5& \le C \sqrt{t} \|\rho_t\|_{L^2} ( \|u\|_{L^\infty} \|\nabla u\|_{L^3} \|\sqrt{t} u_t\|_{L^6} + \|\rho\|_{L^\infty}^{\gamma-1} \|\sqrt{t} \mathop{\mathrm{div}}u_t\|_{L^2}) \\ &\le C\sqrt{t} (1 + \|\nabla^2 u\|_{L^2}^{\frac{1}{2}}) (\|\nabla u\|_{L^2}^{\frac{1}{2}} \|\nabla^2 u\|_{L^2}^{\frac{1}{2}} + 1) \|\sqrt{t} \nabla u_t\|_{L^2}\\ &\le \frac{\nu}{8} \|\sqrt{t} \nabla u_t\|_{L^2}^2 + C(1 + \|\nabla^2 u\|_{L^2}^2 ),\\ Q_4&\le \|\nabla u\|_{L^\infty} \| \sqrt{t} \sqrt\rho u_t\|_{L^2}^2 \\ Q_6 + Q_7&\le C \sqrt{t} \|\sqrt{t} \nabla\chi_t\|_{L^2} \| \nabla\chi\|_{L^\infty} ( \| Du_t\|_{L^2} + \| \mathop{\mathrm{div}}u_t\|_{L^2} ) \\ &\le C \|\sqrt{t} \nabla\chi_t\|_{L^2} \| \nabla\chi\|_{L^\infty} \| \sqrt{t} \nabla u_t \|_{L^2} \\ &\le \frac{\nu}{8} \|\sqrt{t} \nabla u_t\|_{L^2}^2 + C \| \nabla\chi\|_{L^\infty}^2 \|\sqrt{t} \nabla\chi_t\|_{L^2}^2, \end{align}\] where we have used the fact that \[\begin{align} \label{be3:rho5} \|\rho_t\|_{L^2} &\le \|\nabla \rho\|_{L^2} \|u\|_{L^\infty} + \|\rho\|_{L^\infty} \|\mathop{\mathrm{div}}u\|_{L^2}\notag\\ &\le C (1 + \|\nabla u\|_{L^2}^{\frac{1}{2}} \|\nabla^2 u\|_{L^2}^{\frac{1}{2}} )\notag\\ &\le C (1 + \|\nabla^2 u\|_{L^2}^{\frac{1}{2}}). \end{align}\tag{58}\] Substituting the above estimates into \(\eqref{be3:u1}\), we obtain \[\begin{align} \label{be3:u2} \frac{d}{dt}\| \sqrt{t} \sqrt\rho u_{t} \|_{L^2}^{2} + \frac{5\nu}{4} \|\sqrt{t} \nabla u_{t}\|_{L^2}^{2} + 2 ( \lambda + \nu) \|\sqrt{t} \mathop{\mathrm{div}}u_{t} \|_{L^2}^{2} \le C \mathcal{F}_2(t) (1 + \mathcal{E}_2(t)). \end{align}\tag{59}\]
Moreover, by using 50 , 55 , Lemma 10–11, the Young inequality and Gagliardo-Nirenberg inequality, we show \[\begin{align} \|\nabla^2 u\|_{L^2}^2 &\le C (1 + \|\sqrt \rho u_t\|_{L^2}^2 + \|u\|_{L^\infty}^2 + \|\nabla \chi\|_{L^\infty}^2 ) \\ &\le C (1 + \|\sqrt \rho u_t\|_{L^2}^2 + \|\nabla u\|_{L^2} \|\nabla^2 u\|_{L^2} + \|\nabla^2 \chi\|_{L^2} \| \nabla ^3 \chi \|_{L^2} )\\ &\le C (1 + \|\sqrt \rho u_t\|_{L^2}^2 + \|\nabla^2 u\|_{L^2} + (1 + \|\nabla\chi_{t}\|_{L^2}) (1 + \|\nabla \rho\|_{L^q})) \\ &\le \frac{1}{2} \|\nabla^2 u\|_{L^2}^2 + C (1 + \|\sqrt \rho u_t\|_{L^2}^2 + \|\nabla\chi_{t}\|_{L^2}^2 + \|\nabla \rho\|_{L^q}^2 ). \end{align}\] That is, \[\begin{align} \label{be3:u3} \| \sqrt{t} \nabla^2 u \|_{L^2}^2 \le C (1 + \mathcal{E}_2(t)). \end{align}\tag{60}\] This estimate will be used in the next step.
Step 3. Estimates for the phase variable. Recalling 15 , that is \[\begin{align} &\quad\frac{d}{dt} \left( \frac{1}{2}\|\nabla\chi_t\|_{L^2}^2 + \int_{\Omega} \rho_tf(\chi)\chi_t \,dx \right) + \|\rho\chi_{tt}\|_{L^2}^2 \notag \\ &= -2 \int_{\Omega} \rho \rho_t \chi_t \chi_{tt} \,dx - 2 \int_{\Omega} \rho \rho_t ( u \cdot \nabla\chi ) \chi_{tt} \,dx - \int_{\Omega} \rho^2 ( u_t \cdot \nabla \chi ) \chi_{tt} \,dx \notag \\ &\quad - \int_{\Omega} \rho^2 ( u \cdot \nabla\chi_t ) \chi_{tt} \,dx + \int_{\Omega} \rho_t f'(\chi) ( u \cdot \nabla\chi ) \chi_t \,dx + \int_{\Omega} \rho_t f(\chi) (u \cdot \nabla\chi_t ) \,dx \notag \\ &\quad +\int_{\Omega} \rho f'(\chi) ( u_t \cdot \nabla\chi ) \chi_t \,dx + \int_{\Omega} \rho f(\chi) ( u_t \cdot \nabla\chi_t) \,dx + \int_{\Omega} \rho_t f'(\chi) \chi_t^2 \,dx \notag \\ &\quad - \int_{\Omega} \rho f'(\chi) \chi_t \chi_{tt} \,dx. \end{align}\] Multiplying both sides of the above equation by \(t\) and applying the identity \[\begin{align} - 2 \int_{\Omega} \rho \rho_{t} \chi_t \chi_{tt} \,dx &= 2 \int_{\Omega} \rho \mathop{\mathrm{div}}( \rho u ) \chi_t \chi_{tt} \,dx\\ &= \int_{\Omega} ( \nabla ( \rho^2 ) \cdot u + 2 \rho^2 \mathop{\mathrm{div}}u ) \chi_t \chi_{tt} \,dx\\ &= \int_{\Omega} \rho^2 \mathop{\mathrm{div}}u \chi_t \chi_{tt} \,dx - \int_{\Omega} \rho^2 ( u \cdot \nabla \chi_t ) \chi_{tt} \,dx - \int_{\Omega} \rho^2 ( u \cdot \nabla \chi_{tt} ) \chi_t \,dx\\ &= \int_{\Omega} \rho^2 \mathop{\mathrm{div}}u \chi_t \chi_{tt} \,dx - \frac{d}{dt} \int_{\Omega} \rho^2 ( u \cdot \nabla \chi_t ) \chi_t \,dx \\ &\quad+ 2 \int_{\Omega} \rho \rho_t ( u \cdot \nabla \chi_t ) \chi_t \,dx + \int_{\Omega} \rho^2 ( u_t \cdot \nabla \chi_t ) \chi_t \,dx, \end{align}\] we have \[\begin{align} \label{be3:chi2} &\frac{1}{2} \frac{d}{dt} \|\sqrt{t} \nabla\chi_t\|_{L^2}^2 + \| \sqrt{t} \rho\chi_{tt}\|_{L^2}^2 \nonumber\\ =& \frac{d}{dt} [ t \hat{H}(t) ] - \hat{H}(t) + \frac{1}{2}\|\nabla\chi_t\|_{L^2} + t \int_{\Omega} \rho^2 \mathop{\mathrm{div}}u \chi_t \chi_{tt} \,dx \nonumber\\ & + 2 t \int_{\Omega} \rho \rho_t ( u \cdot \nabla \chi_t ) \chi_t \,dx + t\int_{\Omega} \rho^2 ( u_t \cdot \nabla\chi_t ) \chi_t \,dx - 2 t \int_{\Omega} \rho \rho_t ( u \cdot \nabla\chi ) \chi_{tt} \,dx \nonumber\\ & - t \int_{\Omega} \rho^2 ( u_t \cdot \nabla \chi ) \chi_{tt} \,dx - t \int_{\Omega} \rho^2 ( u \cdot \nabla\chi_t ) \chi_{tt} \,dx + t \int_{\Omega} \rho_t f'(\chi) ( u \cdot \nabla\chi ) \chi_t \,dx \\ & + t \int_{\Omega} \rho_t f(\chi) (u \cdot \nabla\chi_t ) \,dx + t \int_{\Omega} \rho f'(\chi) ( u_t \cdot \nabla\chi ) \chi_t \,dx + t \int_{\Omega} \rho f(\chi) ( u_t \cdot \nabla\chi_t) \,dx \nonumber \\ & + t \int_{\Omega} \rho_t f'(\chi) \chi_t^2 \,dx - t \int_{\Omega} \rho f'(\chi) \chi_t^2 \,dx \nonumber\\ =& \frac{d}{dt} [ t \hat{H}(t) ] - \hat{H}(t) + \frac{1}{2}\|\nabla\chi_t\|_{L^2} + \sum_{i=1}^{12}{R}_i, \nonumber \end{align}\tag{61}\] where \[\hat{H}(t) = - \int_{\Omega} \rho^2 ( u \cdot \nabla \chi_t ) \chi_t \,dx - \int_{\Omega} \rho_t f(\chi) \chi_t \,dx.\] Applying Lemma 1–2, Lemma 10–11, it follows from 50 , 55 , 58 , 60 , the Hölder, Young, Poincaré and Gagliardo-Nirenberg inequalities that \[\begin{align} |\hat{H}(t)|&\le \|\rho\|_{L^\infty} \|\rho \chi_t\|_{L^2} \|u\|_{L^\infty} \|\nabla\chi_t\|_{L^2} + \|\rho_t\|_{L^2} \|f(\chi)\|_{L^3} \|\chi_t\|_{6}\\ &\le C \|\nabla u\|_{L^2}^{\frac{1}{2}} \|\nabla^2 u\|_{L^2}^{\frac{1}{2}} \|\nabla\chi_t\|_{L^2} + C ( \|\nabla^2 u\|_{L^2}^{\frac{1}{2}} + 1 ) (\|\rho \chi_t\|_{L^2} + \|\nabla\chi_t\|_{L^2})\\ &\le C (1 + \|\nabla^2 u\|_{L^2} + \|\nabla\chi_t\|_{L^2}^2 ) \le C \mathcal{F}_2(t), \\ R_1 + R_6&\le \|\sqrt{t} \rho\chi_{tt}\|_{L^2} \|\rho\|_{L^\infty} (\|\nabla u\|_{L^3} \|\sqrt{t} \chi_t\|_{L^6} + \|u\|_{L^\infty} \|\sqrt{t} \nabla\chi_t\|_{L^2} ), \\ &\le C \|\sqrt{t} \rho\chi_{tt}\|_{L^2} \|\nabla u\|_{L^2}^{\frac{1}{2}} \|\nabla^2 u\|_{L^2}^{\frac{1}{2}} (\|\rho \chi_t\|_{L^2} + \| \sqrt{t} \nabla\chi_t \|_{L^2}) \\ &\le \frac{1}{8} \|\sqrt{t} \rho\chi_{tt}\|_{L^2}^2 + C \|\nabla^2 u\|_{L^2} (1 + \|\sqrt{t} \nabla\chi_t\|_{L^2}^2) \\ &\le \frac{1}{8} \|\sqrt{t} \rho\chi_{tt}\|_{L^2}^2 + C \mathcal{F}_2(t) (1 + \mathcal{E}_2(t)), \\ R_2&\le 2 \sqrt{t} \|\rho\|_{L^\infty} \|\rho_t\|_{L^3} \|u\|_{L^\infty} \| \sqrt{t}\nabla \chi_t \|_{L^2} \|\chi_t\|_{L^6}\\ &\le C \|\nabla u\|_{L^2}^{\frac{1}{2}} \|\nabla^2 u\|_{L^2} (\|\nabla \rho\|_{L^q} + 1) \|\sqrt{t} \nabla \chi_t\|_{L^2} (\|\rho \chi_t\|_{L^2} + \|\nabla\chi_t\|_{L^2}) \\ &\le C (1 + \|\nabla^2 u\|_{L^2}^2 + \| \nabla \chi_t \|_{L^2}^2) (1 + \|\nabla \rho\|_{L^q}^2 + \|\sqrt{t} \nabla \chi_t\|^2_{L^2})\\ &\le C \mathcal{F}_2(t) ( \mathcal{E}_2(t) + 1 ), \\ R_3 + R_5 &\le \|\rho\|_{L^\infty}^{\frac{1}{2}} \|\sqrt{t} \sqrt\rho u_{t}\|_{L^3} ( \|\rho\|_{L^\infty} \|\sqrt{t} \nabla\chi_t\|_{L^2} \|\chi_t\|_{L^6} + \| \sqrt{t} \rho\chi_{tt}\|_{L^2} \|\nabla\chi\|_{L^6}) \\ &\le C \|\sqrt{t} \sqrt\rho u_{t}\|_{L^2}^{\frac{1}{2}} \|\sqrt{t} \sqrt\rho u_{t}\|_{L^6}^{\frac{1}{2}} \big(\|\sqrt{t} \nabla\chi_t\|_{L^2} (\|\rho \chi_t\|_{L^2} + \|\nabla\chi_t\|_{L^2}) + \| \sqrt{t} \rho\chi_{tt}\|_{L^2}\big) \\ &\le C \|\rho\|_{L^\infty}^{\frac{1}{4}} \|\sqrt{t} \sqrt\rho u_{t}\|_{L^2}^{\frac{1}{2}} \|\sqrt{t} \nabla u_{t}\|_{L^2}^{\frac{1}{2}} \big(\|\sqrt{t} \nabla\chi_t\|_{L^2} (1 + \|\nabla\chi_t\|_{L^2}) + \| \sqrt{t} \rho\chi_{tt}\|_{L^2} \big) \\ &\le \frac{1}{8} \| \sqrt{t} \rho\chi_{tt}\|_{L^2}^2 + \frac{\nu}{16} \| \sqrt{t} \nabla u_t \|_{L^2}^2 + C \| \sqrt{t} \sqrt\rho u_{t} \|_{L^2}^2 + C (1 + \|\nabla\chi_t\|_{L^2}^2) \|\sqrt{t} \nabla\chi_t\|_{L^2}^2 \\ & \le \frac{1}{8} \| \sqrt{t} \rho\chi_{tt}\|_{L^2}^2 + \frac{\nu}{16} \| \sqrt{t} \nabla u_t \|_{L^2}^2 + C \mathcal{F}_2(t) \mathcal{E}_2(t), \\ R_4& \le \sqrt{t} \|\sqrt{t} \rho\chi_{tt}\|_{L^2} \|\rho_t\|_{L^2} \|u\|_{L^\infty} \|\nabla\chi\|_{L^\infty} \\ &\le C \sqrt{t} \|\sqrt{t} \rho\chi_{tt}\|_{L^2} (1 + \|\nabla^2 u\|_{L^2}^{\frac{1}{2}} ) \|\nabla u\|_{L^2}^{\frac{1}{2}} \|\nabla^2 u\|_{L^2}^{\frac{1}{2}} \|\nabla^2\chi\|_{L^2}^{\frac{1}{2}} \|\nabla^3\chi\|_{L^2}^{\frac{1}{2}}\\ &\le C \|\sqrt{t} \rho\chi_{tt}\|_{L^2} (1 + \|\nabla^2 u\|_{L^2} ) \left((1 + \|\sqrt{t} \nabla\chi_{t} \|_{L^2}) (1 + \|\nabla\rho\|_{L^q}^{\frac{q}{3q-6}}) + \|\sqrt{t} \nabla^2 u\|_{L^2}\right)^{\frac{1}{2}} \\ &\le \frac{1}{8} \|\sqrt{t} \rho\chi_{tt}\|_{L^2}^2 + C \mathcal{F}_2(t) (\mathcal{E}_2(t) + 1) , \\ R_7 + R_8&\le C \sqrt{t} \|\rho_{t}\|_{L^2} \left(\|u\|_{L^6} \|f'(\chi)\|_{L^\infty} \|\nabla\chi\|_{L^6}\| \sqrt{t} \chi_t\|_{L^6} + \|u\|_{L^\infty} \|f(\chi)\|_{L^\infty}\|\sqrt{t} \nabla\chi_t\|_{L^2}\right) \\ &\le C (1 + \| \nabla^2 u\|_{L^2}^{\frac{1}{2}}) (\|\nabla^2\chi\|_{L^2} ( \|\rho\chi_t\|_{L^2} + \|\sqrt{t} \nabla\chi_t\|_{L^2}) + \|\sqrt{t} \nabla\chi_t\|_{L^2}) \\ &\le C \mathcal{F}_2(t) (1 + \mathcal{E}_2(t)), \\ R_9 + R_{10}&\le C \|\rho\|_{L^\infty}^{\frac{1}{2}} \|\sqrt{t} \sqrt\rho u_{t}\|_{L^2} ( \|f'(\chi)\|_{L^6} \|\nabla\chi\|_{L^6} \|\sqrt{t} \chi_{t}\|_{L^6} + \|f(\chi)\|_{L^\infty} \|\sqrt{t} \nabla\chi_t\|_{L^2}) \\ & \le C \|\sqrt{t} \sqrt\rho u_{t}\|_{L^2} ( \|\nabla^2\chi\|_{L^2} (\|\rho\chi_t\|_{L^2} + \|\sqrt{t} \nabla\chi_t\|_{L^2}) + \|\sqrt{t} \nabla\chi_t\|_{L^2} ) \\ &\le C (1 + \mathcal{E}_2(t)), \\ R_{11} + R_{12}&\le C ( \|f'(\chi)\|_{L^6} \|\rho_t\|_{L^2} \| \sqrt{t} \chi_t\|_{L^6}^2 + \|f'(\chi)\|_{L^3} \|\sqrt{t} \rho\chi_{tt}\|_{L^2} \|\sqrt{t} \chi_{t}\|_{L^6}) \\ &\le \frac{1}{8} \|\sqrt{t} \rho\chi_{tt}\|_{L^2}^2 + C (1 + \|\nabla^2 u\|_{L^2}^{\frac{1}{2}}) ( \|\rho\chi_t\|_{L^2}^2 + \|\sqrt{t} \nabla\chi_t\|_{L^2}^2) \\ & \le \frac{1}{8} \|\sqrt{t}\rho\chi_{tt}\|_{L^2}^2 + C \mathcal{F}_2(t) (1 + \mathcal{E}_2(t)), \end{align}\] where we have used \[\begin{align} \|\rho_t\|_{L^3} &\le \|\rho\|_{L^\infty} \|\nabla u\|_{L^3} + \|u\|_{L^\infty} \|\nabla\rho\|_{L^3}\\ &\le C \|\nabla u\|_{L^2}^{\frac{1}{2}} \|\nabla^2 u\|_{L^2}^{\frac{1}{2}} (1 + \|\nabla\rho\|_{L^2}^{\frac{2q - 6}{3q - 6}} \|\nabla\rho\|_{L^q}^{\frac{q}{3q - 6}})\\ &\le C \|\nabla^2 u\|_{L^2}^{\frac{1}{2}} (1 + \|\nabla\rho\|_{L^q} ). \end{align}\] Plugging the above estimates into 61 , one has \[\label{be3:chi3} \frac{d}{dt} \|\sqrt{t} \nabla\chi_t\|_{L^2}^2 + \| \sqrt{t} \rho\chi_{tt}\|_{L^2}^2 \le 2\frac{d}{dt} [ t \hat{H}(t) ] + \frac{\nu}{8} \| \sqrt{t} \nabla u_t \|_{L^2}^2 + C \mathcal{F}_2(t) ( \mathcal{E}_2(t) + 1).\tag{62}\]
Step 4. Closure of the estimates. Combining 55 , 56 , 59 with 62 gives \[\begin{align} \mathcal{E}_2'(t) + \mathcal{D}_2(t) \le 2\frac{d}{dt} [ t \hat{H}(t) ] + C \mathcal{F}_2(t) ( \mathcal{E}_2(t) + 1 ). \end{align}\] Integrating over \((0, t)\), one gets \[\begin{align} \mathcal{E}_2(t) + \int_0^t \mathcal{D}_2(s) \, ds \le 2 t \hat{H}(t) + C \int_0^t \mathcal{F}_2(s) ( \mathcal{E}_2(s) + 1 )\, ds. \end{align}\] By using the estimate of \(\| \sqrt{t} \nabla^2 u \|_{L^2}\) in 60 , we have \[\begin{align} 2 t \hat{H}(t) &\le 2 t \|\rho\|_{L^\infty} \|u\|_{L^\infty} \| \nabla \chi_t \|_{L^2} \|\rho \chi_t\|_{L^2} + 2 t \|\rho_t\|_{L^2} \|f(\chi)\|_{L^3} \|\chi_t\|_{L^6}\\ &\le C t^{\frac{1}{4}} \|\nabla u\|_{L^2}^{\frac{1}{2}} \| \sqrt{t} \nabla^2 u \|_{L^2}^{\frac{1}{2}} \| \sqrt{t} \nabla \chi_t \|_{L^2} + C t^{\frac{1}{4}} (1 + \| \sqrt{t} \nabla^2 u \|_{L^2}^{\frac{1}{2}} ) (\|\rho\chi_t\|_{L^2} + \|\sqrt{t} \nabla\chi_t\|_{L^2}) \\ &\le C (1 + \mathcal{E}_2^{\frac{1}{4}}(t)) \mathcal{E}_2^{\frac{1}{2}}(t) + C (1 + \mathcal{E}_2^{\frac{1}{4}}(t)) (1 + \mathcal{E}_2^{\frac{1}{2}}(t))\\ &\le \frac{1}{2} \mathcal{E}_2(t) + C. \end{align}\] Hence, one gets \[\begin{align} \frac{1}{2}\mathcal{E}_2(t) + \int_0^t \mathcal{D}_2(s) \, ds \le C \int_0^t \mathcal{F}_2(s) ( \mathcal{E}_2(s) + 1 )\, ds. \end{align}\] It follows from \(\mathcal{F}_2 \in L^1(0, T^*)\) and the Grönwall inequality that \[\label{be3:1} \sup_{0 \le t < T^*} \mathcal{E}_2(t) + \int_0^{T^*} \mathcal{D}_2(t) \, dt \le C.\tag{63}\] By using 52 , 55 , 60 and 63 , there holds \[\label{be3:2} \sup_{0 \le t < T^*} \left(\mathcal{E}_2(t) + \| \sqrt{t} \nabla^2u \|_{L^2}^2(t) + \|\sqrt{t} \nabla^3\chi\|_{L^2}^2(t)\right) + \int_0^{T^*} \left( \mathcal{D}_2(t) + \|\nabla^2 u\|_{L^q}(t) \right) \,dt \le C.\tag{64}\] Recalling 19 , we have \[\begin{align} \|\nabla^{2}u\|_{L^q}&\le C ( \|\rho u_{t}\|_{L^q} + \|\rho ( u \cdot \nabla ) u\|_{L^q} + \|\nabla (P(\rho))\|_{L^q} + \|\Delta \chi \nabla \chi\|_{L^q}) \\ &\le C (\|u_t\|_{L^6} + \|u\|_{L^\infty} \|\nabla u\|_{L^6} + \|\nabla \rho\|_{L^q} \|\rho\|_{L^\infty}^{\gamma-1} + \|\nabla \chi\|_{L^\infty} \|\nabla^2\chi\|_{L^q}) \\ &\le C (1 + \|\nabla u_t\|_{L^2} + \|\nabla^2 u\|_{L^2}^{\frac{3}{2}} + \|\nabla^3 \chi\|_{L^2}^{\frac{3}{2}}). \end{align}\] Combining the above inequalities and 64 yields \[\begin{align} \int_0^{T^*} \|\sqrt{t} \nabla^2 u\|_{L^q}^2 \,dt &\le C \sup_{0 \le t < T^*} \left(\mathcal{E}_2^{\frac{1}{2}}(t) + \|\sqrt{t} \nabla^2u\|_{L^2}(t) + \|\sqrt{t} \nabla^3\chi\|_{L^2}(t) \right) \int_0^{T^*} \mathcal{D}_2(t)\,dt \\ &\le C. \end{align}\] This completes the proof. ◻
Acknowledgment. Li’s work is supported by the National Natural Science Foundation of China (No.12371205) and the Natural Science Foundation of Guangdong Province (No.2025A151
5012026, 2025A1515040001).
Before proceeding with the proof, we first recall the following fixed point theorem.
Lemma 13 (Schaefer’s fixed point theorem). Let \(X\) be a Banach space and let \(\Lambda : X \to X\) be a continuous and compact operator. Assume further that the set \[\begin{align} \left\{ x \in X: x = \kappa \Lambda x \;\text{for some } 0 \le \kappa \le 1 \right\} \end{align}\] is bounded in \(X\). Then \(\Lambda\) admits at least one fixed point in \(X\).
Now we are ready to give the proof of proposition 2. Given \(\tilde{u} \in C([0, T]; X_N)\). It follows from the classical theory (cf. [44], [45]) that, there exists a unique \(\tilde{\rho} \in C^1(\overline{\Omega}\times [0, T])\), such that \[\begin{align} \partial_t \tilde{\rho} + \mathop{\mathrm{div}}(\tilde{\rho} \tilde{u} ) &= 0 && \text{\rm in } \Omega\times(0,T),\\ \tilde{\rho}|_{t = 0} &= \rho_{0N} && \text{\rm in } \Omega. \end{align}\] In fact, \(\tilde{\rho}\) is given by \[\label{App:rho1} \tilde{\rho}(x, t) = \rho_{0N}\left(U(x, 0; t)\right) \mathop{\mathrm{exp}}\left\{ -\int_0^t \mathop{\mathrm{div}}\tilde{u}\left( U(x, \tau; t), \tau\right) \, d\tau \right\}, \quad (x, t) \in \overline{\Omega} \times [0, T],\tag{65}\] where \(U \in C(\overline{\Omega} \times [0, T] \times [0, T])\) is the solution to \[\begin{cases} \frac{\partial}{\partial t}U(x, t; s) = \tilde{u}(U(x, t; s), t), &\quad 0 \le t \le T,\\ U(x, s; s) = x, &\quad 0 \le s \le T, x \in \overline{\Omega}. \end{cases}\] It follows from 65 and the regularity of the eigenfunctions that \[\label{App:rho2} \begin{align} \tilde{\rho}(x, t) \ge \left(\inf_{\overline{\Omega}} \rho_{0N} \right) \mathop{\mathrm{exp}}\left\{ - \int_0^t \| \nabla \tilde{u} \|_{L^\infty} \, d\tau \right\} \ge \frac{C}{N} \mathop{\mathrm{exp}}\left\{ - \int_0^t \| \nabla \tilde{u} \|_{L^2} \, d\tau \right\}. \end{align}\tag{66}\] Define the map \[\begin{align} D_N: C([0, T]; X_N) &\rightarrow C^1(\overline{\Omega} \times [0, T]), \\ \tilde{u} &\rightarrow \tilde{\rho}. \end{align}\] Owing to the uniform positive lower bound of \(\tilde{\rho}\) and \(\chi_{0N}\in H^2\), the well-posedness of the semi-linear parabolic equation implies that there exists a unique solution \(\tilde{\chi} \in L^2(0, T; H^2) \cap H^1(0, T; L^2)\) satisfies \[\label{App:chi1} \begin{cases} \partial_t \tilde{\chi} + \tilde{u} \cdot \nabla \tilde{\chi} - \frac{1}{\tilde{\rho}^2} \Delta \tilde{\chi} + \frac{1}{\tilde{\rho}} F' (\tilde{\chi}) = 0 &\text{\rm in } \Omega\times (0, T),\\ \partial_{\boldsymbol{n}} \tilde{\chi} = 0 &\text{\rm on } \partial \Omega\times (0, T), \\ \tilde{\chi}|_{t = 0} = \chi_{0N} &\text{\rm in } \Omega. \end{cases}\tag{67}\] Define the map \[\begin{align} C_N: C([0, T]; X_N) &\rightarrow C([0, T]; H^1), \\ \tilde{u} &\rightarrow \tilde{\chi}, \end{align}\] where we have used the continuous embedding \(L^2(0, T; H^2) \cap H^1(0, T; L^2) \hookrightarrow C([0, T]; H^1)\). Within \(D_N(\tilde{u})\) and \(C_N(\tilde{u})\), we seek the solution to the following problem \[\label{App:u1} \begin{cases} D_N(\tilde{u}) \partial_t u_N + P_N[D_N(\tilde{u}) (u_N \cdot \nabla) u_N] - \mathcal{L}u_N = G_N(\tilde{u}), \\ u_N |_{t=0} = u_{0N}, \end{cases}\tag{68}\] where \(G_N(\tilde{u}) = - P_N[\nabla (P(D_N(\tilde{u}))) + \mathop{\mathrm{div}}(\nabla C_N(\tilde{u}) \otimes \nabla C_N(\tilde{u}) - \frac{|\nabla C_N(\tilde{u})|^2}{2}\mathbb{I})]\). To this end, we consider the linearized problem \[\label{App:u2} \begin{cases} D_N(\tilde{u}) \partial_t u_N + P_N[D_N(\tilde{u}) (\tilde{u} \cdot \nabla) u_N] - \mathcal{L}u_N = G_N(\tilde{u}), \\ u_N |_{t=0} = u_{0N}. \end{cases}\tag{69}\]
Once the problem 69 is uniquely solvable for any given \(\tilde{u}\), we can define a solution map \[\begin{align} \Lambda_N: C([0, T]; X_N) &\rightarrow C([0, T]; X_N), \\ \tilde{u} &\rightarrow u_N, \end{align}\] where \(u_N\) is the unique solution to the system 69 . We first claim that \(\Lambda_N\) is well-defined. Furthermore, we claim that the mapping \(\Lambda_N\) satisfies the following properties:
1. \(\Lambda_N\) is continuous.
2. \(\Lambda_N\) is compact.
3. the set \(\{\, u \in C([0,T];X_N) : u = \kappa \Lambda_N(u),\;\kappa \in [0,1] \,\}\) is bounded in \(C([0,T]; X_N)\).
Then one can apply Schaefer’s fixed point theorem (see Lemma 13) to obtain a fixed point \(u\). More precisely, we have found \(u =
\Lambda_N (u)\), \(\rho = D_N(u)\) and \(\chi = C_N(u)\), which together solve 23 . So it remains for us to prove the unique solvability of 69 and Claim 1–3.
Unique solvability of 69 . We reformulate problem 69 in the following equivalent form \[\label{App:u3} \begin{cases} \langle D_N(\tilde{u}) \partial_t u_N, w_k \rangle + \langle D_N(\tilde{u}) (\tilde{u} \cdot \nabla) u_N, w_k \rangle + \nu \langle \nabla u, \nabla w_k \rangle + (\nu + \lambda) \langle \mathop{\mathrm{div}}u, \mathop{\mathrm{div}}w_k \rangle\\ \quad = \langle \nabla P(D_N(\tilde{u})) - \Delta C_N(\tilde{u}) \nabla C_N(\tilde{u}) , w_k \rangle, \quad 0 \le t \le T, \\ \langle u_N(0), w_k \rangle = \langle u_0, w_k \rangle, \quad k = 1, 2, \cdots, N. \end{cases}\tag{70}\] Suppose that \(u_N\) has the form \(u_N(x, t) = \sum\limits_{j=1}^N {\alpha_j}(t) w_j(x)\). It follows from integration by parts and the properties of the eigenfunction that problem 70 is equivalent to the following problem \[\label{App:u4} \begin{cases} \sum_{j=1}^N \langle D_N(\tilde{u}) w_j, w_k \rangle \alpha_j'(t) + \sum_{j=1}^N \langle D_N(\tilde{u}) (\tilde{u} \cdot \nabla) w_j, w_k \rangle \alpha_j(t) + \lambda_k \alpha_k(t) \\ \quad = \langle \nabla P(D_N(\tilde{u})) - \Delta C_N(\tilde{u}) \nabla C_N(\tilde{u}), w_k \rangle, \quad 0 \le t \le T, \\ \alpha_k(0) = \langle u_0, w_k \rangle, \quad k = 1, 2, \cdots, N. \end{cases}\tag{71}\] Denote \[\begin{align} &a(t) = (\alpha_1(t), \cdots, \alpha_N(t))^T, \quad \mathcal{A}(t) = (\mathcal{A}_{ij}(t)), \quad \mathcal{B}(t) = (\mathcal{B}_{ij}(t)), \quad \mathcal{F}(t) = (\mathcal{F}_1(t), \cdots, \mathcal{F}_N(t))^T,\\ &\mathcal{A}_{ij}(t) = \langle D_N(\tilde{u}) w_j, w_i \rangle, \quad \mathcal{B}_{ij}(t) = \langle D_N(\tilde{u}) (\tilde{u} \cdot \nabla) w_j, w_i \rangle + \delta_{ij} \lambda_j , \\ &\mathcal{F}_{j}(t) = \langle \nabla P(D_N(\tilde{u})) - \Delta C_N(\tilde{u}) \nabla C_N(\tilde{u}), w_j \rangle, \quad i,j = 1, \cdots, N. \end{align}\] Then, problem 71 can be rewritten in the compact form \[\begin{cases}\label{Odes1} \mathcal{A}(t) a'(t) + \mathcal{B} (t) a(t) = \mathcal{F}(t), \quad 0 \le t \le T, \\ a(0) = (\langle u_0, w_1 \rangle, \cdots, \langle u_0, w_N \rangle)^{T}. \end{cases}\tag{72}\] It follows from 66 that the matrix \(\mathcal{A}(t)\) is positive definite for any \(t\in [0,T]\). Indeed, for any \(\beta=(\beta_1,\cdots,\beta_N)^T\in\mathbb{R}^N\), we have \[\begin{align} \beta^T \mathcal{A}(t) \beta &= \sum_{i,j=1}^N \langle D_N(\tilde{u}) w_j, w_i \rangle \beta_j \beta_i = \Big\langle D_N(\tilde{u}) \sum_{j=1}^N \beta_j w_j, \sum_{i=1}^N \beta_i w_i \Big\rangle \\ &= \int_{\Omega} D_N(\tilde{u}) \Big| \sum_{j=1}^N \beta_j w_j \Big|^2 dx \ge \inf_{\Omega} D_N(\tilde{u}) {\Big| \sum_{j=1}^N \beta_j^2 \Big|}. \end{align}\] Since \(D_N(\tilde{u})=\tilde{\rho}\) satisfies 66 , it follows that \[\beta^T \mathcal{A}(t) \beta >0 \qquad \text{for all } \beta\neq 0.\] Hence, \(\mathcal{A}(t)\) is positive definite and therefore invertible for all \(t\in[0,T]\).
Consequently, problem 72 can be rewritten as \[\begin{cases}\label{Odes2} a'(t) + \mathcal{A}(t)^{-1} \mathcal{B} (t) a(t) = \mathcal{A}(t)^{-1} \mathcal{F}(t), \quad 0 \le t \le T, \\ a(0) = (\langle u_0, w_1 \rangle, \cdots, \langle u_0, w_N \rangle)^{T}. \end{cases}\tag{73}\] Note that \[\begin{align} \mathcal{F}_{j}(t) &= \langle \nabla P(D_N(\tilde{u})) - \Delta C_N(\tilde{u}) \nabla C_N(\tilde{u}), w_j \rangle \\ &= \langle \nabla P(D_N(\tilde{u})) + \nabla C_N(\tilde{u}) \otimes \nabla C_N(\tilde{u}), \nabla w_j \rangle - \frac{1}{2} \langle |\nabla C_N(\tilde{u})|^2, \mathop{\mathrm{div}}w_j \rangle. \end{align}\] Moreover, \[\mathcal{A},\mathcal{B}\in C([0,T];\mathbb{R}^{N\times N}), \qquad \mathcal{F}\in C([0,T];\mathbb{R}^{N}),\] since \[D_N(\tilde{u})\in C^1(\overline{\Omega}\times[0,T]), \quad C_N(\tilde{u})\in C([0,T];H^1), \quad w_k\in C^2(\overline{\Omega}).\] Together with the fact that \(\mathcal{A}(t)\) is invertible for all \(t\in[0,T]\), we deduce that \[\mathcal{A}^{-1}\in C([0,T];\mathbb{R}^{N\times N}).\] Therefore, by the classical theory of linear ordinary differential equations with continuous coefficients, problem 73 admits a unique global solution \[a\in C^1([0,T];\mathbb{R}^N).\] Consequently, \[u_N(x,t)=\sum_{j=1}^N \alpha_j(t)w_j(x) \in C^1([0,T];X_N).\]
The proof of Claim 2. Let \(K > 0\) be given and define \[\begin{align} B_K=\{\,u\in C([0,T];X_N): \|u\|_{C([0,T];X_N)}\le K \,\}. \end{align}\] For any \(\tilde{u} \in B_K\), it follows from 66 that there exists a positive constant \(\underline{\rho}=\underline{\rho}(K,N)\) such that \[\begin{align} \underline{\rho}=\inf_{\overline{\Omega}\times[0,T]} D_N(\tilde{u})>0. \end{align}\]
Let \(t\in[0,T]\) be arbitrary. By the non-singularity and positive definiteness of \(\mathcal{A}(t)\), we have \[\begin{align} \|\mathcal{A}^{-1}(t)\| &:= \max_{y\neq 0} \frac{|\mathcal{A}^{-1}(t)y|}{|y|} = \max_{z\neq 0} \frac{|z|}{|\mathcal{A}(t)z|} = \left(\min_{z\neq 0} \frac{|\mathcal{A}(t)z|}{|z|}\right)^{-1} \\ &\le \left(\min_{z\neq 0} \frac{(z, \mathcal{A}(t) z)}{|z|^2}\right)^{-1} \le \frac{1}{\underline{\rho}}, \end{align}\] where \(|\cdot|\) and \((\cdot,\cdot)\) denote the Euclidean norm and inner product in \(\mathbb{R}^N\), respectively, and \(\|\cdot\|\) denote the Euclidean norm in \(\mathbb{R}^{N \times N}\). Here we used the fact that for any \(z\in\mathbb{R}^N\), \[\begin{align} (z, \mathcal{A}(t) z) &= \sum_{i,j=1}^{3} z_i z_j \int_\Omega D_N(\tilde{u}) w_i w_j \, dx = \int_\Omega D_N(\tilde{u}) \Big|\sum_{i=1}^{3} z_i w_i\Big|^2 \, dx \\ &\ge \underline{\rho} \int_\Omega \Big|\sum_{i=1}^{3} z_i w_i\Big|^2 \, dx = \underline{\rho} \sum_{i,j=1}^{3} z_i z_j \int_\Omega w_i w_j \, dx = \underline{\rho} |z|^2. \end{align}\] Consequently, we obtain \[\label{App:A} \max_{0\le t \le T} \|\mathcal{A}^{-1}(t)\| \le \frac{1}{\underline{\rho}}.\tag{74}\] In what follows, \(C\) denotes a generic positive constant depending only on \(A\), \(\gamma\), \(K\), \(N\), \(\underline{\rho}\), \(\|\rho_0\|_{L^\infty}\), \(\|\nabla \chi_0\|_{L^2}\), \(T\), and \(|\Omega|\).
Since \(\dim X_N < \infty\), all norms on \(X_N\) are equivalent. Let \[\tilde{\mu} := {- \tilde{\rho}} \Big(\tilde{\chi}_t + (\tilde{u}\cdot\nabla)\tilde{\chi}\Big).\] Multiplying 67 by \({\tilde{\rho}}\tilde{\mu}\) and integrating by parts, we obtain \[\begin{align} \label{App:chi2} \frac{1}{2} \frac{d}{dt} \|\nabla \tilde{\chi}\|_{L^2}^2 + \frac{d}{dt} \int_\Omega \tilde{\rho} F(\tilde{\chi})\,dx + \|\tilde{\mu}\|_{L^2}^2 &= - \int_\Omega \Delta \tilde{\chi} (\tilde{u}\cdot\nabla)\tilde{\chi}\,dx \notag\\ &= \int_\Omega (\nabla \tilde{\chi} \otimes \nabla \tilde{\chi}) : \nabla \tilde{u}\,dx - \frac{1}{2} \int_\Omega |\nabla \tilde{\chi}|^2 \mathop{\mathrm{div}}\tilde{u}\,dx \notag\\ &\le C \|\nabla \tilde{u}\|_{L^\infty} \|\nabla \tilde{\chi}\|_{L^2}^2 \le C \|\nabla \tilde{\chi}\|_{L^2}^2, \end{align}\tag{75}\] where we used the identity \[\int_\Omega \tilde{\rho} f(\tilde{\chi}) \big(\tilde{\chi}_t + (\tilde{u}\cdot\nabla)\tilde{\chi}\big)\,dx = \frac{d}{dt} \int_\Omega \tilde{\rho} F(\tilde{\chi})\,dx.\] Integrating 75 over \((0,t)\) yields the uniform bound \[\|\nabla \tilde{\chi}\|_{L^2}^2(t) + \int_\Omega \tilde{\rho}(x,t) F(\tilde{\chi}(x,t))\,dx + \int_0^t \|\tilde{\mu}\|_{L^2}^2 \,ds \le C, \quad 0\le t\le T.\] Consequently, one gets \[\label{App:chi3} \max_{0\le t\le T} \Big( \|\nabla C_N(\tilde{u})\|_{L^2}^2 + \int_\Omega D_N(\tilde{u}) F(C_N(\tilde{u}))\,dx \Big) + \int_0^T \|\tilde{\mu}\|_{L^2}^2\,dt \le C.\tag{76}\]
Combining 76 with the Hölder inequality gives \[\begin{align} |\mathcal{B}_{ij}(t)| &\le \| D_N(\tilde{u}) \|_{L^\infty} \| \tilde{u} \|_{L^\infty} \| \nabla w_j \|_{L^2} \| w_i \|_{L^2} + \lambda_N\le C,\\ |\mathcal{F}_j(t)| &\le C (\| D_N(\tilde{u}) \|_{L^\infty}^{\gamma} \| \nabla w_j \|_{L^1} + \| \nabla C_N(\tilde{u}) \|_{L^2}^2 \| \nabla \tilde{u} \|_{L^\infty}) \le C, \end{align}\] for any \(i,j = 1, \cdots, N\) and \(t \in [0, T]\). Hence, we have \[\label{App:BF} \max_{0\le t\le T} \big(\|\mathcal{B}(t)\| + |\mathcal{F}(t)|\big) \le C.\tag{77}\]
Multiplying 73 by \(a(t)\) and using 74 , 77 , we obtain \[\frac{1}{2} \frac{d}{dt} |a(t)|^2 \le C \big(|a(t)|^2 + 1\big), \quad 0\le t\le T,\] which, by the Grönwall inequality, gives \[\label{App:u5} \sup_{0\le t\le T} |a(t)|^2 = \sup_{0\le t\le T} \|\Lambda_N(\tilde{u})\|_{L^2}^2 \le C.\tag{78}\] Similarly, multiplying 73 by \(a'(t)\) and using 74 , 77 , 78 , we obtain \[\sup_{0\le t\le T} |a'(t)|^2 = \sup_{0\le t\le T} \|\partial_t \Lambda_N(\tilde{u})\|_{L^2}^2 \le C.\] Combining the above, we conclude \[\label{App:u6} \sup_{0\le t\le T} \big( \|\Lambda_N(\tilde{u})\|_{L^2}^2 + \|\partial_t \Lambda_N(\tilde{u})\|_{L^2}^2 \big) \le C.\tag{79}\] Finally, since \(X_N\) is finite dimensional, the Arzelà-Ascoli theorem implies that \(\overline{\Lambda_N(B_K)}\) is compact, and thus \(\Lambda_N\) is a compact operator.
The proof of Claim 3. Suppose that \(u = \kappa \Lambda_N(u)\) for some \(\kappa \in (0, 1]\), and define \[\begin{align} \rho := D_N(u), \;\chi := C_N(u), \; \mu := - \rho \chi_t - \rho (u \cdot \nabla) \chi, \; u = \sum_{i = 1}^N c_i(t) w_i(x), \end{align}\] where \(c_i = \langle u, w_i \rangle\). It follows from 68 and the definitions of \(D_N\), \(C_N\) that, \((\rho, u, \chi)\) satisfies \[\label{claim3:1} \begin{cases} \rho_t + \mathop{\mathrm{div}}(\rho u)=0,\\ \langle \rho \partial_t u + \rho (u \cdot \nabla) u, w_k \rangle + \nu \langle \nabla u, \nabla w_k \rangle + (\nu + \lambda) \langle \mathop{\mathrm{div}}u, \mathop{\mathrm{div}}w_k \rangle \\ \quad= \kappa \langle \nabla P(\rho) - \Delta\chi \nabla\chi, w_k \rangle, \\ \rho^2 \chi_t + \rho^2 ( u \cdot \nabla ) \chi - \Delta\chi + \rho F' (\chi) = 0, \\ (u, \partial_{\boldsymbol{n}}\chi)|_{\partial \Omega\times (0, T)} = 0, \\ (\rho, u, \chi)|_{t=0} = (\rho_{0N}, \kappa u_{0N}, \chi_{0N}), \quad k = 1, \cdots, N, \end{cases}\tag{80}\] where we have used \(\Lambda_N u = \frac{1}{\kappa} u\). Multiplying \(\eqref{claim3:1}_2\) with \(c_k\) and summing over \(k = 1, \cdots, N\), one deduces by integration by parts and \(\eqref{claim3:1}_1\) that \[\label{claim3:2} \begin{align} &\frac{d}{dt} \left( \frac{1}{2} \| \sqrt{\rho} u \|_{L^2}^2 + \frac{\kappa A}{\gamma-1} \int_{\Omega} \rho^{\gamma} \,dx \right) + \nu \|\nabla u\|_{L^2}^2 + ( \lambda + \nu) \|\mathop{\mathrm{div}}u\|_{L^2}^2 \\ &\quad + \kappa \int_{\Omega} \Delta\chi ( u \cdot \nabla ) \chi \,dx = 0. \end{align}\tag{81}\] where we have used the following identity \[\begin{align} \kappa \langle \nabla P(\rho), u \rangle = - \kappa A \frac{d}{dt} \int_{\Omega} \rho^{\gamma} \,dx + \kappa \gamma \int_{\Omega} \nabla P(\rho) \cdot u \,dx. \end{align}\] Multiplying \(\eqref{claim3:1}_3\) with \(\kappa (\chi_t + (u \cdot \nabla) \chi)\) and integrating over \(\Omega\), it follows from integration by parts and \(\eqref{claim3:1}_1\) that \[\label{claim3:3} \frac{\kappa}{2} \frac{d}{dt} \|\nabla \chi\|_{L^2}^2 + \kappa \frac{d}{dt} \int_{\Omega} \rho F(\chi) dx + \kappa \|\mu\|_{L^2}^2 - \kappa \int_{\Omega} \Delta\chi ( u \cdot \nabla ) \chi \,dx = 0,\tag{82}\] where we have used \[\begin{align} \kappa \int_{\Omega} \rho f(\chi) \big( \chi_t + ( u \cdot \nabla ) \chi \big) \,dx = \kappa \frac{d}{dt} \int_{\Omega} \rho F(\chi) dx. \end{align}\] Combining 81 with 82 , one has \[\begin{align} & \frac{d}{dt} \left( \frac{1}{2} \| \sqrt{\rho} u \|_{L^2}^2 + \frac{\kappa}{2} \|\nabla \chi\|_{L^2}^2 + \frac{\kappa A}{\gamma-1} \int_{\Omega} \rho^{\gamma} \,dx + \kappa \int_{\Omega}\rho F(\chi)dx \right)\\ &\quad + \kappa \|\mu\|_{L^2}^2 + \nu \|\nabla u\|_{L^2}^2 + ( \lambda + \nu) \|\mathop{\mathrm{div}}u\|_{L^2}^2 = 0. \end{align}\] Integrating it over \((0, t)\), it follows that \[\frac{1}{2} \| \sqrt{\rho} u \|_{L^2}^2(t) + \nu \int_0^t \|\nabla u\|_{L^2}^2(s) \, ds \le E_{0,\kappa},\] for any \(t \in [0, T]\), where \[E_{0,\kappa} = \frac{\kappa^2}{2} \| \sqrt{\rho_{0N}} u_{0N} \|_{L^2}^2 + \frac{\kappa A}{\gamma-1} \int_{\Omega} \rho_{0N}^{\gamma}(x) \,dx + \frac{\kappa}{2} \| \nabla\chi_{0N} \|_{L^2}^2 + \kappa \int_{\Omega} \rho_{0N} F(\chi_{0N}) (x, t) \,dx.\] Combining the above estimate with 66 , it follows from the Hölder inequality that \[\begin{align} \|u\|_{L^2}^2(t) &\le \sup_{(x, s) \in \overline{\Omega} \times [0, t]} \frac{1}{\rho(x, s)} \| \sqrt{\rho} u \|_{L^2}^2(t)\\ &\le C N E_{0, \kappa}\mathop{\mathrm{exp}}\left\{ \int_0^t \|\nabla u\|_{L^2}(s) \,ds \right\}\\ &\le C N E_{0, \kappa} \mathop{\mathrm{exp}}(T^{\frac{1}{2}} E_{0, \kappa}), \end{align}\] for any \(t \in [0, T]\). Since \(E_{0, \kappa}\) is non-decreasing with respect to \(\kappa\), we conclude that \[\|u\|_{C([0, T]; X_N)} \le C N E_{0, 1} \mathop{\mathrm{exp}}(T^{\frac{1}{2}} E_{0, 1}).\] This completes the proof of Claim 3.
The proof of Claim 1. Suppose that \[\begin{align} \tilde{u}_m &\rightarrow \tilde{u} \quad \text{in } C([0, T]; X_N) \quad \text{with } \| \tilde{u}_m \|_{C([0, T]; X_N) } \le 2 \| \tilde{u} \|_{C([0, T]; X_N)} . \end{align}\] Let \(K = 2 \| \tilde{u} \|_{C([0, T]; X_N)}\) and denote \[\begin{align} \label{claim1:rho} \tilde{\rho}_m = D_N(\tilde{u}_m),~ \tilde{\chi}_m = C_N(\tilde{u}_m), ~ \tilde{\mu}_m = - \tilde{\rho}_m \partial_t\tilde{\chi}_m - \tilde{\rho}_m (\tilde{u}_m \cdot \nabla) \tilde{\chi}_m, \quad m = 1, 2, \cdots \end{align}\tag{83}\] Then we infer from 65 that \[\tilde{\rho}_m \rightarrow \tilde{\rho} = D_N(\tilde{u}) \quad \text{in } C(\overline{\Omega} \times [0, T]).\] It follows from 76 and 79 that \[\begin{align}\label{claim1:E} &\sup_{0 \le t \le T} \left( \|\Lambda_N(\tilde{u}_m)\|_{L^2}^2 + \|\partial_t\Lambda_N(\tilde{u}_m)\|_{L^2}^2 + \|\nabla \tilde{\chi}_m\|_{L^2}^2 + \int_{\Omega} \tilde{\rho}_m F(\tilde{\chi}_m) \,dx (t) \right) \\ &\quad+ \int_0^T \|\tilde{\mu}_m\|_{L^2}^2 \,dt \le C. \end{align}\tag{84}\] Combining the above estimates with Lemma 1 and using the same argument as in Lemma 1, we deduce \[\sup_{0 \le t \le T} \| \tilde{\chi}_m \|_{H^1}^2 \le C.\] Note that \[\begin{align} \sup_{0 \le t \le T} \| \tilde{\rho}_m (\tilde{u}_m \cdot \nabla) \tilde{\chi}_m \|_{L^2} \le \sup_{0 \le t \le T} \left( \| \tilde{\rho}_m \|_{L^\infty} \| \tilde{u}_m \|_{L^\infty} \| \nabla\tilde{\chi}_m \|_{L^2} \right) \le C, \end{align}\] and \[\begin{align} \inf_{\overline{\Omega} \times [0, T]} \tilde{\rho}_m \ge \frac{1}{N} \mathop{\mathrm{exp}}\left\{ -\int_0^t \| \nabla \tilde{u}_m \|_{L^\infty} \, d\tau \right\} \ge \frac{C}{N}. \end{align}\] Plugging the above estimates into 84 , one gets \[\sup_{0 \le t \le T} \left( \|\Lambda_N(\tilde{u}_m)\|_{L^2}^2 + \|\partial_t\Lambda_N(\tilde{u}_m)\|_{L^2}^2 + \|\tilde{\chi}_m\|_{H^1}^2\right) + \int_0^T ( \|\partial_t\tilde{\chi}_m\|_{L^2}^2 + \|\nabla^2\tilde{\chi}_m\|_{L^2}^2) \,dt \le C.\] By the Banach-Alaoglu theorem, the Arzelà-Ascoli theorem, and the Aubin-Lions lemma, we can extract a subsequence of \(\{(\tilde{\chi}_m, \tilde{u}_m)\}\), still denoted by \(\{(\tilde{\chi}_m, \tilde{u}_m)\}\), such that \[\begin{align} \Lambda_N(\tilde{u}_m) \rightarrow \hat{u} &\quad\text{strongly in }C([0, T]; X_N), \tag{85}\\ \partial_t \Lambda_N(\tilde{u}_m) \rightharpoonup\partial_t\hat{u} &\quad\text{weakly-* in }L^{\infty}(0, T; L^2), \tag{86}\\ \tilde{\chi}_m \rightarrow \hat{\chi} &\quad\text{strongly in }C([0, T]; L^2) \cap L^2(0, T; H^1), \tag{87}\\ \tilde{\chi}_m \rightharpoonup\hat{\chi} &\quad\text{weakly in }L^2((0, T); H^2) \cap H^1(0, T; L^2), \tag{88} \end{align}\] where \((\hat{u}, \hat{\chi})\) satisfies \[\hat{u} \in C([0, T]; X_N) \cap W^{1,\infty}(0, T; X_N), \quad \hat{\chi} \in L^2(0,T; H^2)\cap H^1(0, T; L^2).\] Substituting \(\tilde{\chi}_m\) and \(\tilde{\rho}_m\) into 67 , we obtain \[\begin{cases} \partial_t \tilde{\chi}_m + \tilde{u}_m \cdot \nabla \tilde{\chi}_m - \frac{1}{\tilde{\rho}_m^2} \Delta \tilde{\chi}_m + \frac{1}{\tilde{\rho}_m} F' (\tilde{\chi}_m) = 0 &\text{\rm in } \Omega\times (0, T),\\ \partial_{\boldsymbol{n}} \tilde{\chi}_m = 0 &\text{\rm on } \partial \Omega\times (0, T), \\ \tilde{\chi}_m|_{t = 0} = \chi_{0N} &\text{\rm in } \Omega. \end{cases}\] Passing to the limit \(m\rightarrow\infty\), it follows from 83 , 87 and 88 that \(\hat{\chi}\) satisfies \[\begin{cases} \partial_t \hat{\chi} + \tilde{u} \cdot \nabla \hat{\chi} - \frac{1}{\tilde{\rho}^2} \Delta \hat{\chi} + \frac{1}{\tilde{\rho}} F' (\hat{\chi}) = 0 &\text{\rm in } \Omega\times (0, T),\\ \partial_{\boldsymbol{n}} \hat{\chi} = 0 &\text{\rm on } \partial \Omega\times (0, T), \\ \hat{\chi}|_{t = 0} = \chi_{0N} &\text{\rm in } \Omega. \end{cases}\] Then, the unique solvability of system 67 implies that \(\hat{\chi} = \tilde{\chi}\). Similarly, substituting \(\tilde{u}_m\), \(\tilde{\chi}_m\) and \(\tilde{\rho}_m\) into 70 , it follows from integration by parts that \[\begin{cases} \langle \tilde{\rho}_m \partial_t \Lambda_N(\tilde{u}_m) + \tilde{\rho}_m (\tilde{u}_m \cdot \nabla) \Lambda_N(\tilde{u}_m), w_k \rangle + \langle P(\tilde{\rho}_m), \mathop{\mathrm{div}}w_k \rangle\\ \quad = - \nu \langle \nabla \Lambda_N(\tilde{u}_m), \nabla w_k \rangle - (\nu + \lambda) \langle \mathop{\mathrm{div}}\Lambda_N(\tilde{u}_m), \mathop{\mathrm{div}}w_k \rangle \\ \qquad+ \langle \nabla\tilde{\chi}_m \otimes \nabla\tilde{\chi}_m, \nabla w_k \rangle - \frac{1}{2} \langle |\nabla\tilde{\chi}_m|^2 , \mathop{\mathrm{div}}w_k \rangle, \quad 0 \le t \le T, \\ \langle \Lambda_N(\tilde{u}_m)(\cdot, 0), w_k \rangle = \langle u_0, w_k \rangle, \quad k = 1, 2, \cdots, N. \end{cases}\] Passing to the limit \(m\rightarrow\infty\) and combining 85 –88 with the above convergence results, we have \[\begin{cases} \langle \tilde{\rho} \partial_t \hat{u} + \tilde{\rho} (\tilde{u} \cdot \nabla) \hat{u} , w_k \rangle + \nu \langle \nabla \hat{u} , \nabla w_k \rangle + (\nu + \lambda) \langle \mathop{\mathrm{div}}\hat{u} , \mathop{\mathrm{div}}w_k \rangle\\ \quad = - \langle P(\tilde{\rho}), \mathop{\mathrm{div}}w_k \rangle + \langle \nabla\tilde{\chi} \otimes \nabla\tilde{\chi}, \nabla w_k \rangle - \frac{1}{2} \langle |\nabla\tilde{\chi}|^2 , \mathop{\mathrm{div}}w_k \rangle, \quad \text{a.e. } 0 \le t \le T, \\ \langle \hat{u} (\cdot, 0), w_k \rangle = \langle u_0, w_k \rangle, \quad k = 1, 2, \cdots, N. \end{cases}\] Denote \(\hat{u} = \sum\limits_{j=1}^N {\hat{\alpha}_j}(t) w_j(x)\) and \(\hat{a}(t) = (\hat{\alpha}_1(t), \cdots, \hat{\alpha}_N(t))^T\), where \(\hat{\alpha}_j(t) = \langle \hat{u}(\cdot, t), w_j \rangle\). Using integration by parts, it follows that \(\hat{a}\) satisfies \[\begin{cases} \hat{a}'(t) + \mathcal{A}(t)^{-1} \mathcal{B} (t) \hat{a}(t) = \mathcal{A}(t)^{-1} \mathcal{F}(t), \quad 0 \le t \le T, \\ \hat{a}(0) = (\langle u_0, w_1 \rangle, \cdots, \langle u_0, w_N \rangle)^{T}. \end{cases}\] By the unique solvability of the ODE system 73 , we conclude that \[a(t) = \hat{a}(t) \quad \text{a.e. in }(0, T),\] which implies \[\Lambda_N(\tilde{u}) = \hat{u} \quad \text{a.e. in }(0, T).\] Thus, the proof of the Claim 1 is complete.