The most important features that distinguish fluid flow in the atmosphere and ocean are the effects of rotation and stratification, which are described by the Boussinesq equations. This paper examines the stability threshold at high Reynolds numbers \(\boldsymbol{Re}\) for the three-dimensional Boussinesq equations with rotation on the domain \(\Omega=\{(x,\,y,\,z)\in \mathbb{T} \times \mathbb{R} \times \mathbb{T}\}\) around the Couette flow \((y,0,0)\) and the vertically stratified temperature \(\Theta_s=1+\alpha^2 z\). For the linear system without rotation, stratification not only suppresses the lift-up effect but also exhibits certain dispersion effects, except for some points where degradation occurs, which will bring essential difficulties to nonlinear estimates. In contrast, when rotation is taken into account, we observe that this degeneracy in dispersion effects disappears; furthermore, we can derive dispersive estimates for the second and third components of the simple-zero mode within the velocity field. However, neither the first component of velocity nor temperature exhibits any dispersion characteristics. Fortunately, we find a combined quantity from these components to obtain its dispersive estimates. Additionally, we develop three good unknowns to minimize linear coupling terms as much as possible while mitigating growth induced by linear stretching terms; through constructing a series of multipliers, we achieve enhanced dissipation and inviscid damping effects. In our analysis of the nonlinear system aimed at establishing an improved stability threshold, we utilize quasi-linearization methods to rectify deficiencies in dispersive estimates related to both the first component of velocity and temperature, as well as address regularity issues along vertical directions caused by buoyancy forces and stratification. Consequently, we demonstrate that if initial perturbations in velocity and temperature satisfy \[\left\|u_{\mathrm{in}}\right\|_{H^{N+2}\cap W^{N+3,1}}+\left\|\theta_{\mathrm{in}}\right\|_{H^{N+1}\cap W^{N+3,1}}<\delta \mathbf{Re}^{-\frac{14}{15}},\] for any \(N\geq 11\) and some \(\delta>0\) independent of \(\mathbf{Re}\), then the solution to the 3D Boussinesq equations with rotation is nonlinearly stable without transitioning away from the steady state.

Stability threshold; Boussinesq equations with rotation; Couette flow; Stratification; Dispersive.

35Q35, 76U05, 76E07, 76F10.

1 Introduction↩︎

In this paper, we consider the following three-dimensional (3D) Boussinesq equations with rotation, which model the motion of incompressible fluids under both the Coriolis force and buoyancy: \[\label{1461} \begin{cases} \partial_{t} V+V \cdot \nabla V-\nu \Delta V+\nabla P+\beta \mathrm{e}_3 \times V=\Theta \mathbf{g} \mathrm{e}_3, \\ \partial_{t} \Theta +V \cdot \nabla \Theta-\mu \Delta \Theta=0, \\ \nabla \cdot V=0. \end{cases}\tag{1}\] where \((x_1, y, z) \in \mathbb{T} \times \mathbb{R} \times \mathbb{T}\) (the torus \(\mathbb{T}\) is the periodized interval \([0,2\pi]\)), \(V=\left(V^1, V^2, V^3\right)^{\mathrm{T}}\) denotes the velocity field, \(P\) is the pressure, \(\Theta\) is the temperature, \(\mathrm{e}_3=(0, 0, 1)^{\mathrm{T}}\) is the unit vector in the vertical direction. The term \(\beta \mathrm{e}_3 \times V\) represents the Coriolis force, where \(\beta \in \mathbb{R}\backslash\{0\}\) characterizes the rotation speed \(\beta/2\). \(\Theta \mathbf{g} \mathrm{e}_3\) symbolizes the buoyancy with the gravity \(\mathbf{g}\). \(\nu\) represents the viscosity coefficient, i.e., \(\nu=\mathbf{Re}^{-1}\), where \(\mathbf{Re}\) is the Reynolds number. And \(\mu\) is the thermal diffusivity. For simplicity, we take \(\mu=\nu,\,\mathbf{g}=1\). The derivation of 1 can be referred to [1], [2]. The system 1 is supplemented with initial data: \[\begin{align} \label{invalue} (V, \Theta)(0, x_1, y, z)=(V_{\mathrm{in}}, \Theta_{\mathrm{in}})(x_1, y, z). \end{align}\tag{2}\]

We study the dynamic stability behavior of solutions to 1 around the nontrivial stationary solution, which is known as Couette flow with the linearly stratified temperature profile \[\label{1462} V_s(y)=(y, 0, 0)^{\mathrm{T}},\,\,\,\,\Theta_s(z)=1+\alpha^2 z,\tag{3}\] where \(\alpha > 0\) represents the Brunt-Väisälä frequency, which indicates the fluid’s response strength to displacements in the direction of gravity. This scenario corresponds to a stable stratification where warmer fluid overlays colder fluid, with gravity acting as a restoring force for vertical perturbations and thereby supporting internal gravity waves. The associated pressure of this steady state is \[\label{1464} P_s(y, z)=z+\frac{1}{2} (\alpha^2z^2-\beta y^2).\tag{4}\]

We introduce the perturbations \(u=V-(y, 0, 0)^{\mathrm{T}}, \theta=-\frac{1}{\alpha}\left(\Theta-\Theta_s \right), p=P-P_s\), the perturbed \((u, p, \theta)\) satisfies the following system \[\label{1466} \begin{cases} \partial_{t} u+y \partial_{x_1} u-\nu \Delta u+\nabla p^L+\begin{pmatrix} (1-\beta) u^2 \\ \beta u^1 \\ \alpha \theta \end{pmatrix}=-u \cdot \nabla u-\nabla p^{NL}, \\ \partial_{t} \theta+y \partial_{x_1} \theta-\nu \Delta \theta-\alpha u^3=-u \cdot \nabla \theta, \\ \nabla \cdot u=0, \\ (u, \theta)(0, x_1, y, z)=(u_{\mathrm{in}}, \theta_{\mathrm{in}})(x_1, y, z), \end{cases}\tag{5}\] where the initial data \(u_{\mathrm{in}}=V_{\mathrm{in}}-(y, 0,0)^{\mathrm{T}}\), \(\theta_{\mathrm{in}}=-\frac{1}{\alpha}\left(\Theta_{\mathrm{in}}-(1+\alpha^2 z)\right)\). The pressure \(p^{L}\) and \(p^{NL}\) are determined by \[\begin{align} \Delta p^{L}=-2\partial_{x_1} u^2+\beta\left(\partial_{x_1} u^2 -\partial_{y} u^1\right)-\alpha \partial_{z} \theta, \quad \Delta p^{NL}=-\partial_{i}u^j \partial_{j}u^i. \end{align}\]

The Coriolis force, in its role concerning fluid stability, is influenced by the variation of the parameter \(\beta\). The relevant physical quantity for quantifying the effects of rotation is known as the Bradshaw-Richardson number (see [3], [4]), \[B_{\beta}:= \beta(\beta-1)\in [-\frac{1}{4},\infty).\] This number effectively captures and quantifies the stable or unstable nature of dynamics described in equation 5 .

1.1 Background and Motivation↩︎

Before elaborating on the background and motivation, some relevant notations are introduced. The Fourier transform of a function \(f= f(x_1, y, z)\) will be denoted as follows. Given \((k,\eta,l)\in \mathbb{Z}\times \mathbb{R}\times \mathbb{Z}\), define \[\label{2461} \mathcal{F}(f)=\widehat{f}\left(k, \eta, l\right)=\frac{1}{(2 \pi)^{2}}\int_{x_1 \in \mathbb{T}}\int_{y \in \mathbb{R}} \int_{z \in \mathbb{T}}f\left(x_1, y, z\right)e^{- i \left(k x_1+ \eta y+ l z\right)} dx_1dydz.\tag{6}\] Denoting the inverse operation to the Fourier transform by \(\mathcal{F}^{-1}\) or \(\check{f}\) \[\label{2462} \mathcal{F}^{-1}(f)=\check{f} \left(x_1, y, z\right)=\sum_{k \in \mathbb{Z}}\int_{\eta \in \mathbb{R}} \sum_{l \in \mathbb{Z}} f \left(k, \eta, l\right)e^{ i \left(k x_1+ \eta y+ l z\right)} d\eta.\tag{7}\] In addition, we also define the Fourier transform in the \(z\)-direction of a function \(h= h(y, z)\) by \[h^l (y) = \frac{1}{2 \pi}\int_{z \in \mathbb{T}}h\left(y, z\right)e^{- i l z} dz.\] For a general Fourier multiplier with symbol \(m(k,\eta,l)\), we write \(mf\) to denote \(\mathcal{F}^{-1}(m(k,\eta,l)\hat{f})\). Thus, it holds \[\widehat{m(D)f}=m(k,\eta,l)\widehat{f}.\]

We define the projections on the zero modes and the non-zero modes in \(x_1\) of a function \(f\) as follows: \[\label{HSX333} P_{k=0}f=f_{0}=\frac{1}{2 \pi}\int_{\mathbb{T}} f(x_1,y,z)d{x_1}, \quad P_{\neq}f=f_{\neq}=f-f_{0}.\tag{8}\] Furthermore, the projection onto the zero frequency in \(x_1\) and \(z\) of a function \(f(x_1,y,z)\) is denoted by \[\overline{f}_0=\frac{1}{(2 \pi)^2}\int_{\mathbb{T}} \int_{\mathbb{T}} f(x_1,y,z)d{x_1} dz, \quad \quad \widetilde{f}_0=f_0-\overline{f}_0,\] where \(\overline{f}_{0}\) represents the double-zero modes and \(\widetilde{f}_{0}\) stands for the simple-zero modes of \(f\).

1.1.1 Known results for the Navier-Stokes equation with/without rotation↩︎

The stability of plane Couette flow has been extensively studied since the pioneering works of Rayleigh [5] and Kelvin [6]. It is well established that plane Couette flow remains linearly stable for all Reynolds numbers, as noted in [7], [8]. However, at high Reynolds numbers, the flow can exhibit nonlinear instability and transition to turbulence under small but finite perturbations—a phenomenon known as the Sommerfeld paradox. One approach to resolving this paradox involves examining the transition threshold problem, a concept rigorously formulated by Trefethen et al. [9]. Bedrossian, Germain, and Masmoudi [10], [11] developed a comprehensive mathematical framework for this issue that can be articulated as follows:

Given a norm \(\|\cdot\|_{X}\), determine an exponent \(\gamma = \gamma(X)\) such that \[\begin{align} &\left\| u_{\mathrm{in}} \right\|_{X} \lesssim \; \mathbf{Re}^{-\gamma } \,\, \Rightarrow \text{ stability},\\ & \left\| u_{\mathrm{in}} \right\|_{X} \gg \;\mathbf{Re}^{-\gamma } \,\, \Rightarrow \text{ instability}, \end{align}\] where \(\gamma\) is called the transition threshold.

In the absence of the rotation and temperature, the system 1 reduces to the standard Navier-Stokes equations. Significant progress has been made on the stability threshold problem for Navier-Stokes equations near the Couette flow. For 2D case on the domain \(\mathbb{T}\times \mathbb{R}\), Bedrossian, Masmoudi and Vicol [12] demonstrated that the threshold \(\gamma\leq 0\) for the perturbations in Gevrey space. Bedrossian, Vicol and Wang [13] showed that the threshold \(\gamma\leq 1/2\) for the perturbation in Sobolev space \(H^N\) with \(N>1\). Wei and Zhang [14] made the energy estimates in short and long time scale, and constructed a new multiplier to obtain that the threshold \(\gamma\leq 1/3\) in \(H^3\). On the bounded domain \(\mathbb{T}\times [-1,1]\), the results depend on the boundary conditions. Under Navier-slip boundary conditions, Wei and Zhang [15] proved that \(\gamma\leq {1}/{3}\) for the perturbations in Sobolev class, and Bedrossian, He, Iyer and Wang [16] showed that \(\gamma\leq 1/2\) for the perturbations in Gevrey space; Under no-slip boundary conditions, \(\gamma\leq {1}/{2}\) for the perturbations in Sobolev space [17]. There are still many important works in this area, one can refer to the references [11], [16], [18]–[23] and their related literature. The 3D case is considerably more challenging due to the lift-up effect. Bedrossian, Germain and Masmoudi [10] designed the Fourier multipliers that precisely encode the interplay between the dissipation and possible growth and obtained the stability threshold index \(\gamma\leq 3/2\) for Sobolev space \(H^{\sigma}\) \((\sigma>9/2)\). For a related work also see [24], [25] and so on. Later on, Wei and Zhang [26] showed that the transition threshold \(\gamma \leq1\) of 3D Navier-Stokes equations in \(H^2\). They further provided a mathematically rigorous proof that the regularity of initial perturbations does not affect this transition threshold. Chen, Wei and Zhang [27] demonstrated that the transition threshold index is also \(\gamma\leq 1\) in the finite channels with the non-slip boundary condition.

Without thermal stratification (i.e., \(\Theta=0\)), the system 1 corresponds to the Navier-Stokes equations with rotation. In contrast to the standard Navier-Stokes equations, the Navier-Stokes equation with rotation not only represents a fundamental physical model, but also exhibits stabilizing effects induced by the Coriolis force. Our prior studies [28], [29] addressed the stability of the Navier-Stokes equation with rotation near Couette flow on the domain \(\mathbb{T} \times \mathbb{R} \times \mathbb{T}\). The analysis considered two distinct parameter regimes: one with significant dispersive effects for \(B_{\beta}>0\), and another exhibiting a strong lift-up effect at \(B_{\beta}=0\). In the dispersive regime, we established the stability threshold \(\gamma\leq 1\). Subsequently, under the additional condition that \[\begin{align} \label{0in} \overline{u_0}(t=0,y):=\int_{\mathbb{T}^2} u_{\mathrm{in}}(x_1,y,z) \mathrm{d}{x_1} \mathrm{d}z=0, \end{align}\tag{9}\] Coti Zelati, Del Zotto and Widmayer [30] established nonlinear dispersive estimates, thereby obtaining the threshold \(\gamma \leq 8/9\) for \(B_{\beta}>0\) and \(\gamma \leq 5/6\) for \(B_{\beta}\gtrsim \nu^{-1}\). Recently, Li–Sun–Wang–Wei–Zhang [31] further removed the restricted condition 9 . By deriving suitable Strichartz estimates \[\left\| e^{\mathcal{A} t} h^l \right\|_{L_t^2 (L_y^{\infty}(\mathbb{R}))} + \nu^{1/2} |l| \left\| e^{\mathcal{A} t} h^l \right\|_{L_t^1 (L_y^{\infty}(\mathbb{R}))} \lesssim \nu^{-1/3} |l|^{-\frac{1}{6}} \| h^l \|_{L^2(\mathbb{R})},\] where the dispersive semigroup \(\mathcal{A}:=\sqrt{B_{\beta}} \mathcal{R}_3^l+\nu(\partial_y^2-l^2)\) with \(\mathcal{R}_3^l:=-il(l^2-\partial_y^2)^{-\frac{1}{2}}\) and \(h^l=\sum_{j \in \mathbb{Z}} P_j h^l\), they obtained an improved threshold \(\gamma \leq 5/6\) in anisotropic Sobolev spaces on the domain \(\mathbb{T} \times \mathbb{R}\times\mathbb{T}\). Moreover, they also provided a stability threshold of the form \(2/3+\delta\) with \(\delta>0\) on the domain \(\mathbb{T} \times \mathbb{R}^2\).

1.1.2 Motivation behind 1 with vertical stratification↩︎

The interplay between rotation and vertical thermal stratification establishes a fundamental geophysical framework that underpins numerous atmospheric, oceanic, and astrophysical flows. Consequently, examining their combined effects on the stability of Couette flow is directly pertinent to geophysical applications. Recent experimental research by Oxley and Kerswell [32] has initiated an investigation into the linear stability of the 3D rotating Boussinesq equations in relation to Couette–Poiseuille flow, taking into account the synergistic influences of stratification, rotation, and viscosity. However, the corresponding mathematical theory—particularly regarding the stability of the steady-state solution 3 4 —remains incomplete.

The stability of Couette flow under vertical stratification \(\Theta_s = 1 + \alpha^2 z\) (buoyancy aligned with \(\theta\mathbf{e}_3\)) remains an unsolved problem, particularly when rotation is present. This contrasts with the more well-developed theory for horizontally stratified configurations. We first recall the linear dispersion relations for two canonical non-rotating (\(\beta=0\)) setups:

• Vertical stratification (\(\Theta_s = 1 + \alpha^2 z\) and \(\theta\mathbf{e}_3\)):
  Notice that the simple zero modes satisfy \[\partial_{t} \begin{bmatrix} {\widetilde{U}_{0}^{2}} \\ {\widetilde{\Theta}_{0}} \end{bmatrix}=\begin{bmatrix} \nu \Delta_{L,0} & \alpha \partial_{y} \Delta_{L,0}^{-1} \partial_{z} \\ -\alpha \partial_{z}^{-1} \partial_{y} & \nu \Delta_{L,0} \end{bmatrix}\begin{bmatrix} {\widetilde{U}_{0}^{2}} \\ {\widetilde{\Theta}_{0}} \end{bmatrix},\nonumber\] and the eigenvalues are \[\lambda_{\pm} = -\nu(\eta^2+l^2) \pm i\frac{\alpha|\eta|}{\sqrt{\eta^2+l^2}}.\] The dispersion relation is \(\frac{h}{|\eta,l|} = \frac{\alpha|\eta|}{|\eta,l|}\). And it can be seen that \(h\) will serve as the denominator of the single-zero modes. We encounter a singularity at wavenumber \(\eta = 0\), which poses intrinsic difficulties in establishing the dispersion estimate.

  When the buoyancy term \(\Theta \mathrm{e}_3\) is replaced by \(\Theta \mathrm{e}_2\) with \(\mathrm{e}_2=(0,1,0)^{\mathrm{T}}\), Coti Zelati and Del Zotto [33], [34] first established the linear and nonlinear stability results for the 3D Boussinesq equations near the stably stratified Couette flow, i.e., \(V_s = (y, 0, 0)\), \(\Theta_s = 1 + \alpha^2 y\) with \(\alpha > 0\). In addition, they found that the dispersive structure exhibited by the zero-frequency components as follows:

• Horizontal stratification (\(\Theta_s = 1 + \alpha^2 y\) and \(\theta\mathbf{e}_2\)):
  Coti Zelati, Del Zotto and Widmayer investigated the 3D Boussinesq system with horizontal stratification in [34]. For the simple-zero modes satisfy \[\partial_{t} \begin{bmatrix} {\widetilde{U}_{0}^{2}} \\ {\widetilde{\Theta}_{0}} \end{bmatrix}=\begin{bmatrix} \nu \Delta_{L,0} & -\alpha \partial_{z} \Delta_{L,0}^{-1} \partial_{z} \\ \alpha & \nu \Delta_{L,0} \end{bmatrix}\begin{bmatrix} {\widetilde{U}_{0}^{2}} \\ {\widetilde{\Theta}_{0}} \end{bmatrix},\nonumber\] and the eigenvalues become \[\lambda_{\pm}=-\nu (\eta^2+l^2)\pm i \frac{\alpha |l|}{\sqrt{\eta^2+l^2}}.\] Here, dispersion persists for all simple-zero modes, namely \(l\neq 0\), and no degeneracy occurs. Based on this, under the extra condition in [34] that \[\int_{\mathbb{T}^2} u_{\mathrm{in}} \mathrm{d}{x_1} \mathrm{d}z=\int_{\mathbb{T}^2} \theta_{\mathrm{in}} \mathrm{d}{x_1} \mathrm{d}z=0,\] Coti Zelati, Del Zotto and Widmayer further proved the stability thresholds: \(\gamma\leq 11/12\) for \(\alpha>1/4\) and \(\gamma\leq 8/9\) for \(\alpha\gtrsim \nu^{-1}\). More recently, Cui, Wang and Wang [35] demonstrated that when the initial velocity and temperature satisfy \(\left\| V_{\mathrm{in}}-(y,0,0) \right\|_{H^2} \leqslant \delta \mathbf{Re}^{-1}\) and \(\left\| \Theta_{\mathrm{in}} \right\|_{H^2} \leqslant \delta \mathbf{Re}^{-2}\) for some \(\delta>0\), the 3D Boussinesq equations exhibit nonlinear stability. In their work, the steady state of the temperature is \(\Theta_s=0\), while the steady state of the velocity remains the Couette flow. More specifically, for the relevant results of the 2D Boussinesq equation, one can refer to the references [20], [36]–[41].

When the system incorporates the rotational effect, one finds that rotation can suppress the lift-up effect and have the dispersion effect. It is great surprised that when both the rotational and stratification effects are taken into account, we find the dispersion relation is reshaped, and the dispersive waves is restored at all streamwise frequencies \(\eta \in \mathbb{R}\), thereby eliminating the singularity at \(\eta=0\) present in the non-rotating, vertically stratified case.

• Vertical stratification with rotation (\(\Theta_s = 1 + \alpha^2 z\), and \(\theta\mathbf{e}_3\))
  The linearized equations for \((\widetilde{U}_0^1, \widetilde{U}_0^2, \widetilde{\Theta}_0)\) read: \[\partial_{t} \begin{bmatrix} \widetilde{U}_{0}^{1} \\ \widetilde{U}_{0}^{2} \\ \widetilde{\Theta}_{0} \end{bmatrix} = \begin{bmatrix} \nu \Delta_{L,0} & \beta-1 & 0 \\[4pt] -\beta \partial_{z} \Delta_{L,0}^{-1} \partial_{z} & \nu \Delta_{L,0} & \alpha \partial_{y} \Delta_{L,0}^{-1} \partial_{z} \\[4pt] 0 & -\alpha \partial_{z}^{-1} \partial_{y} & \nu \Delta_{L,0} \end{bmatrix} \begin{bmatrix} \widetilde{U}_{0}^{1} \\ \widetilde{U}_{0}^{2} \\ \widetilde{\Theta}_{0} \end{bmatrix},\]

  and the eigenvalues are \[\lambda_1 = -\nu (\eta^2+l^2), \qquad \lambda_{\pm} = -\nu (\eta^2+l^2) \pm i \, \frac{h(\alpha, \beta, \eta,l)}{\sqrt{\eta^2+l^2}},\] where the dispersion relation \(h(\alpha, \beta, \eta,l)\) is given by \[\frac{h(\alpha, \beta, \eta, l)}{|\eta,l|} = \frac{\sqrt{\alpha^2 \eta^2 + B_{\beta} l^2}}{|\eta, l|}, \quad (\eta, l) \in \mathbb{R} \times (\mathbb{Z}\setminus\{0\}),\] where \(\alpha\) and \(\beta\) represent the strengths of stratification and the speed of rotation, respectively. The key observation is that \(h(\alpha, \beta, \eta,l) > 0\) for all \(l \neq 0\) and \(\eta\in \mathbb{R}\) under the condition \(B_{\beta}>0\) due to the participation of the rotation effect, which can simplify the situation.

The dynamics of fluids are significantly influenced by rotation and temperature stratification. Rapid rotation excites inertial waves, while perturbations in a stably stratified state generate internal gravity waves. It is also important to note that all these waves exhibit dispersive characteristics. As highlighted by Babin et al. [42], one of the primary challenges in understanding geophysical flow dynamics lies in accounting for the oscillatory effects produced by such dispersive waves. This phenomenon has already been applied to investigate the global well-posedness of equation 1 , as referenced in works such as [43]–[47].

1.2 Summary of main result↩︎

We first introduce a coordinate transform described in detail in [10], [24], [33], which is a central tool for our analysis. Due to the characteristics of the transport structure in (5 ), we define a linear change of variable \[\label{2465} x=x_{1}- t y, \quad \quad y=y, \quad \quad z=z.\tag{10}\] In the new coordinates, the corresponding differential operators become \[\label{2466} \nabla =(\partial_{x_1},\partial_{y},\partial_{z}) \mapsto \nabla_{L}:=(\partial_{x},\partial_{y}^{L},\partial_{z}), \quad\quad \Delta\mapsto \Delta_{L}\tag{11}\] where \[\label{24677} \partial_{y}^{L}:=\partial_{y}-t\partial_{x},\quad \Delta_{L}:=\partial_{x}^{2}+(\partial_{y}^{L})+\partial_{z}^{2}.\tag{12}\] Furthermore, we define \[\nabla_{L,h}:=(\partial_{x},\partial_{y}-t\partial_{x}),\, \quad \Delta_{L,h}:=(\partial_x)^2+(\partial_y-t\partial_x)^2,\] which implies the Fourier symbol denoted by (10 ) as \[\label{HSX444} |\widehat{\nabla_{L}}|=|k,\eta-kt,l|,\quad |\widehat{\nabla_{L,h}}|=|k,\eta-kt|,\tag{13}\] and \[\label{2468} p(t, k, \eta, l):=\widehat{-\Delta_{L}}=k^2+(\eta- kt)^2+l^2,\quad p_{h}(t, k, \eta):= \widehat{-\Delta_{L,h}}=k^2+(\eta- kt)^2.\tag{14}\] Note that \(p\) and \(p_{h}\) are time dependent, thus their time derivative is \[\label{2469} \partial_{t} {p}= \partial_{t}{p_{h}} =-2 k (\eta- kt).\tag{15}\]

Combining the definition of the new coordinates as in (10 ) and denoting \(U^i(t, x, y, z)=u^i(t, x_1, y, z)\) \((i=1, 2, 3)\), \(\Theta(t, x, y, z)=\theta(t, x_1, y, z)\). 5 then can be rewritten as follows

\[\begin{align} \label{24611} \begin{cases} \partial_{t} U-\nu \Delta_{L} U+\begin{pmatrix} \left(1-\beta \right) U^2 \\ \beta U^1 \\ \alpha\Theta\end{pmatrix}+\left( \beta-2 \right) \nabla_{L} \Delta_{L}^{-1} \partial_{x} U^2-\beta \nabla_{L} \Delta_{L}^{-1} \partial_{y}^{L} U^1\\ \quad-\alpha\nabla_{L}\Delta_{L}^{-1}\partial_{z}\Theta=-U\cdot \nabla_{L} U+\nabla_{L} \Delta_{L}^{-1}(\partial_{i}^{L} U^j \partial_{j}^{L} U^i),\\ \partial_{t}\Theta-\nu\Delta_{L}\Theta-\alpha U^{3}=-U\cdot \nabla_{L}\Theta,\\ U(t=0)=U_{\text{in}},\quad \Theta(t=0)= \Theta_{\text{in}}. \end{cases} \end{align}\tag{16}\]

It is important to note that under the coordinate transformation 10 , the initial value conditions presented in 16 are fundamentally equivalent to the initial values stated in 5 . Specifically, we have \(U_{\text{in}}=u_{\text{in}}\) and \(\Theta_{\text{in}}=\theta_{\text{in}}\). In the following section, we will not make a special distinction regarding this matter. We will examine the behavior of solutions derived from the linearization of 3 . Our first main result gives the linear stability of 1 near the Couette flow and vertical stratification as follows.

Now, we provide the nonlinear stability result of 1 –2 in Sobolev space.

1.3 Obstacles and strategies↩︎

1.3.1 Linear stability↩︎

We analysis two main difficulties on the linear system.

1.3.1.1 Difficulty 1:

The interactions of stratification, rotation, and Couette flow lead to complex linear coupling phenomena, which prevent us from directly closing the energy estimate.

More precisely, it was observed in 16 that four equations for \((U^1,U^2,U^3,\Theta)\) are coupled with each other and cannot be decoupled directly as in the case of two equations. Moreover, although \(U^2\) plays a significant role in the stability study of the Navier-Stokes system in [10] and [28], we find that it is not suitable for the linear system 22 . To address this difficulty, we find several good unkowns to simplify the system, which are constructed in Strategy 1.

1.3.1.2 Strategy 1: Constructing good unknowns.

Based on the characteristics of stratification and rotation effects, we find three good unknowns about \(U^3,\;\Theta\) and the third component of vorticity \(W^3\). Thanks to the incompressibility condition, we can use \(U^3\) and \(W^3\) to recover \(U^1\) and \(U^2\), thus those three variables are sufficient to describe the original system.

We start the process with the formal analysis in detail. Taking the Fourier transform of 22 in the moving frame, we have \[(\partial_t+\nu p)\begin{pmatrix} \widehat{U^3_{\neq}}\\ \widehat{W^3_{\neq}} \\ \widehat{\Theta_{\neq}} \end{pmatrix}+ \begin{pmatrix} 2\frac{k(\eta-kt)l^2}{pp_h} & - \beta\frac{il}{p} & \alpha\frac{p_h}{p} \\ (1-\beta)il & 0 & 0 \\ -\alpha & 0 & 0 \end{pmatrix}\begin{pmatrix} \widehat{U^3_{\neq}}\\ \widehat{W^3_{\neq}} \\ \widehat{\Theta_{\neq}} \end{pmatrix} + \begin{pmatrix}2i\frac{k^2l}{pp_h} \widehat{W^3_{\neq}} \\ 0 \\ 0 \end{pmatrix} = 0.\] The third term has so many time decay that can be controlled by the Fourier multiplier \(M\). To make full use of the paired linear coupling terms, we design new variables \(Q_{\neq},\;K_{\neq}\) and \(H_{\neq}\) to respectively represent \(U^3_{\neq},\;W^3_{\neq}\) and \(\Theta_{\neq}\). Let us assume that \(\widehat{Q_{\neq}}=C_{Q_{\neq}}\widehat{U^3_{\neq}},\;\widehat{K_{\neq}}=C_{K_{\neq}}\widehat{W^3_{\neq}}\) and \(\widehat{H_{\neq}}=C_{H_{\neq}}\widehat{\Theta_{\neq}}\). The ODE then becomes \[\begin{align} (\partial_t+\nu p-\begin{pmatrix} \frac{\partial_t C_{Q_{\neq}}}{C_{Q_{\neq}}}\\ \frac{\partial_t C_{K_{\neq}}}{C_{K_{\neq}}} \\ \frac{\partial_t C_{H_{\neq}}}{C_{H_{\neq}}} \end{pmatrix}+ \begin{pmatrix} 2\frac{k(\eta-kt)l^2}{pp_h} & - \beta\frac{il}{p}\frac{C_{Q_{\neq}}}{C_{K_{\neq}}} & \alpha\frac{p_h}{p}\frac{C_{Q_{\neq}}}{C_{H_{\neq}}} \\ (1-\beta)il\frac{C_{K_{\neq}}}{C_{Q_{\neq}}} & 0 & 0 \\ -\alpha\frac{C_{H_{\neq}}}{C_{Q_{\neq}}} & 0 & 0 \end{pmatrix})\begin{pmatrix} \widehat{Q_{\neq}} \\ \widehat{K_{\neq}} \\ \widehat{H_{\neq}} \end{pmatrix} + \begin{pmatrix}2i\frac{k^2l}{pp_h}\frac{C_{Q_{\neq}}}{C_{K_{\neq}}} \widehat{K_{\neq}} \\ 0 \\ 0 \end{pmatrix} = 0. \end{align}\] To antisymmetrize the coefficient matrix, set \(\frac{C_{H_{\neq}}^2}{C_{Q_{\neq}}^2}=\frac{p_h}{p},\;\frac{C_{Q_{\neq}}^2}{C_{K_{\neq}}^2}=-p\frac{\beta-1}{\beta}\), and choose \(C_{H_{\neq}}=\sqrt{\frac{p_h}{p}}C_{Q_{\neq}},\;C_{K_{\neq}}=i\sqrt{\frac{\beta}{\beta-1}}\frac{1}{\sqrt{p}}C_{Q_{\neq}}\). We also note that \(\left|\frac{l^2}{p}\frac{\partial_t p_h}{p_h}\right|\left/\left|\frac{l}{\sqrt{p}}\right|\right. \leq 1, \left|\frac{\partial_t p}{p}\right|\left/\left|\sqrt{\frac{p_h}{p}}\right|\right. \leq 1\), which means that by using the linear coupling terms that appear in pairs, specific stretching terms can be eliminated. Finally, we choose \(C_{Q_{\neq}} = -p^{\frac{3}{4}}p_h^{-\frac{1}{2}}\) and construct good unknowns \[\label{hhll} Q_{\neq}\triangleq-|\nabla_L|^{\frac{3}{2}}|\nabla_{L,h}|^{-1}U^3_{\neq},\;K_{\neq}\triangleq-i\sqrt{\frac{\beta}{\beta-1}}|\nabla_L|^{\frac{1}{2}}|\nabla_{L,h}|^{-1}W^3_{\neq}, \;H_{\neq}\triangleq-|\nabla_L|^{\frac{1}{2}}\Theta_{\neq}.\tag{17}\] This approach not only counteracts the mutually coupled linear terms, thereby clearly extracting the enhanced dissipation and inviscid damping mechanisms, but also effectively manages the growth of the linear stretching term \(\partial_{xy} \Delta^{-1}\) that arises from pressure. Proper handling of this growth is typically achieved through a meticulously designed Fourier multiplier \(m\), as demonstrated in 26 , which accurately encodes its dependence on frequency and viscosity. The design of this multiplier \(m\) aims to ensure it possesses a more optimized bound ?? , with three well-chosen variables 17 facilitating this objective. Furthermore, we introduce a cross-term estimate ?? that ensures \(B_{\beta}\) satisfies the condition \(B_{\beta} > 1/4\). For further details, please refer to section 3.1. Of course, if our goal is solely to achieve enhanced dissipation at non-zero frequencies, it suffices for us to have \(B_{\beta} > 0\), which is thoroughly proven in Appendix 5.

1.3.1.3 Difficulty 2:

The single-zero modes of temperature \(\tilde{\theta}_0\) and the first component \(\tilde{u}_0^1\) within the velocity field does not exhibit linear dispersion estimates.

Recall from section 3.2 that the ODE with respect to time variables \(t\) can be obtained and solved by taking the Fourier transform of the single zero frequency system 47 with respect to \((x,y,z)\). It can be seen that \(u_0^2\) and \(\tilde{u}_0^3\) have linear dispersion effects. Unfortunately, \(\tilde{u}_0^1\) and \(\tilde{\theta}_0\) only exhibit pure thermal effects and have no dispersion estimates. To address this difficulty, we find a combined quantify from \(\tilde{u}_0^1\) and \(\tilde{\theta}_0\), and obtain its dispersive estimates, which are constructed in the following Strategy 2.

1.3.1.4 Strategy 2: Constructing the dispersion combined quantities

Based on the structure of single-zero modes in ?? ?? , we choose two combined quantities concerning \(\tilde{u}^1_0\) and \(\tilde{\theta}_0\). \[\begin{align} V_0^2=&-\beta\mathcal{R}^{-1}\partial_{zz}\Delta_{L,0}^{-1}\tilde{U}_0^1 + \alpha\mathcal{R}^{-1}\partial_{yz}\Delta_{L,0}^{-1}\tilde{\Theta}_0=:G_1\tilde{U}_0^1 + G_2\tilde{\Theta}_0,\\ \Lambda_0=& \alpha\sqrt{\frac{\beta}{\beta-1}}\partial_{yz}\Delta_{L,0}^{-1} \tilde{U}_0^1 + \sqrt{B_{\beta}}\partial_{zz}\Delta_{L,0}^{-1}\tilde{\Theta}_0=:G_3\tilde{U}_0^1 + G_4\tilde{\Theta}_0. \end{align}\] By utilizing \(V_0^2\) and \(\Lambda_0\), we can successfully extract out of the dispersion structure from the pure thermal effect structure in \(\tilde{U}_0^1\) and \(\tilde{\Theta}_0\). Furthermore, these two combined quantities allow for the recovery of both \(\tilde{U}_0^1\) and \(\tilde{\Theta}_0\). Additionally, we find a corresponding combined quantity, denoted as \(\tilde{V}_0^3\), which aligns with \(\tilde{U}_0^3\), thus circumventing the need for direct estimation via the incompressible condition given by \(\tilde{U}^3_0 = -\partial_z^{-1}\partial_y U_0^2\).  We then employ the method of oscillatory integrals to analyze the dispersive semigroup, leading to linear dispersive estimates, further details can be found in sections 3.2 and 3.3.

1.3.2 Nonlinear stability↩︎

We deal with two main difficulties on the analysis of nonlinear system.

1.3.2.1 Difficulty 1: Structural issue for zero modes

We revisit the analysis results of the linear part. Among the zero modes, only \(U_0^2\) and \(\tilde{U}_0^3\) exhibit linear dispersion estimates, while \(\tilde{U}_0^1\) and \(\tilde{\Theta}_0\) are characterized solely by thermal decay estimates. In examining nonlinear systems of zero modes, a significant challenge lies in managing the nonlinear interactions between simple-zero modes. Here, we continue to adopt the approach of linear stability and construct the combined quantity \(V_0^2\) to derive additional nonlinear dispersion estimates that satisfy the following equation, \[\begin{align}\label{com11} \partial_t V_0^2 - \nu \Delta_{L,0} V_0^2 + \mathcal{R} U_0^2 &= - G_1\widetilde{(U_0 \cdot\nabla U_0^1)} -G_2\widetilde{(U_0 \cdot\nabla \Theta_0)}\\&\quad+ {\text{non-zero modes contributions}}, \end{align}\tag{18}\] where \(\mathcal{R}\) is the dispersion operator as in 51 . We note that the operators \((G_1, G_2)\) and the convection operator \(U_0 \cdot \nabla\) are non-commutative. Therefore, we can not rewrite \(G_1{(U_0 \cdot \nabla \tilde{U}_0^1)} + G_2{(U_0 \cdot \nabla \tilde{\Theta}_0)}\) as a product of two terms that both have the dispersion property, that is \[G_1{(U_0 \cdot\nabla \tilde{U}_0^1)} +G_2{(U_0\cdot\nabla\tilde{\Theta}_0)} \neq U_0 \cdot\nabla (G_1\tilde{U}_0^1 + G_2\tilde{\Theta}_0)=\underbrace{U_0\cdot\nabla V_0^2}_{\text{ dispersion-dispersion}}.\] It is evident that the right side of the aforementioned equation comprises two types of product terms that are challenging to handle. The first type includes dispersion-thermal type terms, such as \(\tilde{U}_0^{2, 3} \cdot \nabla_{y, z} \{U^1_{0}, \Theta_{0}\}\). The second type consists of thermal-thermal type terms, exemplified by \(\overline{U}_{0}^3\partial_zU^1_0\). Furthermore, similar to the right-hand side terms of the \(V_0^2\) equation, the nonlinear terms of the \(\Lambda_0\) equation also lack the dispersive structure. Consequently, it proves difficult to demonstrate the propagation of zero modes within high Sobolev regularity norms and to derive nonlinear dispersion estimates.

1.3.2.2 Strategy 1: Quasilinear method for zero modes.

To capture more nonlinear dispersive effects as accurately as possible, we adopt a quasi-linear approach decomposing the zero-frequency solution into the main part and the perturbation part: \[\begin{cases} U_0 = U_{0,1} + U_{0,2}, \\ \Theta_0 = \Theta_{0,1} + \Theta_{0,2}, \\ V_0^2 = V_{0,1}^2 + V_{0,2}^2, \\ \tilde{V}_0^3 = \tilde{V}_{0,1}^3 + \tilde{V}_{0,2}^3, \\ \Lambda_0 = \Lambda_{0,1} + \Lambda_{0,2}. \end{cases}\]

The first subsystem (the main component) encompasses all initial value information at zero frequency. The nonlinear terms account for contributions at non-zero frequencies, while the zero-frequency component is meticulously designed to filter out dispersion-dispersion type terms in the equations governing dispersive variables and dispersion-thermal type terms in the equations pertaining to non-dispersive variables, as illustrated in 58 : \[\begin{align} \mathcal{N}_{\neq+0}(U_{0,1}^2)=&2\partial_y\Delta_{L,0}^{-1}\big(\partial_yU_{0,1}^2\partial_yU^2_{0,1}+\partial_y\tilde{U}^3_{0,1}\partial_zU^2_{0,1}\big)-\tilde{U}_{0,1}\cdot\nabla U_{0,1}^2 \\&+ {\text{non-zero modes contributions}}. \end{align}\]

In relation to the second subsystem (perturbation part), it solely encompasses zero initial values along with the remaining nonlinear terms, as illustrated in 61 –63 . Following decomposition, the first subsystem is employed to propagate high Sobolev regularity norms by means of Lemma 17, leading to corresponding nonlinear dispersion estimates and an improved threshold. Subsequently, the energy estimates derived from the first subsystem are introduced as external forces into the second subsystem; this external force dictates the threshold size of the perturbation system. In addressing zero modes within the entire nonlinear system, we note that non-zero frequencies exhibit enhanced dissipation and inviscid damping effects, making their interactions relatively manageable. However, when considering interactions among zero modes, we aim to investigate additional dispersion mechanisms.

Specifically, by applying the Duhamel principle, we derive low-regularity \(L^\infty\) dispersion estimates for the dispersion variables \(U_{0,1}^2\), \(\tilde{U}_{0,1}^3\), \(V_{0,1}^2\), and \(\tilde{V}_{0,1}^3\) pertaining to the first subsystem. Utilizing Lemma [lowDisper], these dispersion variables achieve higher regularity within Sobolev spaces. Through the incorporation of dispersion-dispersion type terms, we obtain nonlinear dispersion estimates for these variables, thereby enhancing our energy estimate results. Under condition ?? , we leverage the dispersion-dispersion type terms associated with \(\overline{U_{0,1}^3}\) and \(\overline{\Theta_{0,1}}\) to refine our energy estimates further. Additionally, employing precise evaluations of the "dispersion-thermal type terms" related to \(\overline{U_{0,1}^1}\) and \(\Lambda_{0,1}\) allows us to improve our energy estimates even more significantly—ultimately closing off the energy estimates for this subsystem.

The nonlinear terms of the second subsystem can be classified into four types of interactions: external force-external force, external force-disturbance, disturbance-external force, and disturbance-disturbance. The lower Sobolev regularity propagating on the system enables us to handle these structures by Lemma 17 as follows:
(1) Contribution of two external force terms, since \(s+4\leq N\), for \(G\in\{U,\Theta\}\) we have \[\langle U_{0,1}\cdot\nabla G_{0,1},G_{0,2}\rangle_{H^{s,s+\frac{1}{2}}}\lesssim (\|\langle\partial_z\rangle^{\frac{1}{2}}\tilde{U}_{0,1}^{2,3}\|_{W^{s,\infty}}+\|\overline{U_{0,1}^3}\|_{H^s})\|\nabla G_{0,1}\|_{H^{s}}^{\frac{1}{2}}F_{0,1}^{\frac{1}{2}}(t)\nu^{-\frac{1}{4}}F_{0,2}^{\frac{1}{2}}(t)\nu^{-\frac{1}{2}}.\] (2) Contribution of only one external force term, for \(G\in\{U,\Theta\}\), \[\begin{align} \langle U_{0,1}\cdot\nabla G_{0,2},\tilde{G}_{0,2}\rangle_{H^{s,s+\frac{1}{2}}}\lesssim& (\|\langle\partial_z\rangle^{\frac{1}{2}}\tilde{U}_{0,1}^{2,3}\|_{W^{s,\infty}}+\|\overline{U_{0,1}^3}\|_{H^s})F_{0,2}^{\frac{3}{4}}(t)\nu^{-\frac{3}{4}}\|\tilde{G}_{0,2}\|_{H^{s}}^{\frac{1}{2}},\\ \langle U_{0,2}\cdot\nabla G_{0,1},\tilde{G}_{0,2}\rangle_{H^{s,s+\frac{1}{2}}}\lesssim& \|\langle\partial_z\rangle^{\frac{1}{2}}\left(U_{0,2}\cdot\nabla G_{0,1}\right)\|_{H^{s-1}}F_{0,2}^{\frac{1}{2}}(t)\nu^{-\frac{1}{2}}\\ \lesssim& (\|\langle\partial_z\rangle^{\frac{1}{2}}\tilde{U}_{0,2}^{2,3}\|_{W^{s-3,\infty}}^{\frac{1}{2}}+\|\overline{U_{0,2}^3}\|_{H^{s-3}}^{\frac{1}{2}})\|\nabla U_{0,2}\|_{H^{s,s+\frac{1}{2}}}^{\frac{1}{2}}\|\nabla G_{0,1}\|_{H^{s+\frac{3}{2}}}F_{0,2}^{\frac{1}{2}}(t)\nu^{-\frac{1}{2}}\\ \lesssim& (\|\langle\partial_z\rangle^{\frac{1}{2}}\tilde{U}_{0,2}^{2,3}\|_{W^{s-3,\infty}}^{\frac{1}{2}}+\|\overline{U_{0,2}^3}\|_{H^{s-3}}^{\frac{1}{2}})F_{0,2}^{\frac{3}{4}}(t)\nu^{-\frac{3}{4}}F_{0,1}^{\frac{1}{4}}(t)\nu^{-\frac{1}{4}}E_{0,1}^{\frac{1}{4}}(t). \end{align}\] (3) No external force term case, for \(G\in\{U,\Theta\}\) \[\langle U_{0,2}\cdot\nabla G_{0,2},G_{0,2}\rangle_{H^{s,s+\frac{1}{2}}}\lesssim E_{0,2}^{\frac{1}{2}}(t)F_{0,2}(t)\nu^{-1}.\]

1.3.2.3 Difficulty 2: Regularity issue for non-zero modes

Recalling the three new variables \(Q_{\neq}^*\), \(K_{\neq}^*\) and \(H_{\neq}^*\) constructed in Appendix 5, they satisfy the linear equation 82 . In terms of nonlinear stability, these three variables can eliminate some linear coupling terms, but the equations they satisfy still contain linear stretching terms, which cause the variables to have a growth effect. To effectively control this growth, we introduce new variables \(Q_{\neq}\), \(K_{\neq}\) and \(H_{\neq}\) as in 23 as well as the cross-estimation of \(Q_{\neq}\) and \(K_{\neq}\), which can eliminate the growth caused by \(-(l^2 \partial_t p_h)/(2 p p_h)\{\widehat{Q_{\neq}}, \widehat{K_{\neq}}\}\). The remaining stretching term \((\partial_t p)/(4p)\{\widehat{Q_{\neq}}, \widehat{K_{\neq}}, \widehat{H_{\neq}}\}\) is handled by using the multiplier \(m\) as shown in 26 . Unfortunately, incorporating rotation and stratification results in coupling among the equation’s variables, leading to regularity issues within the vertical direction for the nonlinear system described by equation 64 . Specifically, we will deal with the analogues of the following nonlinear term: \[\begin{align} &\left|\int_0^t\int_{\mathbb{T}\times\mathbb{R}\times\mathbb{T}}\nabla^N \mathcal{A}|\nabla_L|^{\frac{3}{2}}|\nabla_{L,h}|^{-1}(U_{\neq}^2\partial_y^LU_{\neq}^3)\nabla^N (\mathcal{A}Q_{\neq}){\rm d}x{\rm d}y{\rm d}z{\rm d}\tau \right|\\ & \quad\lesssim \left|\int_0^t\int_{\mathbb{T}\times\mathbb{R}\times \mathbb{T}}\mathcal{A}|\nabla_L|^{\frac{3}{2}}|\nabla_{L,h}|^{-1}(U_{\neq}^2\nabla^N \partial_y^LU_{\neq}^3)\nabla^N (\mathcal{A}Q_{\neq}){\rm d}x{\rm d}y{\rm d}z{\rm d}\tau \right| \\&\qquad+ \left|\int_0^t\int_{ \mathbb{T}\times\mathbb{R}\times \mathbb{T}} \mathcal{A}|\nabla_L|^{\frac{3}{2}}|\nabla_{L,h}|^{-1}(\nabla^N U_{\neq}^2\partial_y^LU_{\neq}^3)\nabla^N (\mathcal{A}Q_{\neq}){\rm d}x{\rm d}y{\rm d}z{\rm d}\tau \right|. \end{align}\] The treatment of the first term does not bring about essential difficulties. Since \(U_{\neq}^2\) has a relatively low regularity, the inviscid damping effect can be fully exploited to handle it, thereby obtaining an improved threshold. However, for the treatment of the second term, \(U_{\neq}^2\) has high regularity, and the potential good factor \(|\nabla_{L,h}|^{-1}\) and the inviscid damping effect of \(U_{\neq}^2\) cannot be utilized. Specifically, \[\begin{align} &\left|\int_0^t\int_{\mathbb{T}\times\mathbb{R}\times\mathbb{T}} \mathcal{A}|\nabla_L|^{\frac{3}{2}}|\nabla_{L,h}|^{-1}(\nabla^N U_{\neq}^2\partial_y^LU_{\neq}^3)\nabla^N (\mathcal{A}Q_{\neq}){\rm d}x{\rm d}y{\rm d}z{\rm d}\tau \right| \\ &\quad \lesssim \|\mathcal{A}\nabla_LQ_{\neq}\|_{L^2_tH^N}(\|\mathcal{A}|\nabla_L|^{\frac{1}{2}}(Q_{\neq},K_{\neq})\|_{L^4_tH^N}^2+\|\mathcal{A}(Q_{\neq,K_{\neq}})\|_{L^\infty_tH^N} \|\mathcal{A}\nabla_LQ_{\neq}\|_{L^2_t H^N})\\ &\quad \lesssim E_{\neq}^{\frac{1}{2}}(t)F_{\neq}(t)\nu^{-1}\\ &\quad \lesssim \delta^3\nu^{3c-1}. \end{align}\] When the frequency in the vertical direction is much greater than that in the horizontal direction, that is, \(|l| \gg \sqrt{k^2 + (\eta - kt)^2}\), a rough estimation yields a threshold of 1. To address the difficulty about regularity issue, we employ the quasi-linearization method.

1.3.2.4 Strategy 2: Quasilinear method for non-zero modes.

We decompose the non-zero frequency solution into a main part and a perturbation part: \[Q_{\neq} = Q_{\neq,1} + Q_{\neq,2}, \quad K_{\neq} = K_{\neq,1} + K_{\neq,2}, \quad H_{\neq} = H_{\neq,1} + H_{\neq,2}.\]

The first subsystem (the main part): It carries all the non-zero frequency initial value information, carefully screens the nonlinear terms to avoid regularity issue, as shown in \(\eqref{Non0NL1}\). By using enhanced dissipation and inviscid damping effects, the corresponding energy estimates can be closed under the high regularity norms in Sobolev space.

The second subsystem (perturbation) is characterized by an initial value of zero and the presence of remaining nonlinear terms. By utilizing the enhanced energy estimates derived from the first subsystem as external force inputs to the second subsystem, this external force effectively determines the threshold size of the perturbation system. The nonlinear terms within the second subsystem can be categorized into four types of interactions: force-force, force-perturbation, perturbation-force, and perturbation-perturbation. This framework allows us to address these nonlinear terms individually under lower regularity norms in Sobolev space. With our designed quasi-linearization method, we are able to resolve these previously challenging terms as follows: \[\begin{align} &\left|\int_0^t\int_{\mathbb{T}\times\mathbb{R}\times\mathbb{T}} \mathcal{B}|\nabla_L|^{\frac{3}{2}}|\nabla_{L,h}|^{-1}(\nabla^s U_{\neq,1}^2\partial_y^LU_{\neq,1}^3)\nabla^s (\mathcal{B}Q_{\neq,2}){\rm d}x{\rm d}y{\rm d}z{\rm d}\tau \right| \\ &\quad \lesssim \|\mathcal{B}Q_{\neq,2}\|_{L^\infty_tH^s}\nu^{-\frac{1}{6}}\nu^{-\frac{1}{6}}\|\mathcal{A}(Q_{\neq,1},K_{\neq,1})\|_{L^2_tH^{s+4}}\|\mathcal{A}Q_{\neq,1}\|_{L^2_tH^{s+\frac{7}{2}}}\\ &\quad \lesssim \nu^{-\frac{2}{3}}E_{\neq,2}^{\frac{1}{2}}(t)\int_0^t F_{\neq,1}(\tau){\rm d}\tau\\&\quad\lesssim \delta^3\nu^{b+2c-\frac{2}{3}}. \end{align}\] Here, we employ the difference in high and low regularity \(N \geq 4 + s\). It suffices to have \(b + 2c - {2}/{3} \geq 2b\) to close the bootstrap argument. Clearly, \(c = {14}/{15}\) and \(b = 1\) meet the requirement.

The rest of the paper is organized as follows. In section 2 we give some preliminaries for later use. In section 3, we discuss the linear stability effects of solutions and give a rigorous mathematical proof of Theorem 1. Section 4 is devoted to the brief idea of proof of Theorem 3, including the bootstrap hypotheses and the weighted energy estimates of non-zero modes and zero modes. The proof details of nonlinear stability are given in sections 4.1–4.5.

2 sec:Preliminaries↩︎

2.1 Notation↩︎

Given two quantities \(A\) and \(B\), we denote \(A \lesssim B\), if there exists a constant \(C\) such that \(A\leqslant CB\) where \(C>0\) depends only on \(N\) and \(\alpha\), but not on \(\delta\), \(\nu\) and \(\beta\). We similarly denote \(A \ll B\) if \(A \leqslant cB\) for a small constant \(c \in(0, 1)\) to emphasize the small size of the implicit constant. \(\sqrt{1+t^2}\) is represented by \({\left\langle {t} \right\rangle}\) and \(\sqrt{k^2+\eta^2+l^2}\) is denoted by \(|k, \eta, l|\). For two functions \(f\) and \(g\) and a norm \(\|\cdot\|_{X}\), we write \[\|(f,g)\|_{X}=\sqrt{\|f\|_{X}^{2}+\|g\|_{X}^{2}}.\] Unless specified otherwise, in the rest of this section \(f\) and \(g\) denote functions from \(\mathbb{T} \times \mathbb{R} \times \mathbb{T}\) to \(\mathbb{R}^{n}\) for some \(n\in \mathbb{N}\).

2.2 Functional spaces↩︎

The Schwartz space \(\mathscr{S}(\mathbb{R}^{n})\) with \(n\) being the spatial dimension is the set of smooth functions \(f\) on \(\mathbb{R}^{n}\), we have \[\|f\|_{\mathscr{S}}\triangleq \sup_{x\in \mathbb{R}^{n} }\left(1+|x|\right)^{\alpha} |\partial^{\beta}f(x)|<\infty,\quad \forall \, \alpha,\beta\in \mathbb{Z}_{+}^{n}.\]

The Sobolev space \(H^{s} (s\geqslant0)\) is given by the norm \[\label{HSX-A} \left\| f \right\|_{H^{s}}=\left\| {\left\langle {D} \right\rangle}^{s} f \right\|_{L^2}=\left\| {\left\langle {k, \eta, l} \right\rangle}^{s} \widehat{f} \right\|_{L^2}.\tag{19}\] To better estimate the zero modulus below, here we define the space \[\label{HSX-B} \left\| f_{0} \right\|_{H_{y,z}^{s,s+\frac{1}{2}}}\triangleq \left\| \left\langle {\partial_{z}} \right\rangle^{\frac{1}{2}} f_{0} \right\|_{H_{y,z}^{s}}=\left\| {\left\langle {\eta, l} \right\rangle}^{s}\langle l\rangle^{\frac{1}{2}} \widehat{f_{0}} \right\|_{L^2}.\tag{20}\]

Recall that, for \(s>\frac{3}{2}\), \(H^{s}\) is an algebra. Hence, for any \(f,g\in H^{s}\), one has \[\begin{align} \label{2463} \left\|f g\right\|_{H^{s}} \lesssim \left\|f \right\|_{H^{s}}\left\|g\right\|_{H^{s}}. \end{align}\tag{21}\] We write the associated inner product as \[\langle f,g \rangle_{H^{s}}=\int \langle\nabla\rangle^{s}f\cdot \langle \nabla \rangle^{s} g dV.\] For function \(f(t,x,y,z)\) of space and time defined on the time interval \((a,b)\), we define the Banach space \(L_{t}^{p}(a,b;H^{s})\) for \(1\leqslant p\leqslant \infty\) by the norm \[\|f\|_{L_{t}^{p}(a,b; H^{s})}=\|\|f\|_{H^{s}}\|_{L_{t}^{p}(a,b)}.\] For simplicity of notation, we usually simply write \(\|f\|_{L_{t}^{p}H^{s}}\) since the time-interval of integrating in this work will be the same basically everywhere.

In the field of harmonic analysis, the Van der Corput lemma is an estimate for oscillatory integrals. In this paper, we use this lemma to obtain the dispersive estimates on the zero frequency of velocity.

3 The linear stability↩︎

In this section, we elaborate on the linear stability results and corresponding proof of Bouessinesq equation with rotation near the Couette flow. Now we neglect the effect of nonlinear terms on the right-hand side of 5 , that is, we only need to consider the stability of the linearized equation \[\label{14666} \begin{cases} \partial_{t} u+y \partial_{x_1} u-\nu \Delta u+\begin{pmatrix} (1-\beta) u^2 \\ \beta u^1 \\ \alpha \theta \end{pmatrix}+ (\beta-2)\nabla \Delta^{-1}\partial_{x_1}u^{2}\\ \hskip5cm -\beta\nabla\Delta^{-1}\partial_{y}u^{1}-\alpha\nabla\Delta^{-1}\partial_{z}\theta =0, \\ \partial_{t} \theta+y \partial_{x_1} \theta-\nu \Delta \theta-\alpha u^3=0, \\ u(0, x_1, y, z)=u_{\mathrm{in}}, \quad \theta(0, x_1, y, z)=\theta_{\mathrm{in}}. \end{cases}\tag{22}\]

By the moving frame 10 , we recall the definition of \(U(t,x,y,z)\) and \(\Theta(t,x,y,z)\), and denote \(W^3\triangleq\partial_xU^2-\partial_y^LU^1\). Here, we introduce the new unknowns \[\label{yyy} Q_{\neq}:=-|\nabla_L|^{\frac{3}{2}}|\nabla_{L,h}|^{-1}U^3_{\neq},\quad K_{\neq}:=-i\sqrt{\frac{\beta}{\beta-1}}|\nabla_L|^{\frac{1}{2}}|\nabla_{L,h}|^{-1}W^3_{\neq}, \quad H_{\neq}:=-|\nabla_L|^{\frac{1}{2}}\Theta_{\neq}.\tag{23}\] One can represent the linearized system of 22 as \[\begin{align} \label{Non0NL-1} \begin{cases} \partial_t Q_{\neq} -\nu\Delta_L Q_{\neq} - \frac{1}{2} \partial_x\partial_y^L|\nabla_L|^{-2} Q_{\neq} - \partial_x\partial_y^L(|\nabla_L|^{-2}-|\nabla_{L,h}|^{-2}) Q_{\neq} + i\sqrt{B_{\beta}}\partial_z|\nabla_L|^{-1} K_{\neq} \\ \qquad + 2i\sqrt{\frac{\beta-1}{\beta}}\partial_z\partial_{xx}|\nabla_L|^{-1}|\nabla_{L,h}|^{-2}K_{\neq} + \alpha|\nabla_{L,h}||\nabla_L|^{-1}H_{\neq} = 0, \\ \partial_t K_{\neq} - \nu\Delta_L K_{\neq} + \frac{1}{2} \partial_x\partial_y^L|\nabla_L|^{-2} K_{\neq} + \partial_x\partial_y^L(|\nabla_L|^{-2}-|\nabla_{L,h}|^{-2}) K_{\neq}\\ \qquad - i\sqrt{B_{\beta}}\partial_z|\nabla_L|^{-1}Q_{\neq} = 0,\\ \partial_t H_{\neq} -\nu\Delta_L H_{\neq} - \frac{1}{2} \partial_x\partial_y^L|\nabla_L|^{-2} H_{\neq} - \alpha|\nabla_{L,h}||\nabla_L|^{-1}Q_{\neq} = 0. \end{cases} \end{align}\tag{24}\] Thus, by taking the Fourier transform of the spatial variable, when \(k\neq 0\), 24 can be written \[\begin{align} \label{Non0L} \begin{cases} \partial_t \widehat{Q_{\neq}} + \nu p \widehat{Q_{\neq}} - \frac{1}{4} \frac{\partial_t p}{p}\widehat{Q_{\neq}} - \frac{1}{2} \frac{l^2}{p}\frac{\partial_t p_h}{p_h} \widehat{Q_{\neq}} \\ \qquad\;\;- \sqrt{B_{\beta}}\frac{l}{p^{1/2}}\widehat{K_{\neq}} + \alpha \frac{p_h^{1/2}}{p^{1/2}}\widehat{H_{\neq}} + 2\sqrt{\frac{\beta-1}{\beta}}\frac{k^2}{p_h}\frac{l}{p^{1/2}}\widehat{K_{\neq}} = 0, \\ \partial_t\widehat{K_{\neq}} + \nu p\widehat{K_{\neq}} + \frac{1}{4}\frac{\partial_t p}{p}\widehat{K_{\neq}} +\frac{1}{2} \frac{l^2}{p}\frac{\partial_t p_h}{p_h} \widehat{K_{\neq}}+ \sqrt{B_{\beta}}\frac{l}{p^{1/2}}\widehat{Q_{\neq}} = 0,\\ \partial_t\widehat{H_{\neq}} + \nu p\widehat{H_{\neq}} - \frac{1}{4}\frac{\partial_t p}{p} \widehat{H_{\neq}} - \alpha \frac{p_h^{1/2}}{p^{1/2}}\widehat{Q_{\neq}} = 0. \end{cases} \end{align}\tag{25}\]

3.1 Dynamics of nonzero modes↩︎

As observed in [10], to capture the associated effects of inviscid damping and enhanced dissipation, it is convenient to use a suitably constructed Fourier multiplier \(m\) with symbol \(m=m(t,k,\eta,l)\), defined by \[\begin{align} \label{446464646464} -\frac{\dot{m}}{m}=&\left\{\begin{array}{ll} \frac{1}{2}\frac{| k \left( \eta- kt \right)|}{k^2+\left(\eta- kt\right)^2+l^2}, & \text{ if } \;|t-\frac{\eta}{k}|\leq 1000\nu^{-\frac{1}{3}}, \\ 0, & \text{ if } \;|t-\frac{\eta}{k}|> 1000\nu^{-\frac{1}{3}}, \end{array}\right. \end{align}\tag{26}\] with the initial data \[\begin{align} m(t=0,k,\eta,l)=&\begin{cases} 1, & \text{ if } \; \frac{\eta}{k}\leq 0,\\ \left(\frac{k^2+\eta^2+l^2}{k^2+l^2}\right)^{\frac{1}{4}}, & \text{ if } \;0<\frac{\eta}{k}<1000\nu^{-\frac{1}{3}},\\ \left(\frac{k^2+(1000\nu^{-\frac{1}{3}}k)^2+l^2}{k^2+l^2}\right)^{\frac{1}{4}},& \text{ if } \;1000\nu^{-\frac{1}{3}}\leq \frac{\eta}{k}. \end{cases} \end{align}\] The central role of \(m\) is to deal with the slowly decaying term \(\partial_{x}\partial_{y}^{L}|\nabla_{L}|^{-2}\{Q,K,H\}\) in 24 , balancing the growth that solutions experience in the transition between the inviscid damping regime \(t<\frac{\eta}{k}\), and the dissipative regime \(t\gtrsim \frac{\eta}{k}+\nu^{-\frac{1}{3}}\). Define the additional multipliers \(M_{i}(t, k, \eta, l), i=1,2,...,7\), by \(M_{i}(t=0, k, \eta, l)=1\) and

• if \(k=0\), \(M_{i}(t, k=0, \eta, l)=1\) for all \(t\).

• if \(k \neq 0\), \[\begin{align} \label{4465} \frac{\dot{M_{1}}}{M_{1}}&= \frac{- \nu^{\frac{1}{3}} }{\left[\nu^{\frac{1}{3}} \big( t-\frac{\eta}{k} \big)\right]^{2}+1}. \end{align}\tag{27}\]

• if \(k \neq 0\), \[\begin{align} -\frac{\dot{M}_2}{M_2}&=\frac{1}{4\sqrt{B_{\beta}}}\left|\frac{\partial_t p}{p}\frac{l}{\sqrt{p}}\frac{\partial_t p_h}{p_h}\right|, \tag{28}\\ -\frac{\dot{M}_3}{M_3}&=\left|\partial_t\left(-\frac{1}{2\sqrt{B_{\beta}}}\frac{l}{\sqrt{p}}\frac{\partial_t p_h}{p_h}\right)\right|, \tag{29} \\ -\frac{\dot{M}_4}{M_4}&=\left|\frac{2}{\beta}\frac{l^2}{p}\frac{k^2}{p_h}\frac{\partial_t p_h}{p_h}\right|, \tag{30} \\ -\frac{\dot{M}_5}{M_5}&=2\sqrt{\frac{\beta-1}{\beta}}\frac{k^2}{p_h}\left|\frac{l}{\sqrt{p}}\right|,\tag{31}\\ -\frac{\dot{M}_6}{M_6}&=\frac{\alpha}{2\sqrt{B_{\beta}}}\left|\frac{l}{p}\frac{\partial_tp_h}{\sqrt{p_h}}\right|, \tag{32} \\ -\frac{\dot{M}_7}{M_7}&=\frac{1}{(k^2+|\eta-kt|^2)^{\frac{1+\kappa}{2}}},\tag{33} \end{align}\]

for any \(\kappa>0\). We then define \[\label{666} M=\prod_{j=1}^{7} M_j.\tag{34}\]

For the expressions and properties of the multiplier \(m\), there is a similar situation as in [10], so we use it as known conclusion and ignore the proof process.

The following lemma embodies some properties of the ghost multipliers \(M_{j}(t, k, \eta, l)\).

Proof.The proof process of ?? and ?? is similar to that of [10], [49] and we omit it. Here we only prove ?? . According to 32 , we have \(\dot{M_6}/{M_6} \leqslant 0\) for any \(k\), \(l \in \mathbb{Z}\), \(\eta \in \mathbb{R}\). Hence, for any \(t \geqslant 0\), we obtain \(M_6(t, k, \eta, l) \leqslant 1\).

If \(k=0\), then \(M_6(t, k=0, \eta, l)=1\). If \(k \neq 0\), by 32 , we have \[\begin{align} -\ln M_6=&\frac{\alpha}{\sqrt{B_{\beta}}}\int_{0}^{t} \frac{|k||l||\eta-kt|}{(k^2+(\eta-kt)^2+l^2)\sqrt{k^2+(\eta-kt)^2}} d \tau \\ \leqslant& \frac{\alpha|l|}{\sqrt{B_{\beta}}}\int_{0}^{t} \frac{|k|}{k^2+(\eta-kt)^2+l^2} d \tau \\ \leqslant& \frac{\alpha|l|}{\sqrt{B_{\beta}}}\int_{-\infty}^{+\infty} \frac{1}{k^2+l^2+s^2} d s \\ \leqslant& \frac{\alpha \pi}{\sqrt{B_{\beta}}}, \end{align}\] where we have used the fact \(\int_{-\infty}^{+\infty} \frac{1}{k^2+l^2+s^2} d s=\frac{\pi}{|k, l|}\). Hence, \[0< \exp(-\frac{\alpha \pi}{\sqrt{B_{\beta}}}) \leqslant M_6 \leqslant 1.\] Hence, we finish the proof of Lemma 7\(\square\)

Using Lemma 7, we directly give the following inference.

The proof of Theorem 1 is directly obtained from the following Lemma 11 and Corollary 14 and the Lemma 27 in Appendix A. Here, We first give the linear enhanced dissipation and inviscid damping on \(U_{\neq}\) when \(B_\beta>{1}/{4}\).

Proof.To control the evolution of the nonzero modes, we introduce the Fourier multiplies \[\label{888} \mathcal{A}:=mMe^{\lambda \nu^{\frac{1}{3}}t},\qquad{(9)}\] which incorporates the multiplier \(m\) already used in the linear analysis and additional ghost weights \({M}\) as in 34 as well as the time weight \(e^{\lambda\nu^{\frac{1}{3}}t}\), which \(\lambda\) can be confirmed later. Recall the system 25 , energy estimates yield that \[\label{999} \begin{align} \frac{1}{2}&\frac{d}{dt}|\mathcal{A}(\widehat{Q_{\neq}},\widehat{K_{\neq}},\widehat{H_{\neq}})|^2 + \nu p |\mathcal{A}(\widehat{Q_{\neq}},\widehat{K_{\neq}},\widehat{H_{\neq}})|^2 +\left[(-\frac{\dot{M}}{M})+(-\frac{\dot{m}}{m})-\lambda\nu^{\frac{1}{3}}\right]|\mathcal{A}(\widehat{Q_{\neq}},\widehat{K_{\neq}},\widehat{H_{\neq}})|^2\\ &-\left(\frac{1}{2}\frac{\partial_t p_h}{p_h}-\frac{1}{4}\frac{\partial_t p}{p}\right)\left(|\widehat{\mathcal{A}Q_{\neq}}|^2-|\widehat{\mathcal{A}K_{\neq}}|^2\right)-\frac{1}{4}\frac{\partial_t p}{p}|\widehat{\mathcal{A}H_{\neq}}|^2+2\sqrt{\frac{\beta-1}{\beta}}\frac{k^2}{p_h}\frac{l}{\sqrt{p}}\widehat{\mathcal{A}K_{\neq}}\overline{\widehat{\mathcal{A}Q_{\neq}}}=0. \end{align}\qquad{(10)}\] Consider the cross terms, we introduce operator \(G\) whose fourier symbol satisfies \[\widehat{G}:=-\frac{1}{2\sqrt{B_{\beta}}}\frac{l}{\sqrt{p}}\frac{\partial_t p_h}{p_h},\] thus one has \[\begin{align} \label{sss} &\partial_t\left(\widehat{G}\widehat{\mathcal{A}Q_{\neq}}\overline{\widehat{\mathcal{A}K_{\neq}}}\right) \nonumber\\ &\quad=(\partial_t \widehat{G})\widehat{\mathcal{A}Q_{\neq}}\overline{\widehat{\mathcal{A}K_{\neq}}} + \left[(\frac{\dot{M}}{M})+(\frac{\dot{m}}{m})+\lambda\nu^{\frac{1}{3}}\right]2\widehat{G}\widehat{\mathcal{A}Q_{\neq}}\overline{\widehat{\mathcal{A}K_{\neq}}} \nonumber\\ &\qquad -\widehat{G}\left[\nu p\widehat{\mathcal{A}Q_{\neq}}-\left(\frac{1}{2}\frac{\partial_t p_h}{p_h}-\frac{1}{4}\frac{\partial_t p}{p}\right)\widehat{\mathcal{A}Q_{\neq}}-\sqrt{B_{\beta}}\frac{l}{\sqrt{p}}\widehat{\mathcal{A}K_{\neq}}+\alpha\sqrt{\frac{p_h}{p}}\widehat{\mathcal{A}H_{\neq}}+2\sqrt{\frac{\beta-1}{\beta}}\frac{k^2}{p_h}\frac{l}{\sqrt{p}}\widehat{\mathcal{A}K_{\neq}}\right]\overline{\widehat{\mathcal{A}K_{\neq}}} \nonumber\\ &\qquad-\widehat{G}\widehat{\mathcal{A}Q_{\neq}}\left[\nu p\overline{\widehat{\mathcal{A}K_{\neq}}}+\left(\frac{1}{2}\frac{\partial_t p_h}{p_h}-\frac{1}{4}\frac{\partial_t p}{p}\right)\overline{\widehat{\mathcal{A}K_{\neq}}}+\sqrt{B_{\beta}}\frac{l}{\sqrt{p}}\overline{\widehat{\mathcal{A}Q_{\neq}}}\right]. \end{align}\qquad{(11)}\] This combined with ?? implies that \[\begin{align} \frac{1}{2}&\frac{d}{dt}\left(|\mathcal{A}(\widehat{Q_{\neq}},\widehat{K_{\neq}},\widehat{H_{\neq}})|^2+2\widehat{G}\widehat{\mathcal{A}Q_{\neq}}\overline{\widehat{\mathcal{A}K_{\neq}}}\right) +\left[ \nu p+\left(-\frac{\dot{M}}{M}\right)\right]\left(|\mathcal{A}(\widehat{Q_{\neq}},\widehat{K_{\neq}},\widehat{H_{\neq}})|^2+2\widehat{G}\widehat{\mathcal{A}Q_{\neq}}\overline{\widehat{\mathcal{A}K_{\neq}}}\right)\\ &=\lambda\nu^{\frac{1}{3}}\left(|\mathcal{A}(\widehat{Q_{\neq}},\widehat{K_{\neq}},\widehat{H_{\neq}})|^2+2\widehat{G}\widehat{\mathcal{A}Q_{\neq}}\overline{\widehat{\mathcal{A}K_{\neq}}}\right)\\ &\quad-\left[(-\frac{\dot{m}}{m})-\frac{1}{4}\frac{\partial_t p}{p}\right]|\widehat{\mathcal{A}Q_{\neq}}|^2-\left[(-\frac{\dot{m}}{m})+\frac{1}{4}\frac{\partial_t p}{p}\right]|\widehat{\mathcal{A}K_{\neq}}|^2-\left[(-\frac{\dot{m}}{m})-\frac{1}{4}\frac{\partial_t p}{p}\right]\|\widehat{\mathcal{A}H_{\neq}}|^2-(-\frac{\dot{m}}{m})2\widehat{G}\widehat{\mathcal{A}Q_{\neq}}\overline{\widehat{\mathcal{A}K_{\neq}}}\\ &\quad+(\partial_t \widehat{G})\widehat{\mathcal{A}Q_{\neq}}\overline{\widehat{\mathcal{A}K_{\neq}}}+\frac{1}{\beta}\frac{l^2}{p}\frac{k^2}{p_h}\frac{\partial_t p_h}{p_h}|\widehat{\mathcal{A}K_{\neq}}|^2-2\sqrt{\frac{\beta-1}{\beta}}\frac{k^2}{p_h}\frac{l}{\sqrt{p}}\widehat{\mathcal{A}K_{\neq}}\overline{\widehat{\mathcal{A}Q_{\neq}}}+\frac{\alpha}{2\sqrt{B_{\beta}}}\frac{l}{p}\frac{\partial_tp_h}{\sqrt{p_h}}\widehat{\mathcal{A}H_{\neq}}\overline{\widehat{\mathcal{A}K_{\neq}}}. \end{align}\]

Donate that \(e_{\neq}(t):=|\mathcal{A}(\widehat{Q_{\neq}},\widehat{K_{\neq}},\widehat{H_{\neq}})|^2+2\widehat{G}\widehat{\mathcal{A}Q_{\neq}}\overline{\widehat{\mathcal{A}K_{\neq}}}\), then using the expression of \(\hat{G}\) yields that \[\begin{align}\label{00} \left(1-\frac{1}{2\sqrt{B_{\beta}}}\right)|\mathcal{A}(\widehat{Q_{\neq}},\widehat{K_{\neq}},\widehat{H_{\neq}})|^2\leq e_{\neq}(t)\leq \left(1+\frac{1}{2\sqrt{B_{\beta}}}\right)|\mathcal{A}(\widehat{Q_{\neq}},\widehat{K_{\neq}},\widehat{H_{\neq}})|^2, \end{align}\tag{35}\] Notice that the definition of \(m\) as in 26 , we have \(-\frac{\dot{m}}{m}+\frac{\nu p}{2}-\frac{1}{4} \big|\frac{\partial_t p}{p}\big|\gtrsim 0\). A direct calculation yields \[\begin{align}\label{1010} &\frac{d}{dt}e_{\neq}(t) + \left[\nu p+(-\frac{\dot{M}}{M})\right]\left(1-\frac{1}{2\sqrt{B_{\beta}}}\right)|\mathcal{A}(\widehat{Q_{\neq}},\widehat{K_{\neq}},\widehat{H_{\neq}})|^2\\ &\quad\leq 2\lambda\nu^{\frac{1}{3}}\left(1+\frac{1}{2\sqrt{B_{\beta}}}\right)|\mathcal{A}(\widehat{Q_{\neq}},\widehat{K_{\neq}},\widehat{H_{\neq}})|^2-\left[(-\frac{\dot{M_2}}{M_2})+(-2\frac{\dot{m}}{m})\widehat{G}\right]2\widehat{\mathcal{A}Q_{\neq}}\overline{\widehat{\mathcal{A}K_{\neq}}}\\ &\qquad-\left[(-\frac{\dot{M_3}}{M_3})-\partial_t \widehat{G}\right]2\widehat{\mathcal{A}Q_{\neq}}\overline{\widehat{\mathcal{A}K_{\neq}}}-\left[(-\frac{\dot{M_4}}{M_4})-\frac{2}{\beta}\frac{l^2}{p}\frac{k^2}{p_h}\frac{\partial_t p_h}{p_h}\right]|\widehat{\mathcal{A}K_{\neq}}|^2\\ &\qquad-\left[(-\frac{\dot{M_5}}{M_5})+2\sqrt{\frac{\beta-1}{\beta}}\frac{k^2}{p_h}\frac{l}{\sqrt{p}}\right]2\widehat{\mathcal{A}K_{\neq}}\overline{\widehat{\mathcal{A}Q_{\neq}}}-\left[(-\frac{\dot{M_6}}{M_6})-\frac{\alpha}{2\sqrt{B_{\beta}}}\frac{l}{p}\frac{\partial_tp_h}{\sqrt{p_h}}\right]2\widehat{\mathcal{A}H_{\neq}}\overline{\widehat{\mathcal{A}K_{\neq}}}, \end{align}\tag{36}\] where we have used 35 and \(M=\prod_{j=1}^{7} M_j\) and \(M_j(t,k,\eta, l),\;1\leq j\leq 7\) are defined by 27 33 . In the following, we introduce two new notation \[\begin{align} E_{\neq}(t):=&\|\mathcal{A}(Q_{\neq},K_{\neq},H_{\neq})\|_{H^r}^2 +2\langle G\mathcal{A}Q_{\neq},\mathcal{A}K_{\neq}\rangle_{H^r},\\ F_{\neq}(t):=&\nu\|\mathcal{A}\nabla_L (Q_{\neq},K_{\neq},H_{\neq})\|_{H^r}^2 +2\nu\langle G\mathcal{A}\nabla_L Q_{\neq},\mathcal{A}\nabla_L K_{\neq}\rangle_{H^r}\\ &+\left\|\mathcal{A}\sqrt{-\frac{\dot{M}}{M}} \left(Q_{\neq},K_{\neq},H_{\neq}\right)\right\|_{H^r}^2 +2\left\langle G\mathcal{A}\sqrt{-\frac{\dot{M}}{M}} Q_{\neq},\mathcal{A}\sqrt{-\frac{\dot{M}}{M}} K_{\neq}\right\rangle_{H^r}, \end{align}\] for any \(r \geq 0\). Using 35 , \(E_{\neq}(t)\) and \(F_{\neq}(t)\) satisfy \[\begin{align} & \left(1-\frac{1}{2\sqrt{B_{\beta}}}\right)\|\mathcal{A}(Q_{\neq},K_{\neq},H_{\neq})\|_{H^r}^2\leq E_{\neq}(t)\leq \left(1+\frac{1}{2\sqrt{B_{\beta}}}\right)\|\mathcal{A}(Q_{\neq},K_{\neq},H_{\neq})\|_{H^r}^2,\\ & \left(1-\frac{1}{2\sqrt{B_{\beta}}}\right)\left(\nu\|\mathcal{A}\nabla_L(Q_{\neq},K_{\neq},H_{\neq})\|_{H^r}^2 + \left\|\mathcal{A}\sqrt{-\frac{\dot{M}}{M}}(Q_{\neq},K_{\neq},H_{\neq})\right\|_{H^r}^2\right)\\ &\quad \leq F_{\neq}(t)\leq \left(1+\frac{1}{2\sqrt{B_{\beta}}}\right)\left(\nu\|\mathcal{A}\nabla_L(Q_{\neq},K_{\neq},H_{\neq})\|_{H^r}^2 + \left\|\mathcal{A}\sqrt{-\frac{\dot{M}}{M}}(Q_{\neq},K_{\neq},H_{\neq})\right\|_{H^r}^2\right), \end{align}\] thus, recalling 36 and going back to the physical space, we obtain that \[\begin{align} &\frac{\rm d}{{\rm d}t}E_{\neq}(t) + \left(1-\frac{1}{2\sqrt{B_{\beta}}}\right)\nu\|\mathcal{A}\nabla_L(Q_{\neq},K_{\neq},H_{\neq})\|_{H^r}^2 + \left(1-\frac{1}{2\sqrt{B_{\beta}}}\right)\|\mathcal{A}\sqrt{-\frac{\dot{M}}{M}}(Q_{\neq},K_{\neq},H_{\neq})\|_{H^r}^2\\ &\quad\leq2\lambda\nu^{\frac{1}{3}}\left(1+\frac{1}{2\sqrt{B_{\beta}}}\right)\|\mathcal{A}(Q_{\neq},K_{\neq},H_{\neq})\|_{H^r}^2. \end{align}\] Taking \(\lambda=\frac{1}{16}\frac{2\sqrt{B_{\beta}}-1}{2\sqrt{B_{\beta}}+1}\) and integrating the above inequality in time yields that \[\begin{align} &\|\mathcal{A}(Q_{\neq},K_{\neq},H_{\neq})\|_{H^r}^2 + \nu\|\mathcal{A}\nabla_L(Q_{\neq},K_{\neq},H_{\neq})\|_{L^2_tH^r}^2 + \left\|\mathcal{A}\sqrt{-\frac{\dot{M}}{M}}(Q_{\neq},K_{\neq},H_{\neq})\right\|_{L^2_tH^r}^2\\ & \quad\leq \frac{2\sqrt{B_{\beta}}+1}{2\sqrt{B_{\beta}}-1}\|\mathcal{A}(Q_{\neq},K_{\neq},H_{\neq})(t=0)\|_{H^r}^2 +\frac{\nu^{\frac{1}{3}}}{8}\|\mathcal{A}(Q_{\neq},K_{\neq},H_{\neq})\|_{L^2_tH^r}^2. \end{align}\] Due to \(\mathcal{A}=e^{\lambda \nu^{\frac{1}{3}} t }mM\), ?? and ?? , we have \[\label{3x460} \begin{align} &\|m M(Q_{\neq},K_{\neq},H_{\neq})\|_{H^r}^2 + \frac{ \nu}{2}\|mM\nabla_L(Q_{\neq},K_{\neq},H_{\neq})\|_{L^2 H^r}^2 \\&\qquad + \frac{1}{2} \left\|\sqrt{-\dot{M} M}m(Q_{\neq},K_{\neq},H_{\neq})\right\|_{L^2 H^r}^2 \\ & \quad\leq \frac{2\sqrt{B_{\beta}}+1}{2\sqrt{B_{\beta}}-1} e^{-2\lambda \nu^{\frac{1}{3}} t} \| (Q_{\mathrm{in}},K_{\mathrm{in}},H_{\mathrm{in}})\|_{H^{r+{\frac{1}{2}}}}^2. \end{align}\tag{37}\]

Next, recalling the definition of \(Q\), \(K\) and \(H\) and combining with ?? , we have the following fact \[\begin{align} \label{3x461} |\widehat{U^3_{\neq}}|= \frac{p_h^{\frac{1}{2}}}{p^{\frac{3}{4}}}|\widehat{Q_{\neq}}| \leq \frac{1}{c_{\alpha}}\frac{p_h^{\frac{1}{2}}}{p^{\frac{1}{2}}(k^2+l^2)^{\frac{1}{4}}}|mM\widehat{Q_{\neq}}|\leq \frac{1}{c_{\alpha}(k^2+l^2)^{\frac{1}{4}}}|mM\widehat{Q_{\neq}}|, \end{align}\tag{38}\] and \[\begin{align} \label{3x462} |\widehat{\Theta_{\neq}}|=\frac{1}{p^{\frac{1}{4}}}|H_{\neq}|\leq \frac{1}{c_{\alpha}(k^2+l^2)^{\frac{1}{4}}}|mM\widehat{H_{\neq}}|. \end{align}\tag{39}\] Hence, together 37 with 38 39 gives \[\begin{align}\label{3x463} \left\|{U^3_{\neq}}\right\|_{L^2}^2\leq& \frac{1}{c_{\alpha}^{2}}\left\| mM{Q_{\neq}} \right\|_{L^2}^2 \\ \leq& \frac{1}{c_{\alpha}^{2}}\frac{2\sqrt{B_{\beta}}+1}{2\sqrt{B_{\beta}}-1} e^{-2\lambda \nu^{\frac{1}{3}} t} \| (Q_{\mathrm{in}},K_{\mathrm{in}},H_{\mathrm{in}})\|_{H^{\frac{1}{2}}}^2 \\ \leq& \frac{1}{c_{\alpha}^{2}}\frac{2\sqrt{B_{\beta}}+1}{2\sqrt{B_{\beta}}-1} e^{-2\lambda \nu^\frac{1}{3} t} \left(\| u^3_{\mathrm{in}}\|_{H^2}^2+{\frac{\beta}{\beta-1}} \left\| \partial_xu^2_{\mathrm{in}}-\partial_yu^1_{\mathrm{in}}\right\|_{H^1}^2+\| \theta_{\mathrm{in}}\|_{H^1}^2 \right), \end{align}\tag{40}\] similarly, we have \[\begin{align}\label{3x464} \left\|{\Theta_{\neq}}\right\|_{L^2}^2 \leq \frac{1}{c_{\alpha}^{2}}\frac{2\sqrt{B_{\beta}}+1}{2\sqrt{B_{\beta}}-1} e^{-2\lambda \nu^{\frac{1}{3}} t} \left(\| u^3_{\mathrm{in}}\|_{H^2}^2+{\frac{\beta}{\beta-1}} \left\| \partial_xu^2_{\mathrm{in}}-\partial_yu^1_{\mathrm{in}}\right\|_{H^1}^2+\| \theta_{\mathrm{in}}\|_{H^1}^2 \right). \end{align}\tag{41}\]

Using the incompressible condition \(\partial_{x} U^1+\partial_{y}^L U^2=-\partial_{z} U^3\) and the definition of \(W^3\), one gets \[\label{34620} \begin{cases} \left(\partial_{x x}+\partial_{y y}^L\right) U^{1}=-\partial_{y}^{L} W^{3}-\partial_{x z} U^{3}, \\ \left(\partial_{x x}+\partial_{y y}^L\right) U^{2}=\partial_{x} W^{3}-\partial_{y z}^{L} U^{3}, \end{cases}\tag{42}\] By 42 and \(\eqref{44611-2}\), we get \(\widehat{U_{\neq}^1}\) and \(\widehat{U_{\neq}^2}\) which satisfy \[\label{ll6} \begin{align} |\widehat{U_{\neq}^1}|\leq &\frac{1}{p_h}\left( \sqrt{\frac{\beta-1}{\beta}} \frac{|\eta-kt| p_h^{\frac{1}{2}}}{p^{\frac{1}{4}}} |\widehat{K_{\neq}}|+ \frac{|k||l| p_h^{\frac{1}{2}}}{p^{\frac{3}{4}}} |\widehat{Q_{\neq}}|\right)\\ \leq & \frac{1}{c_{\alpha}}\frac{1}{\left(k^2+l^2\right)^{\frac{1}{4}}} \left(\sqrt{\frac{\beta-1}{\beta}}|mM\widehat{K_{\neq}}| + |mM\widehat{Q_{\neq}}|\right), \end{align}\tag{43}\] and \[\label{ll7} \begin{align} |\widehat{U_{\neq}^2}|\leq &\frac{1}{p_h}\left( \sqrt{\frac{\beta-1}{\beta}} \frac{|k| p_h^{\frac{1}{2}}}{p^{\frac{1}{4}}} |\widehat{K_{\neq}}|+ \frac{|\eta-kt||l| p_h^{\frac{1}{2}}}{p^{\frac{3}{4}}} |\widehat{Q_{\neq}}|\right)\\ \leq & \frac{1}{c_{\alpha}}\left(\sqrt{\frac{\beta-1}{\beta}} \frac{|k|^{\frac{1}{2}}}{p_h^{\frac{1}{2}}}|mM\widehat{K_{\neq}}| +\frac{|l|^{\frac{1}{2}}}{p^{\frac{1}{2}}} |mM\widehat{Q_{\neq}}|\right). \end{align}\tag{44}\] Thus, this together 37 leads to \(L^2\)-norm of \(U_{\neq}^1(t)\) and \(U_{\neq}^2(t)\) \[\label{4x465} \begin{align} \left\|{U^1_{\neq}}\right\|_{L^2}^2\leq& \frac{1}{c_{\alpha}^{2}}\left({\frac{\beta-1}{\beta}}\left\|mM\widehat{K_{\neq}}\right\|_{L^2}^2 + \left\|mM\widehat{Q_{\neq}}\right\|_{L^2}^2\right) \\ \lesssim& \frac{1}{c_{\alpha}^{2}}\frac{2\sqrt{B_{\beta}}+1}{2\sqrt{B_{\beta}}-1} \max \left\{ {\frac{\beta-1}{\beta}}, 1 \right\} e^{-2\lambda \nu^{\frac{1}{3}} t} \| (Q_{\mathrm{in}},K_{\mathrm{in}},H_{\mathrm{in}})\|_{H^{\frac{1}{2}}}^2 \\ \lesssim& \frac{1}{c_{\alpha}^{2}}\frac{2\sqrt{B_{\beta}}+1}{2\sqrt{B_{\beta}}-1} \max \left\{ {\frac{\beta-1}{\beta}}, {\frac{\beta}{\beta-1}} \right\} e^{-2\lambda \nu^{\frac{1}{3}} t} \left(\| u_{\mathrm{in}}\|_{H^2}^2+\| \theta_{\mathrm{in}}\|_{H^1}^2 \right), \end{align}\tag{45}\] and \[\label{4x466} \begin{align} \left\|{U^2_{\neq}}\right\|_{L^2}^2\leq& {\left\langle {t} \right\rangle}^{-2} \frac{1}{c_{\alpha}^{2}}\left({\frac{\beta-1}{\beta}}\left\|mM\widehat{K_{\neq}}\right\|_{H^{\frac{5}{2}}}^2 + \left\|mM\widehat{Q_{\neq}}\right\|_{H^{\frac{5}{2}}}^2\right) \\ \lesssim& {\left\langle {t} \right\rangle}^{-2} \, \frac{1}{c_{\alpha}^{2}}\frac{2\sqrt{B_{\beta}}+1}{2\sqrt{B_{\beta}}-1} \max \left\{ {\frac{\beta-1}{\beta}}, 1 \right\} e^{-2\lambda \nu^{\frac{1}{3}} t} \| (Q_{\mathrm{in}},K_{\mathrm{in}},H_{\mathrm{in}})\|_{H^3}^2 \\ \lesssim& {\left\langle {t} \right\rangle}^{-2} \,\frac{1}{c_{\alpha}^{2}}\frac{2\sqrt{B_{\beta}}+1}{2\sqrt{B_{\beta}}-1} \max \left\{ {\frac{\beta-1}{\beta}}, {\frac{\beta}{\beta-1}} \right\} e^{-2\lambda \nu^{\frac{1}{3}} t} \left(\| u_{\mathrm{in}}\|_{H^{\frac{9}{2}}}^2+\| \theta_{\mathrm{in}}\|_{H^{\frac{7}{2}}}^2 \right), \end{align}\tag{46}\] where we have used the fact \(p_h^{-1} \lesssim {\left\langle {t} \right\rangle}^{-2} |k, \eta, l|^2\). Hence, together 40 , 41 , 45 with 46 gives ?? ?? and the proof of Lemma 10 is finished. It is worth noting that in Lemma 10 where we require \(B_{\beta} > 1/4\), but in fact, obtaining the enhanced dissipation of \(U_{\neq}\) and \(\Theta_{\neq}\) does not necessitate this restriction. We have provided a detailed explanation of this point in Appendix 5 later on. \(\square\)

3.2 Analysis of the zero-modes \((u_0, \theta_0)\)↩︎

This section is devoted to the dynamics of the zero modes in the \(x\) frequency, which satisfy \[\label{146467} \begin{cases} \partial_{t} u_0^1-\nu \Delta_{y, z} u_0^1+ (1-\beta)u_0^2=0, \\ \partial_{t} u_0^2-\nu \Delta_{y, z} u_0^2+ \beta \partial_{z} \Delta_{y, z}^{-1} \partial_{z} u_0^1-\alpha \partial_{y} \Delta_{y, z}^{-1} \partial_{z} \theta_0=0, \\ \partial_{t} u_0^3-\nu \Delta_{y, z} u_0^3-\beta \partial_{z} \Delta_{y, z}^{-1} \partial_{y} u_0^1+\alpha \partial_{y} \Delta_{y, z}^{-1} \partial_{y} \theta_{0}=0, \\ \partial_{t} \theta_0-\nu \Delta_{y, z} \theta_0- \alpha u_0^3=0, \\ u(t=0)=u_{\mathrm{in}}, \quad \theta(t=0)=\theta_{\mathrm{in}}. \end{cases}\tag{47}\] The zero modes can be decomposed into simple-zero modes and double-zero modes, that is, \(u_0(t, y, z)=\overline{u}_0(t, y)+\widetilde{u}_{0}(t, y, z)\) and \(\theta_0(t, y, z)=\overline{\theta}_0(t, y)+\widetilde{\theta}_{0}(t, y, z)\). For the linearized system 47 , we express the solution in an explicit form via Fourier transform and Duhamel’s principle. One gives the following lemma.

Proof.First, we give the proof of \((1)\). When \(l=0\), \(\overline{u}_0^1\) satisfies the standard heat equation which gives the estimate ?? \(_{1}\). The coupling of \(\overline{u}_0^3\) and \(\overline{\theta}_0\) in a zero-order manner indicates that this does not directly lead to energy dissipation or enhancement. Due to the symmetry of the coupling relationship, the Fourier transform is obtained \[\begin{cases} \partial_{t} \widehat{\overline{u}_0^3}+\nu \eta^2 \widehat{\overline{u}_0^3}+\alpha \widehat{\overline{\theta}_0}=0, \\ \partial_{t} \widehat{\overline{\theta}_0}+\nu \eta^2 \widehat{\overline{\theta}_0}- \alpha \widehat{\overline{u}_0^3}=0,\\ \widehat{\overline{u^3_0}}(t=0)=\widehat{\overline{({u}^3_{\mathrm{in}})_0}}, \widehat{\overline{\theta_0}}(t=0)=\widehat{\overline{({\theta}_{\mathrm{in}})_0}}. \end{cases}\] Let \(\phi=\widehat{\overline{u}_0^3}+i \widehat{\overline{\theta}_0}\) and \(\psi=\widehat{\overline{u}_0^3}-i \widehat{\overline{\theta}_0}\) with the initial data \(\phi_{\mathrm{in}}=\widehat{\overline{({u}^3_{\mathrm{in}})_0}}+i \widehat{\overline{({\theta}_{\mathrm{in}})_0}}\) and \(\psi_{\mathrm{in}}=\widehat{\overline{({u}^3_{\mathrm{in}})_0}}-i \widehat{\overline{({\theta}_{\mathrm{in}})_0}}\), then we have \[\partial_{t} \phi=(-\nu \eta^2+i \alpha) \phi, \quad \partial_{t} \psi=(-\nu \eta^2-i \alpha) \psi.\] Thus, a direct calculation yields \[\phi(t, \eta)=e^{-\nu \eta^2 t+ i \alpha t}\phi_{\mathrm{in}}, \quad \psi(t, \eta)=e^{-\nu \eta^2 t- i \alpha t}\psi_{\mathrm{in}}.\] Noticing that \(\widehat{\overline{u}_0^3}=\frac{\phi+\psi}{2}\) and \(\widehat{\overline{\theta}_0}=\frac{\phi-\psi}{2i}\), we immediately have \[\begin{align}\label{546x} \widehat{\overline{u}_0^3}(t, \eta)=&\frac{1}{2}e^{-\nu \eta^2 t}\left[ e^{i \alpha t}\left(\widehat{\overline{({u}^3_{\mathrm{in}})_0}}+i \widehat{\overline{({\theta}_{\mathrm{in}})_0}}\right)+e^{- i \alpha t}\left(\widehat{\overline{({u}^3_{\mathrm{in}})_0}}-i \widehat{\overline{({\theta}_{\mathrm{in}})_0}}\right)\right]\\ =&e^{-\nu \eta^2 t} \left[ \cos(\alpha t) \widehat{\overline{({u}^3_{\mathrm{in}})_0}}-\sin(\alpha t) \widehat{\overline{({\theta}_{\mathrm{in}})_0}}\right]. \end{align}\qquad{(13)}\] Similarly, it holds \[\label{646x} \widehat{\overline{\theta}_0}(t, \eta)=e^{-\nu \eta^2 t} \left[ \sin(\alpha t) \widehat{\overline{({u}^3_{\mathrm{in}})_0}}+\cos(\alpha t) \widehat{\overline{({\theta}_{\mathrm{in}})_0}}\right].\qquad{(14)}\] ?? and ?? implies ?? and ?? .

Next, we prove the case when \(l\neq 0\). By using the incompressible condition, \(\widetilde{u}_{0}^{3}\) can be expressed as \(-\partial_{z}^{-1} \partial_{y} \widetilde{u}_{0}^{2}(t, y, z)\). To simplify the calculation, taking the Fourier transform to the \(\widetilde{u}_{0}^{1}\), \(\widetilde{u}_{0}^{2}\) and \(\widetilde{\theta}_{0}\) equations in 47 , it yields \[\partial_{t} \begin{bmatrix} \widehat{\widetilde{u}_{0}^{1}} \\ \widehat{\widetilde{u}_{0}^{2}} \\ \widehat{\widetilde{\theta}_{0}} \end{bmatrix}=\begin{bmatrix} -\nu(\eta^2+l^2) & \beta-1 & 0 \\ -\beta \frac{l^2}{\eta^2+l^2} & -\nu(\eta^2+l^2) & \alpha \frac{\eta l}{\eta^2+l^2} \\ 0 & -\alpha \frac{\eta}{l} & -\nu(\eta^2+l^2) \end{bmatrix}\begin{bmatrix} \widehat{\widetilde{u}_{0}^{1}} \\ \widehat{\widetilde{u}_{0}^{2}} \\ \widehat{\widetilde{\theta}_{0}} \end{bmatrix}=:\mathcal{A}\begin{bmatrix} \widehat{\widetilde{u}_{0}^{1}} \\ \widehat{\widetilde{u}_{0}^{2}} \\ \widehat{\widetilde{\theta}_{0}} \end{bmatrix}.\] Then, the single zero frequency satisfies the following expression \[\begin{align} \label{746xx} \begin{bmatrix} \widetilde{u}_{0}^{j}(t, \eta, l) \\ \widehat{\widetilde{\theta}_{0}}(t, \eta, l) \end{bmatrix}=e^{\mathcal{A}t}\begin{bmatrix} \widehat{\widetilde{({u^j_{\mathrm{in}}})_0}}\\ \widehat{\widetilde{({\theta_{\mathrm{in}}})_0}} \end{bmatrix}, \quad \mathrm{for} \quad j=1,2. \end{align}\tag{48}\]

We compute the fundamental matrix \(e^{\mathcal{A} t}\) explicitly. The characteristic polynomial associated with \(\mathcal{A}\) is given by \[\left(\lambda+ \nu \left( \eta^2+l^2 \right)\right)\left[ \left(\lambda+ \nu \left( \eta^2+l^2 \right)\right)^2+\alpha^2 \frac{\eta^2}{\eta^2+l^2}+\beta \left( \beta-1\right) \frac{l^2}{\eta^2+l^2}\right]=0,\] and thus the spectra of \(\mathcal{A}\) as follows \[\lambda_1=-\nu |\eta, l|^2, \quad \lambda_2=-\nu |\eta, l|^2+i \frac{h}{|\eta, l|}, \quad \quad \lambda_3=-\nu |\eta, l|^2-i \frac{h}{|\eta, l|},\] where \(h=h(\alpha, \beta, \eta, l)=\sqrt{\alpha^2 \eta^2+ \beta (\beta-1) l^2}\) with \(B_{\beta}>0\). Note that the eigenvalues \(\lambda_2\) and \(\lambda_3\) are a pair of conjugate complex roots. The eigenvectors \(\mathcal{V}_{i}\) \((i=1, 2, 3)\) associated with \(\lambda_{i}\) satisfy \[\begin{align} \left( \lambda_{i} \boldsymbol{I} - \mathcal{A} \right) \mathcal{V}_{i}=\begin{bmatrix} \lambda_{i}+\nu(\eta^2+l^2) & 1-\beta & 0 \\ \beta \frac{l^2}{\eta^2+l^2} & \lambda_{i}+\nu(\eta^2+l^2) & -\alpha \frac{\eta l}{\eta^2+l^2} \\ 0 & \alpha \frac{\eta}{l} & \lambda_{i}+\nu(\eta^2+l^2) \end{bmatrix} \begin{bmatrix} \mathcal{V}_{i 1}\\ \mathcal{V}_{i 2}\\ \mathcal{V}_{i 3} \end{bmatrix}=0, \end{align}\] which gives \[\mathcal{V}_{1}=\begin{bmatrix} \alpha \frac{\eta}{l} \\ 0 \\ \beta \end{bmatrix}, \quad \quad \mathcal{V}_{2}=\begin{bmatrix} i(1-\beta) \\ \frac{h}{|\eta, l|}\\ i \alpha \frac{\eta}{l} \end{bmatrix}, \quad \quad \mathcal{V}_{3}=\begin{bmatrix} i(\beta-1) \\ \frac{h}{|\eta, l|}\\ -i \alpha \frac{\eta}{l} \end{bmatrix} .\] Let \[\label{3462469} \mathcal{P}:=[\mathcal{V}_{1}, \mathcal{V}_{2}, \mathcal{V}_{3}]=\begin{bmatrix} \alpha \frac{\eta}{l} & \quad i(1-\beta) & \quad i(\beta-1) \\ 0 & \quad \frac{h}{|\eta, l|} & \quad \frac{h}{|\eta, l|}\\ \beta & \quad i \alpha \frac{\eta}{l} & \quad -i \alpha \frac{\eta}{l} \end{bmatrix},\quad \mathcal{J}:=\begin{bmatrix} e^{\lambda_1 t} & & \\ & e^{\lambda_2 t} & \\ & & e^{\lambda_3 t} \end{bmatrix},\tag{49}\] then we can calculate the adjoint matrix \(\mathcal{P^*}\) and determinant \(\det(\mathcal{P})\) of \(\mathcal{P}\), respectively, \[\mathcal{P^*}=\begin{bmatrix} -2 i\alpha \frac{h^2}{l (\eta^2+l^2)} & \quad 0 & \quad 2 i(1-\beta) \frac{h}{|\eta, l|} \\ \beta \frac{h}{|\eta, l|} & \quad -i \alpha^2 \frac{\eta^2}{l^2}-iB_{\beta} & \quad -\alpha \frac{\eta h}{l |\eta, l|} \\ -\beta \frac{h}{|\eta, l|} & \quad -i \alpha^2 \frac{\eta^2}{l^2}-iB_{\beta} & \quad \alpha \frac{\eta h}{l |\eta, l|} \end{bmatrix},\] and \(\label{34624611} det(\mathcal{P})=-2i \frac{h^3}{l^2 |\eta, l|}.\) Hence, we obtain that \(e^{\mathcal{A} t}=\frac{1}{\det(\mathcal{P})} \mathcal{P} \mathcal{J} \mathcal{P^*}\), which combined with 48 gives ?? –?? . \(\square\)

Contrast to the classical Navier-Stokes equations, by means of ?? and ?? we find that although the combined effect of rotation and stratification brings about fluid oscillation, it cancels the lift-up effect. In other words, in the zero frequency part, the oscillation effect caused by the rotation and stratification is significantly stronger than the lift-up effect caused by the Couette flow.

3.3 Linear dispersive estimates↩︎

Recall the analysis of the zero frequency in section 3.2, we know that the simple-zero modes of \(u^{2}\) and \(u^{3}\) exhibit a dispersion mechanism from the expression. However, the simple-zero modes of \(u^{1}\) and \(\theta\) has no dispersion attenuation estimation, which brings difficulties to our research on the nonlinear stability threshold problem. Fortunately we introduce some new variables, which brought about a good structure. \[\begin{align}\label{000} V_0^2=&-\beta\mathcal{R}^{-1}\partial_{zz}\Delta_{L,0}^{-1}\tilde{U}_0^1 + \alpha\mathcal{R}^{-1}\partial_{yz}\Delta_{L,0}^{-1}\tilde{\Theta}_0=:G_1\tilde{U}_0^1 + G_2\tilde{\Theta}_0,\\ {V}_0^3=&-\partial_z^{-1}\partial_yV_0^2=\beta\mathcal{R}^{-1}\partial_{yz}\Delta_{L,0}^{-1}\tilde{U}_0^1 - \alpha\mathcal{R}^{-1}\partial_{yy}\Delta_{L,0}^{-1}\tilde{\Theta}_0=:G_1'\tilde{U}_0^1 + G_2'\tilde{\Theta}_0,\\ \Lambda_0=& \alpha\sqrt{\frac{\beta}{\beta-1}}\partial_{yz}\Delta_{L,0}^{-1} \tilde{U}_0^1 + \sqrt{B_{\beta}}\partial_{zz}\Delta_{L,0}^{-1}\tilde{\Theta}_0=:G_3\tilde{U}_0^1 + G_4\tilde{\Theta}_0, \end{align}\tag{50}\] where \(l\neq 0\), \(\Delta_{L,0}:=\partial_y^2+\partial_z^2\) and its symbol \(p_0:=\eta^2 + l^2\) and \[\label{ss11} \mathcal{R}:=i\left(B_{\beta}\partial_{zz}\Delta_{L,0}^{-1}+\alpha^2\partial_{yy}\Delta_{L,0}^{-1}\right)^{1/2}.\tag{51}\] Then using the linearized system 47 , we give \[\begin{align} \label{S0L} \begin{cases} \partial_t\widehat{V_0^j}+\nu p_0\widehat{V_0^j} + \sqrt{B_{\beta}l^2p_0^{-1}+\alpha^2\eta^2p_0^{-1}}\widehat{U_0^j} =0,\\ \partial_t\widehat{U_0^j}+\nu p_0\widehat{U_0^j} - \sqrt{B_{\beta}l^2p_0^{-1}+\alpha^2\eta^2p_0^{-1}}\widehat{V_0^j} =0,\\ \partial_t\widehat{\Lambda_0}+\nu p_0\widehat{\Lambda_0} =0, \end{cases} \end{align}\tag{52}\] or equivalently \[\begin{align} \label{sol1} \begin{cases} \partial_t(U_0^j+iV_0^j) = \mathcal{L}(U_0^j+iV_0^j),\,\, \mathcal{L}:=\nu\Delta_{L,0}-\mathcal{R},\\ (\partial_t - \nu\Delta_{L,0})\Lambda_0=0, \end{cases} \end{align}\tag{53}\] for \(j\in\{2,3\}\).

Recalling the definition of operator \(\mathcal{R}\) as in 51 is of dispersive nature. To prove this, we employ the Fourier transform.

For \(f\in\mathbb{R}\times\mathbb{T}\to\mathbb{C}\), we have \[\mathcal{R}f(y,z)=\sum_{l\neq 0}e^{ilz}\mathcal{R}^lf_l(y),\] with \(f_l(y):=\frac{1}{2\pi}\int_{\mathbb{T}}e^{-ilz}f(y,z){\rm d}z\), where for \(h\in\mathscr{S}(\mathbb{R})\), we set \[\mathcal{R}^l h(y,z):=\mathcal{F}_{\eta,l}^{-1}\left(i\sqrt\frac{B_{\beta}l^2+\alpha^2\eta^2}{l^2+\eta^2}\hat{h}(l,\eta)\right)(y,z).\]

Then we can calculate the extra decay from the semigroup generated by \(\mathcal{R}^l\) as follows.

Proof.Recall the Littlewood-Paley decomposition, there exists a smooth function \(\phi:\mathbb{R}\to[0,1]\) with \(\operatorname{supp}\phi\subset[-2,2]\) and \(\phi\big|_{[-\frac{3}{2} ,\frac{3}{2}]}=1\), such that \[h=\sum_{j\in\mathbb{Z}} P_j h,\;\mathcal{F}(P_j h)(\eta)=\varphi_j(\eta)\hat{h}(\eta),\;\varphi_j(\eta):=\varphi(2^{-j}\eta),\;\varphi(\eta) = \phi(\eta)-\phi(2\eta).\] By Young’s convolution inequality, we have \[\begin{align}\label{ll1} \left\|e^{-t\mathcal{R}^l}h\right\|_{L^{\infty}} \leq& \sum_{j\in\mathbb{Z}}\|e^{-t\mathcal{R}^l}P_j h\|_{L^{\infty}} \\ \leq& \sum_{j\in\mathbb{Z}}\left\|\mathcal{F}^{-1}\left(e^{-it\sqrt{\frac{\beta(\beta-1)l^2+\alpha^2\eta^2}{l^2+\eta^2}}}\varphi(2^{-j}\eta)\right)*\mathcal{F}^{-1}\left(\varphi^{\dagger}(2^{-j}\eta)\hat{h}(\eta)\right)\right\|_{L^{\infty}}\\ \leq& \sum_{j\in\mathbb{Z}}\left\|e^{-t\mathcal{R}^l}\check{\varphi_j} \right\|_{L^\infty}\left\|P_j^{\dagger} h\right\|_{L^1}, \end{align}\qquad{(15)}\] where \(\varphi^{\dagger}\) has similar support properties as \(\varphi\) and \(\varphi=\varphi\varphi^{\dagger}\), and \(\;P_j^{\dagger}\) is the associated Littlewood-Paley projection. We next calculate carefully and use stationary phase method to handle the behaviour of semigroup: \[\label{ll2} \begin{align} e^{-t\mathcal{R}^l}\check{\varphi_j}(y)&=\int_\mathbb{R}e^{- it\sqrt{\frac{\beta(\beta-1)l^2 + \alpha^2\eta^2}{l^2+\eta^2}}}e^{iy\eta}\varphi(2^{-j}\eta){\rm d}\eta\\ &\stackrel{\eta=|l|\xi}{=}|l|\int_{\mathbb{R}} e^{iy|l|\xi-it\sqrt{\frac{\beta(\beta-1)l^2 + \alpha^2\eta^2}{l^2+\eta^2}}}\varphi(2^{-j}|l|\xi){\rm d}\xi\\ &=|l|\int_{\mathbb{R}}e^{it\Phi(\xi)}\varphi(2^{-j}|l|\xi){\rm d}\xi=:|l|I(t,\alpha,\beta,l,j), \end{align}\qquad{(16)}\] where the phase function \[\label{ll3} \Phi(\xi):=\frac{yl}{t}\xi-\sqrt{\frac{\beta(\beta-1)+\alpha^2\xi^2}{1+\xi^2}}.\qquad{(17)}\] Direct calculation shows that \[\label{l4} \Phi''(\xi)=[\beta(\beta-1)-\alpha^2]\frac{-3\alpha^2\xi^4-2\beta(\beta-1)\xi^2+\beta(\beta-1)}{(1+\xi^2)^{5/2}(\beta(\beta-1)+\alpha^2\xi^2)^{3/2}}.\qquad{(18)}\] When \(\beta(\beta-1)\neq\alpha^2,\) with \(\alpha>0\) and \(\beta\notin[0,1]\), then one has \[\Phi''(\xi)=0\Rightarrow \xi=\pm\xi_0,\;\xi_0=\sqrt{\frac{\sqrt{\mu^2+3\mu}-\mu}{3}}\leq\frac{1}{\sqrt{2}},\] here \(\mu=\frac{\beta(\beta-1)}{\alpha^2}\). We decompose \[\begin{align} I(t,\alpha,\beta,l,j)=& I_{-}(t,\alpha,\beta,l,j)+I_{+}(t,\alpha,\beta,l,j),\;I_{+}(t,\alpha,\beta,l,j):=\int_{\mathbb{R}^+}e^{it\Phi(\xi)}\varphi(2^{-j}|l|\xi){\rm d}\xi, \end{align}\] and we focus on \(I_{+}\). Let \(2^{j_0+2}l^{-1}=\xi_0\), then \(j_0=-2+\log_2 l + \frac{1}{2} \log_{2}\frac{\sqrt{\mu^2+3\mu}-\mu}{3}\).

Case 1. \(j\in[j_0,j_0+4].\) Let \(\delta>0\) be a small parameter to be chosen later. We decompose \[I_{+}(t,\alpha,\beta,l,j)=\int_{[\xi_0-\delta,\xi_0+\delta]}e^{it\Phi(\xi)}\varphi(2^{-j}|l|\xi){\rm d}\xi+\int_{\mathbb{R}_{+}\backslash[\xi_0-\delta,\xi_0+\delta]}e^{it\Phi(\xi)}\varphi(2^{-j}|l|\xi){\rm d}\xi=: I_{+}^1+ I_{+}^2.\] It is easy to see that \(|I_{+}^1|\leq 2\delta\). To bound \(I_{+}^{2}\), there exists a small constant \(c_0>0\), independent of \(l,j\), such that \[|\Phi''(\xi)|=|\beta(\beta-1)-\alpha^2|\frac{3\alpha^2\left(\xi^2+\frac{\sqrt{\mu^2+3\mu}+\mu}{3}\right)}{(1+\xi^2)^{5/2}\left(\beta(\beta-1)+\alpha^2\xi^2\right)^{3/2}}|\xi^2-\xi_0^2|\geq c_0q\delta.\] Then applying Van der Corput lemma, we have \[\begin{align} |I_{+}^2|\leq& C(\delta qt)^{-\frac{1}{2}}\left[\|\varphi\|_{L^\infty}+\int_{\mathbb{R}^+\slash[\xi_0-\delta,\xi_0+\delta]}|\frac{\rm d}{{\rm d}\xi}\left(\varphi(2^{-j}|l|\xi)\right)|{\rm d}\xi\right]\leq 2C(\delta qt)^{-\frac{1}{2}}. \end{align}\] Let \(\delta\sim (\delta qt)^{-\frac{1}{2}}\), we choose \(\delta=(qt)^{-\frac{1}{3}}\) and obtain that \[|I_+(t,\alpha,\beta,l,j)|\leq C(qt)^{-\frac{1}{3}}.\]

Case 2. \(j\neq[j_0,j_0+4]\), then \(|\Phi''(\xi)|\geq c>0\). When \(j<j_0\), we can easily get \[|I_{+}(t,\alpha,\beta,l,j)|\leq \operatorname{supp}_{\xi} \varphi(2^{-j}l\xi)=C2^j|l|^{-1}.\] Notice that in this case \(|\xi|< \frac{\xi_{0}}{4}\), there exists a constant \(C>0\) independent of \(j,l\) such that \(\Phi''(\xi)>C\). Applying Lemma 5 again yields \[|I_{+}(t,\alpha,\beta,l,j)|\leq C(qt)^{-\frac{1}{2}}.\] On the other hand, when \(j>j_0+4\), using the fact that \(|\xi|\leq C2^j|l|^{-1}\), we obtain \(|\Phi''(\xi)|\geq C|\xi|^{-4}\geq C2^{-4j}|l|^4\). We have by Lemma 5 \[|I_{+}(t,\alpha,\beta,l,j)|\leq C(qt)^{-\frac{1}{2}}2^{2j}|l|^{-2}.\] Summarizing the estimates above, we arrive at \[\begin{align} |I(t,\alpha,\beta,j,l)|\lesssim&\begin{cases} \min\{2^j|l|^{-1},(qt)^{-\frac{1}{2}}\},\;j<j_0,\\ (qt)^{-\frac{1}{3}},\;j_0\leq j\leq j_0+4,\\ (qt)^{-\frac{1}{2}}2^{2j}|l|^{-2},\;j_0+4<j. \end{cases} \end{align}\]

Inserting this into ?? , we find that \[\begin{align} \|e^{-t\mathcal{R}^l}h\|_{L^\infty}\lesssim& |l|\sum_{j<j_0}\min\{2^j|l|^{-1},(qt)^{-\frac{1}{2}}\}\|P_j^{\dagger}h\|_{L^1} \\ &+ |l|(qt)^{-\frac{1}{3}}\sum_{j_0}^{j_0+4}\|P_j^{\dagger}h\|_{L^1} + |l|^{-1}(qt)^{-\frac{1}{2}}\sum_{j>j_0+4}2^{2j}\|P_j^{\dagger}h\|_{L^1} \\ \lesssim& |l|(qt)^{-\frac{1}{3}}\|h\|_{L^1} + |l|^{-1}(qt)^{-\frac{1}{2}}\|h\|_{W^{2+\sigma,1}}. \end{align}\]

The proposition is proved in the case when \(a=0\). For \(a>0\), using the fact that \(\|(l^2-\partial_y^2)^{-\frac{a}{2}}P_j f\|_{L^1}\lesssim|l|^{-a}\|P_j^{\dagger}f\|_{L^1}\) and we finish the proof.\(\square\)

Proof.According to Proposition 12, for \(0<\sigma<\frac{1}{2}\) small enough, we have \[\begin{align} \left||\partial_z|^a(-\Delta_{L,0})^{-\frac{a}{2}}e^{t\mathcal{L}}f(y,z)\right|=&\left|\sum_{l\neq0}|l|^a e^{izl-\nu l^2t}\int_{\eta}e^{iy\eta-\nu \eta^2t - t\mathcal{R}^l}(\eta^2+l^2)^{-\frac{a}{2}}\hat{f_l}(\eta){\rm d}\eta\right| \\ \leq& \sum_{l\neq0}e^{-\nu l^2 t}|l|^a \left\|e^{-\mathcal{R}^lt}\left(e^{\nu t\partial_y^2}(l^2-\partial_y^2)^{-\frac{a}{2}}f_l\right)\right\|_{L^\infty(\mathbb{R})}\\ \leq& (qt)^{-\frac{1}{3}}\sum_{l\neq0}e^{-\nu l^2t} \left(|l|\|e^{\nu t\partial_y^2}f_l\|_{L^1_y(\mathbb{R})}+|l|^{-1}\|e^{\nu t\partial_y^2}f_l\|_{W^{2+\sigma,1}_y(\mathbb{R})}\right) \\ \leq& e^{-\nu t}(qt)^{-\frac{1}{3}}\left(\sum_{l\neq 0}|l|^{-1-\sigma}\|f\|_{L^1_yW^{2+\sigma,1}_z(\mathbb{R}\times\mathbb{T})}+\sum_{l\neq 0}|l|^{-1-\sigma}\|f\|_{W^{2+\sigma,1}_yW^{\sigma,1}_z(\mathbb{R}\times\mathbb{T})}\right)\\ \leq& C e^{-\nu t}(qt)^{-\frac{1}{3}}\|f\|_{W^{3,1}(\mathbb{R}\times\mathbb{T})}, \end{align}\] thus, we finish the proof of the corollary.0◻

Based on Corollary 14, we directly obtain the linear dispersion estimate ?? of \(u_0^2\) and \(\tilde{u}_0^3\).

4 The nonlineqar stability↩︎

4.1 Quasi-linearization method↩︎

In this section, we discuss the quasi-linearization method, which has been used in many papers, see [15], [17], [27], [39], [40]. The quasi-linearization method is used to extract the nonlinear dispersive effect of simple-zero modes and improve the stability threshold of non-zero modes, simple-zero modes and double-zero modes. The main idea of quasi-linearization method is to decompose the system into two or more systems: a good system that carries the main structure, regularity and the size of original initial data, and some quasi-linearization and nonlinear systems that contain the smaller nonlinear part, which start from zero initial data. Let’s recall the equation 47 and 53 satisfied by zero modes and consider the nonlinear terms, which gives \[\begin{align} \tag{54} &\begin{cases} \partial_t U_0^2 - \nu \Delta_{L,0} U_0^2 - \mathcal{R}V_0^2 ={\mathcal{N}_{0+\neq}(U_0^2)}:=-(U\cdot\nabla_L U^2)_0 + \partial_y\Delta_{L,0}^{-1}(\partial_i^LU^j\partial_j^LU^i)_0,\\ \partial_t V_0^2 - \nu \Delta_{L,0} V_0^2 + \mathcal{R} U_0^2 ={\mathcal{N}_{0+\neq}(V_0^2)}:= - G_1\widetilde{(U\cdot\nabla_L U^1)}_0 -G_2\widetilde{(U\cdot\nabla_L\Theta)}_0 ,\\ \end{cases}\\ & \quad \partial_t \Lambda_0 - \nu\Delta_{L,0} \Lambda_0 ={\mathcal{N}_{0+\neq}(\Lambda_0)}:= - G_3\widetilde{(U\cdot\nabla_L U^1)}_0 -G_4\widetilde{(U\cdot\nabla_L\Theta)}_0,\tag{55}\\ &\begin{cases}\tag{56} \partial_t \tilde{U}_0^3 - \nu \Delta_{L,0} \tilde{U}_0^3 - \mathcal{R}\tilde{V}_0^3 ={\mathcal{N}_{0+\neq}(\tilde{U}_0^3)}:=-(\widetilde{U\cdot\nabla_L U^3})_0 + \partial_z\Delta_{L,0}^{-1}(\widetilde{\partial_i^LU^j\partial_j^LU^i})_0,\\ \partial_t \tilde{V}_0^3 - \nu \Delta_{L,0} \tilde{V}_0^3 + \mathcal{R} \tilde{U}_0^3 ={\mathcal{N}_{0+\neq}(\tilde{V}_0^3)}:= - \widetilde{G_1'(U\cdot\nabla_L U^1)}_0 -\widetilde{G_2'(U\cdot\nabla_L\Theta)}_0, \end{cases}\\ &\begin{cases}\tag{57} \partial_t \overline{U_0^1} - \nu \partial_{yy} \overline{U_0^1} = -\partial_y(\overline{U^2 U^1})_0,\\ \partial_t \overline{U_0^3} - \nu \partial_{yy} \overline{U_0^3} +\alpha\overline{\Theta_0}= -\partial_y(\overline{U^2 U^3})_0,\\ \partial_t \overline{\Theta_0} - \nu \partial_{yy} \overline{\Theta_0} -\alpha\overline{U_0^3}= -\partial_y(\overline{U^2 \Theta})_0. \end{cases} \end{align}\] Due to the complex structure of the system, we cannot directly obtain the nonlinear dispersion estimates of simple zero modes. We will use a quasi-linearization method to decompose the system into two parts, one of which retains the regularity and size of the initial data and has a good nonlinear dispersion effect. Another part contains smaller nonlinear terms, and their initial data starts from zero. More precisely, we decompose \[U_0=U_{0,1}+U_{0,2},\;\Theta_0=\Theta_{0,1} + \Theta_{0,2},\;V_0^2=V_{0,1}^2+V_{0,2}^2,\;\tilde{V}_0^3=\tilde{V}_{0,1}^3+\tilde{V}_{0,2}^3,\;\Lambda_0=\Lambda_{0,1}+\Lambda_{0,2},\] which satisfies the following systems: \[\begin{align} \label{1stQuasi1} &\;\left\{\begin{array}{ll} \partial_t U^2_{0,1} - \nu\Delta_{L,0}U^2_{0,1} - \mathcal{R}V_{0,1}^2 ={\mathcal{N}_{0+\neq}(U_{0,1}^2)}\\ := -\tilde{U}_{0,1}\cdot\nabla U_{0,1}^2 + 2\partial_y\Delta_{L,0}^{-1}(\partial_yU_{0,1}^2\partial_yU^2_{0,1}+\partial_y\tilde{U}^3_{0,1}\partial_zU^2_{0,1})- (U_{\neq,1}\cdot\nabla_L U^2_{\neq,1})_0 \\ \quad + \partial_y\Delta_{L,0}^{-1}(\partial_i^L U_{\neq,1}\cdot\nabla_L U^i_{\neq,1})_0 ,\\ \partial_t V^2_{0,1} - \nu\Delta_{L,0}V^2_{0,1} + \mathcal{R}U_{0,1}^2 ={\mathcal{N}_{0+\neq}(V_{0,1}^2)}\\ := -\tilde{U}_{0,1}\cdot\nabla V_{0,1}^2 - G_1\widetilde{(U_{\neq,1}\cdot\nabla_L U^1_{\neq,1})}_0 - G_2\widetilde{(U_{\neq,1}\cdot\nabla_L \Theta_{\neq,1})}_0,\\ \partial_t\Lambda_{0,1} - \nu\Delta_{L,0}\Lambda_{0,1} ={\mathcal{N}_{0+\neq}(\Lambda_{0,1})}\\ := {- G_3(\tilde{U}_{0,1}\cdot\nabla_L U^1_{0,1}) - G_4(\tilde{U}_{0,1}\cdot\nabla_L \Theta_{0,1})} - G_3\widetilde{(U_{\neq,1}\cdot\nabla_L U^1_{\neq,1})}_0 - G_4\widetilde{(U_{\neq,1}\cdot\nabla_L \Theta_{\neq,1})}_0. \end{array}\right. & \end{align}\tag{58}\] \[\begin{align} \label{1stQuasi2} &\;\left\{\begin{array}{ll} \partial_t \tilde{U}^3_{0,1} - \nu\Delta_{L,0}\tilde{U}^3_{0,1} - \mathcal{R} \tilde{V}_{0,1}^3={\mathcal{N}_{0+\neq}(\tilde{U}_{0,1}^3)} \\ := -\widetilde{\tilde{U}_{0,1}\cdot\nabla\tilde{U}_{0,1}^3} + 2\partial_z\Delta_{L,0}^{-1}(\widetilde{\partial_yU_{0,1}^2\partial_yU^2_{0,1}}+\widetilde{\partial_y\tilde{U}^3_{0,1}\partial_zU^2_{0,1}}) - \widetilde{(U_{\neq,1}\cdot\nabla_L U^3_{\neq,1})}_0 \\ \quad+ \partial_z\Delta_{L,0}^{-1}\widetilde{(\partial_i^L U_{\neq,1}\cdot\nabla_L U^i_{\neq,1})}_0 ,\\ \partial_t \tilde{V}^3_{0,1} - \nu\Delta_{L,0} \tilde{V}^3_{0,1} + \mathcal{R}\tilde{U}_{0,1}^3 ={\mathcal{N}_{0+\neq}(\tilde{V}_{0,1}^3)} \\ := {\partial_z^{-1}\partial_y(\tilde{U}_{0,1}\cdot\nabla V_{0,1}^2)} - G_1'\widetilde{(U_{\neq,1}\cdot\nabla_L U^1_{\neq,1})}_0 - G_2'\widetilde{(U_{\neq,1}\cdot\nabla_L \Theta_{\neq,1})}_0. \end{array}\right. & \end{align}\tag{59}\] \[\begin{align} \label{1stQuasi3} &\;\left\{\begin{array}{ll} \partial_t\overline{U_{0,1}^1} - \nu\partial_{yy}\overline{U_{0,1}^1} ={\mathcal{N}_{0+\neq}(\overline{U_{0,1}^1})} := -\partial_y\overline{(U_{0,1}^2\tilde{U}_{0,1}^1)} - \partial_y(\overline{U^2_{\neq,1}U^1_{\neq,1}})_0,\\ \partial_t\overline{U_{0,1}^3} - \nu\partial_{yy}\overline{U_{0,1}^3} + \alpha\overline{\Theta_{0,1}} ={\mathcal{N}_{0+\neq}(\overline{U_{0,1}^3})} := -\partial_y\overline{(U_{0,1}^2\tilde{U}_{0,1}^3)} - \partial_y(\overline{U^2_{\neq,1}U^3_{\neq,1}})_0,\\ \partial_t\overline{\Theta_{0,1}} - \nu\partial_{yy}\overline{\Theta_{0,1}} - \alpha\overline{U^3_{0,1}} = {\mathcal{N}_{0+\neq}(\overline{\Theta_{0,1}})} := - \partial_y(\overline{U^2_{\neq,1}\Theta_{\neq,1}})_0. \end{array}\right.& \end{align}\tag{60}\] with the initial data \[\begin{align} U_{0,1}^2(t=0)=&U_0^2(t=0),\;V_{0,1}^2(t=0)=V_0^2(t=0),\;\Lambda_{0,1}(t=0)=\Lambda_0(t=0),\; \tilde{U}_{0,1}^3(t=0)=\tilde{U}_0^3(t=0),\\ \tilde{V}_{0,1}^3(t=0)=&\tilde{V}_0^3(t=0),\;\overline{U_{0,1}^1}(t=0)=\overline{U_0^1}(t=0),\;\overline{U_{0,1}^3}(t=0)=\overline{U_0^3}(t=0),\;\overline{\Theta_{0,1}}(t=0)=\overline{\Theta_0}(t=0), \end{align}\] and \[\begin{align} \label{2ndQuasi1} &\;\left\{\begin{array}{ll} \partial_t U^2_{0,2} - \nu\Delta_{L,0}U^2_{0,2} - \mathcal{R}V_{0,2}^2 \\ ={\mathcal{N}_{0+\neq}(U_{0,2}^2)}:= -U_0\cdot\nabla U_0^2 + 2\partial_y\Delta_{L,0}^{-1}(\partial_yU_0^2\partial_yU^2_0+\partial_yU^3_0\partial_zU^2_0)\\ \quad -\left[-\tilde{U}_{0,1}\cdot\nabla U_{0,1}^2 + 2\partial_y\Delta_{L,0}^{-1}(\partial_yU_{0,1}^2\partial_yU^2_{0,1}+\partial_y\tilde{U}^3_{0,1}\partial_zU^2_{0,1})\right]\\ \quad -(U_{\neq}\cdot\nabla_L U^2_{\neq})_0+ (U_{\neq,1}\cdot\nabla_L U^2_{\neq,1})_0 + \partial_y\Delta_{L,0}^{-1}(\partial_i^L U_{\neq}\cdot\nabla_L U^i_{\neq})_0-\partial_y\Delta_{L,0}^{-1}(\partial_i^L U_{\neq,1}\cdot\nabla_L U^i_{\neq,1})_0,\\ \partial_t V^2_{0,2} - \nu\Delta_{L,0}V^2_{0,2} + \mathcal{R}U_{0,2}^2 \\ ={\mathcal{N}_{0+\neq}(V_{0,2}^2)}:= - G_1\widetilde{(U_0\cdot\nabla U^1_0)} - G_2\widetilde{(U_0\cdot\nabla\Theta_0)} + \tilde{U}_{0,1}\cdot\nabla V_{0,1}^2 \\ \quad - G_1\widetilde{(U_{\neq}\cdot\nabla_L U^1_{\neq})}_0 + G_1\widetilde{(U_{\neq,1}\cdot\nabla_L U^1_{\neq,1})}_0 - G_2\widetilde{(U_{\neq}\cdot\nabla_L \Theta_{\neq})}_0 + G_2\widetilde{(U_{\neq,1}\cdot\nabla_L \Theta_{\neq,1})}_0,\\ \partial_t\Lambda_{0,2} - \nu\Delta_{L,0}\Lambda_{0,2} \\ ={\mathcal{N}_{0+\neq}({\Lambda_{0,2}})}:= - G_3\widetilde{(U_0\cdot\nabla U_0^1)} - G_4\widetilde{(U_0\cdot\nabla\Theta_0)} +{G_3(\tilde{U}_{0,1}\cdot\nabla U^1_{0,1}) + G_4(\tilde{U}_{0,1}\cdot\nabla\Theta_{0,1})} \\ \quad - G_3\widetilde{(U_{\neq}\cdot\nabla_L U^1_{\neq})}_0 + G_3\widetilde{(U_{\neq,1}\cdot\nabla_L U^1_{\neq,1})}_0 - G_4\widetilde{(U_{\neq}\cdot\nabla_L \Theta_{\neq})}_0 + G_4\widetilde{(U_{\neq,1}\cdot\nabla_L \Theta_{\neq,1})}_0. \end{array} \right.& \end{align}\tag{61}\] \[\begin{align} \label{2ndQuasi2} &\;\left\{\begin{array}{ll} \partial_t \tilde{U}^3_{0,2} - \nu\Delta_{L,0}\tilde{U}^3_{0,2} - \mathcal{R}V_{0,2}^3 \\ ={\mathcal{N}_{0+\neq}(\tilde{U}_{0,2}^3)}:= -\widetilde{U_{0}\cdot\nabla U_{0}^3} + 2\partial_z\Delta_{L,0}^{-1}(\widetilde{\partial_yU_{0}^2\partial_yU^2_{0}}+\widetilde{\partial_yU^3_{0}\partial_zU^2_{0}})\\ \quad -\left[-\widetilde{\tilde{U}_{0,1}\cdot\nabla\tilde{U}_{0,1}^3} + 2\partial_z\Delta_{L,0}^{-1}(\widetilde{\partial_yU_{0,1}^2\partial_yU^2_{0,1}} + \widetilde{\partial_y\tilde{U}^3_{0,1}\partial_zU^2_{0,1}})\right]\\ \quad - (\widetilde{U_{\neq}\cdot\nabla_L U^3_{\neq}})_0 + \widetilde{(U_{\neq,1}\cdot\nabla_L U^3_{\neq,1})}_0 + \partial_z\Delta_{L,0}^{-1}(\widetilde{\partial_i^L U_{\neq}\cdot\nabla_L U^i_{\neq}})_0 - \partial_z\Delta_{L,0}^{-1}\widetilde{(\partial_i^L U_{\neq,1}\cdot\nabla_L U^i_{\neq,1})}_0,\\ \partial_t \tilde{V}^3_{0,2} - \nu\Delta_{L,0}\tilde{V}^3_{0,2} + \mathcal{R}\tilde{U}_{0,2}^3 \\ ={\mathcal{N}_{0+\neq}(\tilde{V}_{0,2}^3)}:= - \widetilde{G_1'(U_{0}\cdot\nabla U^1_0)}- \widetilde{G_2'(U_{0}\cdot\nabla\Theta_0)} - {\partial_z^{-1}\partial_y(\tilde{U}_{0,1}\cdot\nabla V_{0,1}^2)} \\ \quad - \widetilde{G_1'(U_{\neq}\cdot\nabla_L U^1_{\neq})_0} + G_1'\widetilde{(U_{\neq,1}\cdot\nabla_L U^1_{\neq,1})}_0 - \widetilde{G_2'(U_{\neq}\cdot\nabla_L \Theta_{\neq})_0}- G_2'\widetilde{(U_{\neq,1}\cdot\nabla_L \Theta_{\neq,1})}_0. \end{array}\right.& \end{align}\tag{62}\] \[\begin{align} \label{2ndQuasi3} &\;\left\{\begin{array}{ll} \partial_t\overline{U_{0,2}^1} - \nu\partial_{yy}\overline{U_{0,2}^1} = -\partial_y\overline{(U_{0}^2\tilde{U}_0^1)} + \partial_y\overline{(U_{0,1}^2\tilde{U}_{0,1}^1)}- \partial_y(\overline{U^2_{\neq}U^1_{\neq}})_0 + \partial_y(\overline{U^2_{\neq,1}U^1_{\neq,1}})_0,\\ \partial_t\overline{U_{0,2}^3} - \nu\partial_{yy}\overline{U_{0,2}^3} + \alpha\overline{\Theta_{0,2}} = -\partial_y\overline{(U_{0}^2\tilde{U}_0^3)} + \partial_y\overline{(U_{0,1}^2\tilde{U}_{0,1}^3)} - \partial_y(\overline{U^2_{\neq}U^3_{\neq}})_0 + \partial_y(\overline{U^2_{\neq,1}U^3_{\neq,1}})_0,\\ \partial_t\overline{\Theta_{0,2}} - \nu\partial_{yy}\overline{\Theta_{0,2}} - \alpha\overline{U^3_{0,2}} = -\partial_y\overline{(U_{0}^2\tilde{\Theta}_0)} - \partial_y(\overline{U^2_{\neq}\Theta_{\neq}})_0 + \partial_y(\overline{U^2_{\neq,1}\Theta_{\neq,1}})_0 \end{array} \right.& \end{align}\tag{63}\] with zero initial data \[\begin{align} &U_{0,2}^2(t=0)=\tilde{U}_{0,2}^3(t=0)=V_{0,2}^2(t=0)=\tilde{V}_{0,2}^3(t=0)=\Lambda_{0,2}(t=0)=0,\\ &\overline{U_{0,2}^1}(t=0)=\overline{U_{0,2}^3}(t=0)=\overline{\Theta_{0,2}}(t=0)=0, \end{align}\] where we have frequently used the fact that \[(\partial_i^L U^j\partial_j^L U^i)_0=\partial_i U^j_0\partial_jU^i_0 + (\partial_i^LU^j_{\neq}\partial_j^LU^i_{\neq})_0,\,\, \partial_iU^j_0\partial_j U^i_0=2(\partial_yU_0^2\partial_yU^2_0+\partial_y\tilde{U}^3_0\partial_zU^2_0) + 2\partial_y\overline{U^3_0}\partial_zU^2_0.\]

Similarly, for non-zero modes, we recall that 24 and consider the nonlinear terms, which yields \[\begin{align} \label{Non0NL} \begin{cases} \partial_t Q_{\neq} -\nu\Delta_L Q_{\neq} - \frac{1}{2} \partial_x\partial_y^L|\nabla_L|^{-2} Q_{\neq} - \partial_x\partial_y^L(|\nabla_L|^{-2}-|\nabla_{L,h}|^{-2}) Q_{\neq} \\ \qquad \;+ i\sqrt{\beta(\beta-1)}\partial_z|\nabla_L|^{-1} K_{\neq} + 2i\sqrt{\frac{\beta-1}{\beta}}\partial_z\partial_{xx}|\nabla_L|^{-1}|\nabla_{L,h}|^{-2}K_{\neq} + \alpha|\nabla_{L,h}||\nabla_L|^{-1}H_{\neq} \\ = |\nabla_L|^{\frac{3}{2}}|\nabla_{L,h}|^{-1}(U\cdot\nabla_L U^3)_{\neq} - \partial_z|\nabla_{L,h}|^{-1}|\nabla_L|^{-\frac{1}{2}}(\partial_i^L U^j\partial_j^L U^i)_{\neq}, \\ \partial_t K_{\neq} - \nu\Delta_L K_{\neq} + \frac{1}{2} \partial_x\partial_y^L|\nabla_L|^{-2} K_{\neq} + \partial_x\partial_y^L(|\nabla_L|^{-2}-|\nabla_{L,h}|^{-2}) K_{\neq} - i\sqrt{\beta(\beta-1)}\partial_z|\nabla_L|^{-1}Q_{\neq} \\ = i\sqrt{\frac{\beta}{\beta-1}}|\nabla_L|^{\frac{1}{2}}|\nabla_{L,h}|^{-1}\left[\partial_x(U\cdot\nabla_L U^2)_{\neq} - \partial_y^L(U\cdot\nabla_L U^1)_{\neq} \right],\\ \partial_t H_{\neq} -\nu\Delta_L H_{\neq} - \frac{1}{2} \partial_x\partial_y^L|\nabla_L|^{-2} H_{\neq} - \alpha|\nabla_{L,h}||\nabla_L|^{-1}Q_{\neq} = |\nabla_L|^{\frac{1}{2}}(U\cdot\nabla_L\Theta)_{\neq}. \end{cases} \end{align}\tag{64}\] We do some decomposition via quasi-linearization method as follows \[Q_{\neq}=Q_{\neq,1}+Q_{\neq,2},\;K_{\neq}=K_{\neq,1}+K_{\neq,2},\;H_{\neq}=H_{\neq,1}+H_{\neq,2}.\] Then the first part \((Q_{\neq,1}, K_{\neq,1}, H_{\neq,1})\) enjoys the fisrt subsystem \[\begin{align} \label{Non0NL1} \begin{cases} \partial_t Q_{\neq,1} -\nu\Delta_L Q_{\neq,1} - \frac{1}{2} \partial_x\partial_y^L|\nabla_L|^{-2} Q_{\neq,1} - \partial_x\partial_y^L(|\nabla_L|^{-2}-|\nabla_{L,h}|^{-2}) Q_{\neq,1} \\ \qquad \;+ i\sqrt{\beta(\beta-1)}\partial_z|\nabla_L|^{-1} K_{\neq,1} + 2i\sqrt{\frac{\beta-1}{\beta}}\partial_z\partial_{xx}|\nabla_L|^{-1}|\nabla_{L,h}|^{-2}K_{\neq,1} + \alpha|\nabla_{L,h}||\nabla_L|^{-1}H_{\neq,1} \\ = |\nabla_L|^{\frac{3}{2}}|\nabla_{L,h}|^{-1}(U^1_{\neq,1}\partial_x U^3_{\neq,1}+U^1_{0,1}\partial_x U^3_{\neq,1}) \\ \quad - \partial_z|\nabla_{L,h}|^{-1}|\nabla_L|^{-\frac{1}{2}}(\partial_{x,z} U^{1,3}_{\neq,1}\cdot\nabla_{x,z} U^{1,3}_{\neq,1}+2\nabla_{L,h}U_{\neq,1}^2\partial_y^LU_{\neq,1}^{1,2}+2\partial_z U^{1,3}_{0,1}\cdot\nabla_{x,z} U^3_{\neq,1}+2\partial_yU_{0,1}^1\partial_xU_{\neq,1}^2), \\ \partial_t K_{\neq,1} - \nu\Delta_L K_{\neq,1} + \frac{1}{2} \partial_x\partial_y^L|\nabla_L|^{-2} K_{\neq,1} + \partial_x\partial_y^L(|\nabla_L|^{-2}-|\nabla_{L,h}|^{-2}) K_{\neq,1} - i\sqrt{\beta(\beta-1)}\partial_z|\nabla_L|^{-1}Q_{\neq,1} \\ = i\sqrt{\frac{\beta}{\beta-1}}|\nabla_L|^{\frac{1}{2}}|\nabla_{L,h}|^{-1}\big[\partial_xU_{\neq,1}\cdot\nabla_L U^2_{\neq,1}-\partial_y^LU_{\neq,1}\cdot\nabla_LU^1_{\neq,1} +U_{\neq,1}\cdot\nabla_LW^3_{\neq,1}\\ \quad +\partial_xU_{\neq,1}\cdot\nabla U_{0,1}^2-\partial_y^LU_{\neq,1}^2\partial_y U_{0,1}^1-\partial_yU_{0,1}^{1,3}\cdot\nabla_{x,z}U^1_{\neq,1}+U_{0,1}^{1,3}\cdot\nabla_{x,z}W^3_{\neq,1} \big],\\ \partial_t H_{\neq,1} -\nu\Delta_L H_{\neq,1} - \frac{1}{2} \partial_x\partial_y^L|\nabla_L|^{-2} H_{\neq,1} - \alpha|\nabla_{L,h}||\nabla_L|^{-1}Q_{\neq,1}\\ = |\nabla_L|^{\frac{1}{2}}(U_{\neq,1}\cdot\nabla_L\Theta_{\neq,1}+U_{0,1}^{1,3}\cdot\nabla_{x,z}\Theta_{\neq,1}+U_{\neq,1}^3\partial_z\Theta_{0,1}),\\ Q_{\neq,1}(t=0)=Q_{\neq}(t=0),\;K_{\neq,1}(t=0)=K_{\neq}(t=0),\;H_{\neq,1}(t=0)=H_{\neq}(t=0), \end{cases} \end{align}\tag{65}\] and the rest part \((Q_{\neq,2}, K_{\neq,2}, H_{\neq,2})\) satisfies the second subsystem \[\begin{align} \label{Non0NL2} \begin{cases} \partial_t Q_{\neq,2} -\nu\Delta_L Q_{\neq,2} - \frac{1}{2} \partial_x\partial_y^L|\nabla_L|^{-2} Q_{\neq,2} - \partial_x\partial_y^L(|\nabla_L|^{-2}-|\nabla_{L,h}|^{-2}) Q_{\neq,2} \\ \quad \;+ i\sqrt{\beta(\beta-1)}\partial_z|\nabla_L|^{-1} K_{\neq,2} + 2i\sqrt{\frac{\beta-1}{\beta}}\partial_z\partial_{xx}|\nabla_L|^{-1}|\nabla_{L,h}|^{-2}K_{\neq,2} + \alpha|\nabla_{L,h}||\nabla_L|^{-1}H_{\neq,2} \\ = |\nabla_L|^{\frac{3}{2}}|\nabla_{L,h}|^{-1}\left(U^{2,3}_{\neq,1}\nabla_{y,z}^LU^3_{\neq,1}+U^3_{\neq,1}\partial_zU^3_{\neq,2}+U^3_{\neq,2}\partial_zU^3_{\neq,1}+U^h_{\neq,1}\cdot\nabla_{L,h} U^3_{\neq,2}+ U^h_{\neq,2}\cdot\nabla_{L,h} U^3_{\neq,1}\right.\\ \quad +U_{\neq,2}\cdot\nabla_L U^3_{\neq,2}+U^2_{0,1}\partial_y^L U^3_{\neq,1}+U^3_{0,1}\partial_z U^3_{\neq,1}+U^2_{\neq,1}\partial_y U^3_{0,1}+U^3_{\neq,1}\partial_zU^3_{0,1}+U^2_{0,2}\partial_y^L U^3_{\neq,1}+U^3_{0,2}\partial_z U^3_{\neq,1}\\ \quad +U_{0,1}\cdot\nabla_L U_{\neq,2}^3 + U^1_{0,2}\partial_x U^3_{\neq,1} + U^2_{\neq,1}\partial_y U^3_{0,2} +U^2_{\neq,2}\partial_y U^3_{0,1}+U^3_{\neq,1}\partial_zU^3_{0,2} +U^3_{\neq,2}\partial_zU^3_{0,1}+U_{\neq,2}^{2,3}\cdot\nabla_{y,z} U^3_{0,2} \\ \quad \left. +U_{0,2} \cdot \nabla_L U^3_{\neq,2} \right) - \partial_z|\nabla_{L,h}|^{-1}|\nabla_L|^{-\frac{1}{2}}\left(2\partial_zU_{\neq,1}^2\partial_y^LU_{\neq,1}^3 +\partial_i^L U^j_{\neq,1}\partial_j^L U^i_{\neq,2}+\partial_i^L U^j_{\neq,2}\partial_j^L U^i_{\neq,1}+\partial_i^L U^j_{\neq,2}\partial_j^L U^i_{\neq,2}\right. \\ \quad +2\partial_yU_{0,1}^3\partial_zU_{\neq,1}^2+ \partial_i U^2_{0,1}\partial_y^L U^i_{\neq,1} + \partial_y^L U^i_{\neq,1}\partial_i U^2_{0,1}+\partial_i U^j_{0,1}\partial_j^L U^i_{\neq,2}+ \partial_i^L U^j_{\neq,2}\partial_j U^i_{0,1}+ \partial_i U^j_{0,2}\partial_j^L U^i_{\neq,1}\\ \quad \left.+ \partial_i^L U^j_{\neq,1}\partial_j U^i_{0,2} + \partial_i U^j_{0,2}\partial_j^L U^i_{\neq,2} + \partial_i^L U^j_{\neq,2}\partial_j U^i_{0,2}\right), \\ \partial_t K_{\neq,2} - \nu\Delta_L K_{\neq,2} + \frac{1}{2} \partial_x\partial_y^L|\nabla_L|^{-2} K_{\neq,2} + \partial_x\partial_y^L(|\nabla_L|^{-2}-|\nabla_{L,h}|^{-2}) K_{\neq,2} - i\sqrt{\beta(\beta-1)}\partial_z|\nabla_L|^{-1}Q_{\neq,2} \\ = i\sqrt{\frac{\beta}{\beta-1}}|\nabla_L|^{\frac{1}{2}}|\nabla_{L,h}|^{-1}\left(\partial_xU_{\neq,1}\cdot\nabla_L U^2_{\neq,2}+\partial_xU_{\neq,2}\cdot\nabla_L U^2_{\neq,1}-\partial_y^LU_{\neq,1}\cdot\nabla_LU^1_{\neq,2}-\partial_y^LU_{\neq,2}\cdot\nabla_LU^1_{\neq,1}\right. \\ \quad +U_{\neq,1}\cdot\nabla_LW^3_{\neq,2}+U_{\neq,2}\cdot\nabla_LW^3_{\neq,1}+\partial_xU_{\neq,2}\cdot\nabla_L U^2_{\neq,2}-\partial_y^LU_{\neq,2}\cdot\nabla_LU^1_{\neq,2}+U_{\neq,2}\cdot\nabla_LW^3_{\neq,2}\\ \quad +{U_{0,1}^2\partial_y^LW_{\neq,1}^3}-U_{\neq,1}^2\partial_{yy}U^1_{0,1}-\partial_y U_{0,1}^2\partial_y^L U^1_{\neq,1}-\partial_y^LU_{\neq,1}^3\partial_z U_{0,1}^1 - U_{\neq,1}^3\partial_z\partial_yU^1_{0,1}+ \partial_xU_{\neq,2}\cdot\nabla U_{0,1}^2\\ \quad -\partial_y^LU_{\neq,2}\cdot\nabla U_{0,1}^1-\partial_yU_{0,1}\cdot\nabla_LU^1_{\neq,2}+U_{0,2}^2\partial_y^L W^3_{\neq,1}-U_{\neq,1}^2\partial_{yy}U^1_{0,2}+U_{0,1}\cdot\nabla_LW^3_{\neq,2} - U_{\neq,2}\cdot\nabla\partial_yU^1_{0,1} \\ \quad + \partial_xU_{\neq,1}\cdot\nabla U_{0,2}^2-\partial_y^LU_{\neq,1}\cdot\nabla U_{0,2}^1-\partial_yU_{0,2}\cdot\nabla_LU^1_{\neq,1}+U_{0,2}^{1,3}\cdot\nabla_{x,z}W^3_{\neq,1} - U_{\neq,1}^3\partial_z\partial_yU^1_{0,2} \\ \quad \left.+ \partial_xU_{\neq,2}\cdot\nabla U_{0,2}^2-\partial_y^LU_{\neq,2}\cdot\nabla U_{0,2}^1-\partial_yU_{0,2}\cdot\nabla_LU^1_{\neq,2}+U_{0,2}\cdot\nabla_LW^3_{\neq,2}- U_{\neq,2}\cdot\nabla\partial_yU^1_{0,2}\right),\\ \partial_t H_{\neq,2} -\nu\Delta_L H_{\neq,2} - \frac{1}{2} \partial_x\partial_y^L|\nabla_L|^{-2} H_{\neq,2} - \alpha|\nabla_{L,h}||\nabla_L|^{-1}Q_{\neq,2}\\ = |\nabla_L|^{\frac{1}{2}}(U_{\neq,1}\cdot\nabla_L\Theta_{\neq,2}+U_{\neq,2}\cdot\nabla_L\Theta_{\neq,1}+U_{\neq,2}\cdot\nabla_L\Theta_{\neq,2} +U^2_{\neq,1}\partial_y\Theta_{0,1}+ {U_{0,1}^2\partial_y^L\Theta_{\neq,1}}+ U_{0,2}^2\partial_y^L\Theta_{\neq,1}\\ \quad +U^2_{\neq,1}\partial_y\Theta_{0,2}+U_{0,1}\cdot\nabla_L\Theta_{\neq,2}+U_{\neq,2}\cdot\nabla\Theta_{0,1}+U_{0,2}^{1,3}\cdot\nabla_{x,z}\Theta_{\neq,1}+U_{\neq,1}^3\partial_z\Theta_{0,2}+U_{0,2}\cdot\nabla_L\Theta_{\neq,2}+U_{\neq,2}\cdot\nabla\Theta_{0,2} ),\\ Q_{\neq,2}(t=0)=0,\;K_{\neq,2}(t=0)=0,\;H_{\neq,2}(t=0)=0. \end{cases} \end{align}\tag{66}\]

With the help of analysis above and the relationships between variables, such as 38 , 39 , 43 and 44 , theorem 3 is mainly proved by using the following bootstrap argument. We set \(N\geq s+4\) with \(s>6\) and define \[\mathcal{B}:= mMe^{\frac{\lambda}{2} \nu^{\frac{1}{3}}t}.\]

Proof.For the sake of simplicity in calculation, we assume that \(c\leq a\) and \(c\leq b\). We will maintain this assumption throughout the following proof. Proposition 15 is directly derived from the following propositions 16, 22, 23 and 25. For specific details, see subsection 4.2-4.5. \(\square\)

For convenience, let us define some energy functionals which will be used in the following sections.

\((1)\) the energy functionals for non-zero modes of the systems 65 and 66 : \[\begin{align} E_{\neq,1}(t)=&\|\mathcal{A}(Q_{\neq,1},K_{\neq,1},H_{\neq,1})\|_{H^N}^2,\tag{67} \\ F_{\neq,1}(t)=&\nu\|\mathcal{A}\nabla_L(Q_{\neq,1},K_{\neq,1},H_{\neq,1})\|_{H^N}^2 + \left\|\mathcal{A}\sqrt{-\frac{\dot{M}}{M}}(Q_{\neq,1},K_{\neq,1},H_{\neq,1})\right\|_{H^N}^2 \tag{68}, \end{align}\] \[\begin{align} E_{\neq,2}(t)=&\|\mathcal{B}(Q_{\neq,2},K_{\neq,2},H_{\neq,2})\|_{H^s}^2,\tag{69}\\ F_{\neq,2}(t)=&\nu\|\mathcal{B}\nabla_L(Q_{\neq,2},K_{\neq,2},H_{\neq,2})\|_{H^s}^2 + \left\|\mathcal{B}\sqrt{-\frac{\dot{M}}{M}}(Q_{\neq,2},K_{\neq,2},H_{\neq,2})\right\|_{H^s}^2 \tag{70}. \end{align}\]

\((2)\) the energy functionals for zero modes of the systems 58 60 and 61 63 : \[\begin{align} E_{0,1}(t)=&\|(U^2_{0,1},\tilde{U}^3_{0,1}, V^2_{0,1},\tilde{V}^3_{0,1},\Lambda_{0,1} )\|_{H^{N,N+\frac{1}{2}}}^2 + \|(\overline{U^1_{0,1}},\overline{U^3_{0,1}},\overline{\Theta_{0,1}})\|_{H^{N,N+\frac{1}{2}}}^2,\tag{71} \\ F_{0,1}(t)=&\nu\left(\|\nabla(U^2_{0,1},\tilde{U}^3_{0,1},V^2_{0,1},\tilde{V}^3_{0,1},\Lambda_{0,1})\|_{H^{N,N+\frac{1}{2}}}^2 + \|\nabla(\overline{U^1_{0,1}},\overline{U^3_{0,1}},\overline{\Theta_{0,1}})\|_{H^{N,N+\frac{1}{2}}}^2\right),\tag{72}\\ E_{0,2}(t)=&\|(U^2_{0,2},\tilde{U}^3_{0,2}, V^2_{0,2},\tilde{V}^3_{0,2},\Lambda_{0,2} )\|_{H^{s,s+\frac{1}{2}}}^2 + \|(\overline{U^1_{0,2}},\overline{U^3_{0,2}},\overline{\Theta_{0,2}})\|_{H^{s,s+\frac{1}{2}}}^2, \tag{73} \\ F_{0,2}(t)=&\nu\left(\|\nabla(U^2_{0,2},\tilde{U}^3_{0,2},V^2_{0,2},\tilde{V}^3_{0,2},\Lambda_{0,2})\|_{H^{s,s+\frac{1}{2}}}^2 + \|\nabla(\overline{U^1_{0,2}},\overline{U^3_{0,2}},\overline{\Theta_{0,2}})\|_{H^{s,s+\frac{1}{2}}}^2\right).\tag{74} \end{align}\]

4.2 Energy estimates on \(U_{0,1}\) and \(\Theta_{0,1}\)↩︎

In this section, we aim to prove that under the bootstrap hypotheses of Proposition 15, the estimates on \(U_{0,1}\), and \(\Theta_{0,1}\) hold (i.e., ?? ), with 100 replaced by 50 on the right-hand side.

Proof.Proposition 16 is proved by the following Lemmas 18, 19 and 20. \(\square\)

We first established the low regularity \(L^\infty\) dispersion estimates of \(U_0^2\)and \(\tilde{U}_0^3\), which provide improved index estimates.

Proof.Let \(\Gamma_0^2:= U_0^2+iV_0^2,\;\Gamma_0^3:= \tilde{U}_0^3+i\tilde{V}_0^3\), and donate \(g_1,\;g_2,\;g_1',\;g_2'\) as the symbol of \(G_1,\;G_2,\;G_1',\;G_2'\), respectively. According to the Duhamel’s principle, \[\begin{align} \partial_t \Gamma_0^{i} = \mathcal{L}\Gamma_0^i + \mathcal{N}(U,\Gamma_0^i) \Rightarrow \Gamma_0^i(t)=e^{t\mathcal{L}}\Gamma^i_0(0) + \int_0^t e^{(t-\tau)\mathcal{L}}\mathcal{N}_{0+\neq}(\Gamma_0^i)(\tau){\rm d}\tau,\;i\in\{2,3\}, \end{align}\] where \[\begin{align} \mathcal{N}_{0+\neq}(\Gamma_{0}^2)= \mathcal{N}_{0+\neq}(U_{0}^2) + i\mathcal{N}_{0+\neq}(V_{0}^2),\;\mathcal{N}_{0+\neq}(\Gamma_{0}^3)= \mathcal{N}_{0+\neq}(\tilde{U}_{0}^3) + i\mathcal{N}_{0+\neq}(\tilde{V}_{0}^2). \end{align}\] It is easy to see that \(\int_{\mathbb{T}}\Gamma_0^i(t,y,z){\rm d}z = \int_{\mathbb{T}}\mathcal{N}_{0+\neq}(\Gamma_0^i)(t,y,z){\rm d}z =0\) for \(t\geq 0,\;i\in\{2,3\}\). Notice that \(|g_1|+|g_2|+|g_1'|+|g_2'|\lesssim 1\), then we can apply Corollary 14 and obtain \[\begin{align} &\|\nabla^{\gamma}_{y,z}\Gamma_0^{2,3}(t)\|_{L^\infty_{y,z}} \lesssim e^{-\nu t}(qt)^{-\frac{1}{3}}\|(U^{in}_0,\Theta_0^{in})\|_{W^{s,1}(\mathbb{R}\times\mathbb{T})} + \int_{0}^te^{-\nu(t-\tau)}(q(t-\tau))^{-\frac{1}{3}}\|\mathcal{N}_{0+\neq}(\Gamma_0^{2,3})(\tau)\|_{W^{s,1}(\mathbb{R}\times\mathbb{T})}{\rm d}\tau. \end{align}\] Notice that \(\|t^{-\frac{1}{3}}e^{-\nu t}\|_{L^2_t}\lesssim \nu^{-\frac{1}{6}}\). By bootstrap assumptions, for \(i\in\{2,3\}\) we have \[\begin{align} \|\mathcal{N}_{0+\neq}(\Gamma_0^i)\|_{L^2_tW^{s,1}(\mathbb{R}\times\mathbb{T})} \lesssim& \sum_{F\in\{U^1,U^2,U^3\}}\left(\|F_0\|_{L^{\infty}_tH^{s}} + \|\mathcal{A}F_{\neq}\|_{L^{\infty}_tH^s}\right) \\ &\times \left(\sum_{F\in\{U^1,U^2,U^3,\Theta\}}\left(\|\nabla F_0\|_{L^2_tH^{s}} + \|\nabla_L\mathcal{A}F_{\neq}\|_{L^2_tH^s}\right) + \|\partial_y(\overline{\Theta_0},\overline{U^1_0},\overline{U^3_0})\|_{L^2_tH^{s}}\right) \\ \lesssim& \nu^{-\frac{1}{2}}\delta^2\nu^{2c}, \end{align}\] thus, one has \[\int_{0}^te^{-\nu(t-\tau)}(q(t-\tau))^{-\frac{1}{3}}\|\mathcal{N}_{0+\neq}(\Gamma_0^i)(\tau)\|_{W^{s,1}(\mathbb{R}\times\mathbb{T})}{\rm d}\tau\lesssim q^{-\frac{1}{3}}\nu^{-\frac{2}{3}}\delta^2\nu^{2c}.\] The estimates of \(U^2_{0,1},\tilde{U}^3_{0,1}\) and \(U^2_{0,2},\tilde{U}^3_{0,2}\) are similar to that of \(U^2_{0},\tilde{U}^3_{0}\), so we omit the proof. Note that we can propagation \(\partial_z^{\frac{1}{2}}\) in our estimats, we finish the proof. \(\square\)

For the main system 58 60 , we establish the nonlinear dispersive estimate of \(U_{0,1}^2\) and \(\tilde{U}_{0,1}^3\). Let \(\Gamma_{0,1}^2:= U_{0,1}^2+iV_{0,1}^2,\;\Gamma_{0,1}^3:= \tilde{U}_{0,1}^3+i\tilde{V}_{0,1}^3\), thus \[\begin{align} \partial_t \Gamma_{0,1}^2 - \nu\Delta_{L,0}\Gamma_{0,1}^2 + i\mathcal{R}\Gamma_{0,1}^2 = \mathcal{N}_{0+\neq}(\Gamma_{0,1}^2),\;\partial_t \Gamma_{0,1}^3 - \nu\Delta_{L,0}\Gamma_{0,1}^3 + i\mathcal{R}\Gamma_{0,1}^3 = \mathcal{N}_{0+\neq}(\Gamma_{0,1}^3), \end{align}\] where \[\begin{align} \mathcal{N}_{0+\neq}(\Gamma_{0,1}^2)= \mathcal{N}_{0+\neq}(U_{0,1}^2) + i\mathcal{N}_{0+\neq}(V_{0,1}^2),\;\mathcal{N}_{0+\neq}(\Gamma_{0,1}^3)= \mathcal{N}_{0+\neq}(\tilde{U}_{0,1}^3) + i\mathcal{N}_{0+\neq}(\tilde{V}_{0,1}^2) \end{align}\] are the nonlinear terms. We then decompose \[U_{0,1}^2=U_{0,1}^{2,in}+U_{0,1}^{2,nl},\;\tilde{U}_{0,1}^3=\tilde{U}_{0,1}^{3,in}+\tilde{U}_{0,1}^{3,nl},\;V_{0,1}^2=V_{0,1}^{2,in}+V_{0,1}^{2,nl},\;\tilde{V}_{0,1}^3=\tilde{V}_{0,1}^{3,in}+\tilde{V}_{0,1}^{3,nl}.\]

Proof.To give the nonlinear dispersive estimate of \(U_{0,1}^{2}\) and \(U_{0,1}^{3}\), we only estimate \(\Gamma_{0,1}^2\) and \(\Gamma_{0,1}^3\), since \[U_{0,1}^2=\mathfrak{Re}\Gamma_{0,1}^2,\;\tilde{U}_{0,1}^3=\mathfrak{Re}\Gamma_{0,1}^3,\;V_{0,1}^2=\mathfrak{Im}\Gamma_{0,1}^2,\;\tilde{V}_{0,1}^3=\mathfrak{Im}\Gamma_{0,1}^3.\] Thanks to the Duhamel’s formula, we have \[\Gamma_{0,1}^{k,in}(t)=e^{t\mathcal{L}}\Gamma_{0,1}^k(0),\;\Gamma_{0,1}^{k,nl}(t)=\int_0^t e^{(t-\tau)\mathcal{L}}\mathcal{N}_{0+\neq}(\Gamma_{0,1}^k)(\tau){\rm d}\tau,\;k\in\{2,3\},\] with \(\mathcal{L}\) being the same as in 53 . The estimate of \(\Gamma_{0,1}^{k,in}\) follows directly from the Corollary 14. For estimation of \(\Gamma_{0,1}^{k,nl}\), we mainly divide it into the contribution from single zero frequency and double zero frequency. Specifically. Using Lemma 17, we derive the nonlinear dispersion estimates which is more precise than one obtains by the direct energy method. Recall the system 58 and 59 , we calculate the term \(\Gamma_{0,1}^{k,nl}\) which is is equivalent to requiring the estimation of the following terms. For the estimates of simple zero contributions, one has \[\begin{align} &\|\tilde{U}_{0,1}\cdot\nabla\Gamma_{0,1}^2\|_{H^{N,N+\frac{1}{2}}} + {\|\tilde{U}_{0,1}\cdot\nabla\tilde{U}_{0,1}^3\|_{H^{N,N+\frac{1}{2}}} + \|\partial_z^{-1}\partial_y(\tilde{U}_{0,1}\cdot\nabla V_{0,1}^2)\|_{H^{N,N+\frac{1}{2}}}} \\ &\quad\leq \|\tilde{U}_{0,1}\cdot\nabla\Gamma_{0,1}^2\|_{H^{N,N+\frac{1}{2}}} + \|\tilde{U}_{0,1}\cdot\nabla\tilde{U}_{0,1}^3\|_{H^{N,N+\frac{1}{2}}} + \|\partial_z^{-1}(\partial_y\tilde{U}_{0,1}\cdot\nabla V_{0,1}^2)\|_{H^{N,N+\frac{1}{2}}}\\ &\qquad + \|\partial_z^{-1}(\tilde{U}_{0,1}\cdot\nabla\partial_z\tilde{V}_{0,1}^3)\|_{H^{N,N+\frac{1}{2}}}\\&\quad \lesssim \|(U_{0,1}^2,\tilde{U}_{0,1}^3,V_{0,1}^2,\tilde{V}_{0,1}^3)\|_{W^{2,\infty}}\|\nabla(U_{0,1}^2,\tilde{U}_{0,1}^3,V_{0,1}^2,\tilde{V}_{0,1}^3)\|_{H^{N,N+\frac{1}{2}}} \\&\quad \lesssim q^{-\frac{1}{3}}\left(t^{-\frac{1}{3}}e^{-\nu t}\epsilon_0 + \delta^2\nu^{2c-\frac{2}{3}}\right)\|\nabla(U_{0,1}^2,\tilde{U}_{0,1}^3,V_{0,1}^2,\tilde{V}_{0,1}^3)\|_{H^{N,N+\frac{1}{2}}}, \end{align}\] and \[\begin{align} &\|\nabla_{y,z}\Delta_{L,0}^{-1}(\partial_yU^2_{0,1}\partial_yU^2_{0,1}+\partial_y\tilde{U}^3_{0,1}\partial_zU^2_{0,1})\|_{H^{N,N+\frac{1}{2}}}\\ &\quad\lesssim \|\partial_yU^2_{0,1}\partial_yU^2_{0,1}\|_{H^{N-1,N-\frac{1}{2}}}+\|\partial_y\tilde{U}^3_{0,1}\partial_zU^2_{0,1}\|_{H^{N-1,N-\frac{1}{2}}}\\&\quad \lesssim \|(U^2_{0,1}, \tilde{U}_{0,1}^3)\|_{W^{2,\infty}}(\| U^2_{0,1}\|_{H^{N,N+\frac{1}{2}}} + \|\tilde{U}^3_{0,1}\|_{H^{N,N+\frac{1}{2}}}) \\ &\quad \lesssim q^{-\frac{1}{3}}\left(t^{-\frac{1}{3}}e^{-\nu t}\epsilon_0 + \delta^2\nu^{2c-\frac{2}{3}}\right)\|\nabla(U_{0,1}^2,\tilde{U}_{0,1}^3,V_{0,1}^2,\tilde{V}_{0,1}^3)\|_{H^{N,N+\frac{1}{2}}}. \end{align}\] Setting, \(I\in\{\tilde{U}_0\cdot\nabla\Gamma_0^k,\;\nabla_{y,z}\Delta_{L,0}^{-1}(\partial_yU^2_{0,1}\partial_yU^2_{0,1}+\partial_y\tilde{U}^3_{0,1}\partial_zU^2_{0,1},\partial_z^{-1}\partial_y(\tilde{U}_{0,1}\cdot\nabla V_{0,1}^2)\}\), then \[\begin{align} \int_0^{t} \|e^{(t-\tau)\mathcal{L}}I(\tau)\|_{H^{N,N+\frac{1}{2}}}{\rm d}\tau \lesssim& \int_0^{t} e^{-\nu(t-\tau)}q^{-\frac{1}{3}}\left(\tau^{-\frac{1}{3}}e^{-\nu \tau}\epsilon_0 + \delta^2\nu^{2c-\frac{2}{3}}\right)\|\nabla(U_{0,1}^2,\tilde{U}_{0,1}^3,V_{0,1}^2,\tilde{V}_{0,1}^3)\|_{H^{N,N+\frac{1}{2}}}{\rm d}\tau\\ \lesssim& q^{-\frac{1}{3}}\cdot \delta\nu^c\nu^{-\frac{1}{2}}(\nu^{-\frac{1}{6}}\epsilon_0+ \delta^2\nu^{-\frac{2}{3}}\nu^{2c}\nu^{-\frac{1}{2}})=q^{-\frac{1}{3}}\delta\nu^c\left(\nu^{-\frac{2}{3}}\epsilon_0 + \delta^2\nu^{2c-\frac{5}{3}}\right). \end{align}\] For the nonzero contributions \(\mathcal{N}_{\neq}(U_{0,1}^2,\tilde{U}_{0,1}^3,V_{0,1}^2,\tilde{V}_{0,1}^3)\), we have for \(F\in\{U^1,U^2,U^3,\Theta\}\), \[\begin{align} &\|\mathcal{N}_{\neq}(U_{0,1}^2,\tilde{U}_{0,1}^3,V_{0,1}^2,\tilde{V}_{0,1}^3)\|_{H^{N,N+\frac{1}{2}}}\\ &\quad\lesssim \|U_{\neq,1}\|_{W^{1,\infty}}\|\nabla_L F_{\neq,1}\|_{H^{N,N+\frac{1}{2}}} + \| U_{\neq,1}\|_{H^{N,N+\frac{1}{2}}}\|\nabla_L F_{\neq,1}\|_{L^{\infty}}\\ &\quad\lesssim \|\mathcal{A}U_{\neq,1}\|_{H^{N}}\|\nabla_L \mathcal{A}F_{\neq,1}\|_{H^{N}}, \end{align}\] and thus we obtain \[\begin{align} \int_0^{t} \|e^{(t-\tau)\mathcal{L}}\mathcal{N}_{\neq}(U_{0,1}^2,\tilde{U}_{0,1}^3,V_{0,1}^2,\tilde{V}_{0,1}^3)(\tau)\|_{H^{N,N+\frac{1}{2}}}{\rm d}\tau \lesssim \int_0^{t} \|\mathcal{A}U_{\neq,1}\|_{H^{N}}\|\nabla_L \mathcal{A}F_{\neq,1}\|_{H^{N}}{\rm d}\tau \lesssim \delta\nu^c\cdot \delta\nu^{c-\frac{2}{3}}, \end{align}\] Thus we finish the proof of Lemma 18.\(\square\)

We then establish the nonlinear estimates of \(\overline{U^1_{0,1}},\overline{U^3_{0,1}}\) and \(\overline{\Theta_{0,1}}\).

Proof.The energy estimates tell us that \[\begin{align} &\|\overline{U^1_{0,1}}\|_{H^{N}}^2 + \nu\|\partial_y \overline{U^1_{0,1}}\|_{L^2_t H^{N}}^2\\&\quad \lesssim \|\overline{U^1_{0,1}}(0)\|_{H^{N}}^2 + \nu^{-1}\int_0^{t} \|\overline{U_{0,1}^2\tilde{U}^1_{0,1}}\|_{H^{N}}^2 {\rm d}\tau+ \nu^{-1}\int_0^{t} \|(\overline{U_{\neq,1}^2U^1_{\neq,1}})_0\|_{H^{N}}^2 {\rm d}\tau, \end{align}\] and \[\begin{align} &\|(\overline{U^3_{0,1}},\overline{\Theta_{0,1}})\|_{H^{N}}^2 + \nu\|\partial_y (\overline{U^3_{0,1}},\overline{\Theta_{0,1}})\|_{L^2_t H^{N}}^2 \\ & \quad\lesssim \|(\overline{U^3_{0,1}},\overline{\Theta_{0,1}})(0)\|_{H^{N}}^2 + \nu^{-1}\int_0^{t} \|\overline{U_{0,1}^2\tilde{U}^3_{0,1}}\|_{H^{N}}^2 {\rm d}\tau\\&\qquad+ \nu^{-1}\int_0^{t} (\|(\overline{U_{\neq,1}^2U^3_{\neq,1}})_0\|_{H^{N}}^2 + \|(\overline{U_{\neq,1}^2\Theta_{\neq,1}})_0\|_{H^{N}}^2) {\rm d}\tau. \end{align}\]

On the one hand, the zero modes contributions can be expressed as \[\begin{align} \nu^{-1}\int_0^{t} \|\overline{U_{0,1}^2\tilde{U}_{0,1}^1}\|_{H^{N}}^2 {\rm d}\tau \lesssim& \nu^{-1}\int_0^{t} \left(\|\langle\partial_z\rangle^{\frac{1}{2}}U_{0,1}^{2,in}\|_{W^{N,\infty}}^2 + \|U_{0,1}^{2,nl}\|_{H^{N,N+\frac{1}{2}}}^2\right)\|\tilde{U}_{0,1}^1\|_{H^{N,N+\frac{1}{2}}}^2 {\rm d}\tau \\ \lesssim& q^{-\frac{2}{3}}\delta^2\nu^{2c}\nu^{-1}\int_0^{t} \tau^{-\frac{2}{3}}e^{-2\nu \tau}\|\tilde{U}_{0,1}^1\|_{H^{N,N+\frac{1}{2}}}^2{\rm d}\tau \\ &+\nu^{-1}\int_0^{t} \left(\delta\nu^c(\delta\nu^{c-\frac{2}{3} }) + q^{-\frac{1}{3}} \delta\nu^c \left(\delta\nu^{c-\frac{2}{3}} + \delta^2\nu^{2c-\frac{5}{3}}\right)\right)^2\|\tilde{U}_{0,1}^1\|_{H^{N,N+\frac{1}{2}}}^2 {\rm d}\tau \\ \lesssim& \delta^2\nu^{2c}\delta^4\nu^{4c-\frac{10}{3}} + q^{-\frac{2}{3}}\delta^2\nu^{2c}(\delta^2\nu^{2c-\frac{4}{3}} + \delta^4\nu^{4c-\frac{10}{3}} + \delta^6\nu^{6c-\frac{16}{3}}),\\ \nu^{-1}\int_0^{t} \|\overline{U_{0,1}^2\tilde{U}^3_{0,1}}\|_{H^{N}}^2 {\rm d}\tau\lesssim& \nu^{-1}\int_0^{t} \|\tilde{U}_{0,1}^{2,3}\|_{L^\infty}^2\|\tilde{U}_{0,1}^{2,3}\|_{H^{N,N+\frac{1}{2}}}^2 {\rm d}\tau\\ \lesssim& \nu^{-1}\int_0^{t} q^{-\frac{2}{3}}\delta^2\nu^{2c}\left(\tau^{-\frac{1}{3}}e^{-\nu\tau}+\delta\nu^{c-\frac{2}{3}}\right)^2\|\tilde{U}_{0,1}^{2,3}\|_{H^{N,N+\frac{1}{2}}}^2 {\rm d}\tau\\ \lesssim& q^{-\frac{2}{3}}\delta^2\nu^{2c}(\delta^2\nu^{2c-\frac{4}{3}} + \delta^4\nu^{4c-\frac{10}{3}}). \end{align}\] On the other hand, the nonzero modes contributions can be expressed as \(\mathcal{N}_{\neq}(\overline{U^1_{0,1}},\overline{U^3_{0,1}},\overline{\Theta_{0,1}})\), and one has \[\nu^{-1}\int_0^{t}\mathcal{N}_{\neq}(\overline{U^1_{0,1}},\overline{U^3_{0,1}},\overline{\Theta_{0,1}}){\rm d}\tau\lesssim \nu^{-1}\int_0^{t}\|\mathcal{A}F_{\neq,1}\|_{H^N}^4{\rm d}\tau \lesssim \delta^2\nu^{2c}(\delta^2\nu^{2c-\frac{4}{3}}).\] Thus we finish the proof of Lemma 19.\(\square\)

In the following, we then establish the nonlinear estimates of \(\Lambda_{0,1}\).

Proof.Recall 58 \(_{1}\), the energy estimates tell us that for \(F\in\{U^1,\Theta\}\), \[\begin{align} \|\Lambda_{0,1}\|_{H^{N,N+\frac{1}{2}}}^2 + \nu\|\nabla\Lambda_{0,1}\|_{L^2_t H^{N,N+\frac{1}{2}}}^2 \lesssim& \|\Lambda_{0,1}(t=0)\|_{H^{N,N+\frac{1}{2}}}^2 + \nu^{-1}\int_0^{t} \sum_{F\in\{U^1,\Theta\}}\|\tilde{U}_{0,1}^{2,3}F_{0,1}\|_{H^{N,N+\frac{1}{2}}}^2{\rm d}\tau \\ & + \nu^{-1}\int_0^{t} \left( \|U_{\neq,1}U_{\neq,1}^1\|_{H^{N,N+\frac{1}{2}}}^2 + \|U_{\neq,1}\Theta_{\neq,1}\|_{H^{N,N+\frac{1}{2}}}^2\right){\rm d}\tau. \end{align}\] For the zero modes contributions, the estimates are the same as the processing of double-zero modes in Lemma 19 \[\begin{align} \nu^{-1}\int_0^{t} \sum_{F\in\{U^1,\Theta\}}\|\tilde{U}_{0,1}^{2,3}F_{0,1}\|_{H^{N,N+\frac{1}{2}}}^2{\rm d}\tau \lesssim \delta^2\nu^{2c}( \delta^4\nu^{4c-\frac{10}{3}}) + q^{-\frac{2}{3}}\delta^2\nu^{2c}(\delta^2\nu^{2c-\frac{4}{3}} + \delta^4\nu^{4c-\frac{10}{3}} +\delta^6 \nu^{6c-\frac{16}{3}}). \end{align}\] We donate the nonzero modes contributions as \(\mathcal{N}_{\neq}(\Lambda_{0,1})\). By ?? , we have \[\nu^{-1}\int_0^{t} \mathcal{N}_{\neq}(\Lambda_{0,1}){\rm d}\tau \lesssim \nu^{-1}\int_0^{t}\|\mathcal{A}F_{\neq,1}\|_{H^N}^4{\rm d}\tau \lesssim \delta^2\nu^{2c}(\delta^2\nu^{2c-\frac{4}{3}}).\] Thus, we obtain the estimation of \(\Lambda_{0,1}\). By using 50 and Lemma 18, we can return to the estimates of \(\tilde{U}^1_{0,1}\) and \(\tilde{\Theta}_{0,1}\). We finish the proof of Lemma 20. \(\square\)

4.3 Energy estimates on \(Q_{\neq,1}, K_{\neq,1}\) and \(H_{\neq,1}\)↩︎

In this section, we aim to prove that under the bootstrap hypotheses of Proposition 15, the estimates on \(U_{\neq,1}\), and \(\Theta_{\neq,1}\) hold (i.e., ?? ), with 100 replaced by 50 on the right-hand side. To this end, we present some inequalities, which are frequently employed in non-zero modes estimates.

Proof.For any \(3<r\leq N\), by Corollary 9, it holds

\[\begin{align} \|\mathcal{A}|\nabla_L|^{\frac{1}{2}}(Q_{\neq,1},K_{\neq,1},H_{\neq,1})\|_{L^2_tH^r} &\lesssim \|\mathcal{A}(Q_{\neq,1},K_{\neq,1},H_{\neq,1})\|_{L^2_tH^r}^{\frac{1}{2}}\|\mathcal{A}\nabla_L (Q_{\neq,1},K_{\neq,1},H_{\neq,1})\|_{L^2_tH^r}^{\frac{1}{2}}\\& \lesssim \nu^{-\frac{1}{3}}(\int_0^t F_{\neq,1}{\rm d}\tau)^{\frac{1}{2}},\\ \|\mathcal{A}|\nabla_{L,h}|^{-\frac{1}{2}}(Q_{\neq,1},K_{\neq,1},H_{\neq,1})\|_{L^2_tH^r} &\lesssim \|\mathcal{A}(Q_{\neq,1},K_{\neq,1},H_{\neq,1})\|_{L^2_tH^r}^{\frac{\kappa}{1+\kappa}}\left\|\mathcal{A}\sqrt{-\frac{\dot{M_7}}{M_7}} (Q_{\neq,1},K_{\neq,1},H_{\neq,1})\right\|_{L^2_tH^r}^{\frac{1}{1+\kappa}}\\ &\lesssim \|\mathcal{A}(Q_{\neq,1},K_{\neq,1},H_{\neq,1})\|_{L^2_tH^r}^{\frac{\kappa}{1+\kappa}}\left\|\mathcal{A}\sqrt{-\frac{\dot{M}}{M}} (Q_{\neq,1},K_{\neq,1},H_{\neq,1})\right\|_{L^2_tH^r}^{\frac{1}{1+\kappa}}\\ &\lesssim \nu^{-\frac{\kappa}{6(1+\kappa)}}(\int_0^t F_{\neq,1}{\rm d}\tau)^{\frac{1}{2}}, \end{align}\] which gives ?? and ?? . The proof of ?? and ?? is similar. Direct calculation gives \[\begin{align} \|\widehat{\nabla_{x,z} (U_{\neq},\Theta_{\neq})}\|_{L^\infty_tL^1}&\lesssim \|\langle k,\eta,l\rangle^r\widehat{\nabla_{x,z} (U_{\neq},\Theta_{\neq})}\|_{L^\infty_tL^2}\|\langle k,\eta,l\rangle^{-r}\|_{L^2}\\ &\lesssim \|\mathcal{A}(Q_{\neq},K_{\neq},H_{\neq})\|_{L^\infty_tH^{r+1}},\\ \|\widehat{\nabla_L (U_{\neq},\Theta_{\neq})}\|_{L^\infty_tL^1}&\lesssim \|e^{\lambda\nu^{\frac{1}{3}}t}\widehat{\nabla(U_{\neq},\Theta_{\neq})}\|_{L^\infty_tL^1}\|\langle t\rangle e^{-\lambda\nu^{\frac{1}{3}}t}\|_{L^\infty_t} \\ &\lesssim \nu^{-\frac{1}{3}}\|\mathcal{A}(Q_{\neq},K_{\neq},H_{\neq})\|_{L^\infty_tH^r}. \end{align}\] Using the relationships between variables, such as 38 , 39 , 43 and 44 , we have \[\begin{align} &\|\nabla_L U_{\neq,1}^2\|_{L^\infty_tL^\infty}\lesssim \|\mathcal{A}(Q_{\neq,1},K_{\neq,1})\|_{L^\infty_tH^r},\\ &\|\widehat{U^2_{\neq}}\|_{L^2_tL^1}\lesssim \|\mathcal{A}(Q_{\neq},K_{\neq})\|_{L^\infty_tH^r}\|\langle t\rangle^{-1}\|_{L^2_t}\lesssim \|\mathcal{A}(Q_{\neq},K_{\neq})\|_{L^\infty_tH^r},\\ &\|p^{\frac{1}{4}}\widehat{mH_{\neq}}\|_{L^\infty_tL^1}\lesssim \||\nabla|^{\frac{1}{2}}\widehat{\mathcal{A}H_{\neq}}\|_{L^\infty_tL^1}\|t^{\frac{1}{2}}e^{-\lambda\nu^{\frac{1}{3}}t}\|_{L^\infty_t}\lesssim\nu^{-\frac{1}{6}}\|\mathcal{A} H_{\neq}\|_{L^\infty_tH^r},\\ &\||k,l|^{\frac{3}{2}}p_h^{\frac{1}{4}}\widehat{K_{\neq}}\|_{L^2_tL^1}\lesssim \||k,l|p^{\frac{1}{2}}\widehat{mK_{\neq}}\|_{L^2_tL^1}\lesssim \|\mathcal{A} \nabla_L K_{\neq,1}\|_{L^2_tH^r}. \end{align}\] Thus, we finish the proof of Lemma 21 \(\square\)

Proof.According to the linear analysis, an energy estimate yields \[\begin{align}\label{kkk} &\|\mathcal{A}(Q_{\neq,1},K_{\neq,1},H_{\neq,1})\|_{H^N}^2 + \nu\|\mathcal{A}\nabla_L(Q_{\neq,1},K_{\neq,1},H_{\neq,1})\|_{L^2_tH^N}^2 + \left\|\mathcal{A}\sqrt{-\frac{\dot{M}}{M}}(Q_{\neq,1},K_{\neq,1},H_{\neq,1})\right\|_{L^2_tH^N}^2\\ &\quad\lesssim \frac{2\sqrt{\beta(\beta-1)}+1}{2\sqrt{\beta(\beta-1)}-1}\|\mathcal{A}(Q_{\neq,1},K_{\neq,1},H_{\neq,1})(t=0)\|_{H^N}^2 +\frac{\nu^{\frac{1}{3}}}{16}\|\mathcal{A}(Q_{\neq,1},K_{\neq,1},H_{\neq,1})\|_{L^2_tH^N}^2\\ &\qquad+\frac{4\sqrt{\beta(\beta-1)}}{2\sqrt{\beta(\beta-1)}-1}\int_0^t\mathcal{N}_{\mathcal{A}1}{\rm d}\tau+\frac{4\sqrt{\beta(\beta-1)}}{2\sqrt{\beta(\beta-1)}-1}\int_0^tG\mathcal{N}_{\mathcal{A}2}{\rm d}\tau, \end{align}\qquad{(24)}\] where \[\begin{align} \int_0^t\mathcal{N}_{\mathcal{A}1}{\rm d}\tau &=\int_0^t\big|\langle\mathcal{A}Q_{\neq,1},\mathcal{A}|\nabla_{L}|^{\frac{3}{2}}|\nabla_{L,h}|^{-1}(U^1_{\neq,1}\partial_xU^3_{\neq,1})\rangle_{H^N}\big|{\rm d}\tau\\ &\quad + \int_0^t\big|\langle\mathcal{A}Q_{\neq,1},\mathcal{A}\partial_z|\nabla_{L,h}|^{-1}|\nabla_L|^{-\frac{1}{2}}(\partial_{x,z} U^{1,3}_{\neq,1}\cdot\nabla_{x,z} U^{1,3}_{\neq,1}+2\nabla_{L,h}U_{\neq,1}^2\partial_y^LU_{\neq,1}^{1,2})\rangle_{H^N}\big|{\rm d}\tau\\ &\quad + \int_0^t\big|\langle\mathcal{A}K_{\neq,1},\mathcal{A}\sqrt{\frac{\beta}{\beta-1}}|\nabla_L|^{\frac{1}{2}}|\nabla_{L,h}|^{-1}\big[\partial_xU_{\neq,1}\cdot\nabla_L U^2_{\neq,1}-\partial_y^LU_{\neq,1}\cdot\nabla_LU^1_{\neq,1} \\ &\quad+U_{\neq,1}^{1,3}\cdot\nabla_{x,z}W^3_{\neq,1}+U_{\neq,1}^{2}\partial_y^L W^3_{\neq,1}\big]\rangle_{H^N}\big|{\rm d}\tau\\&\quad +\int_0^t\big|\langle\mathcal{A}H_{\neq,1},\mathcal{A}|\nabla_L|^{\frac{1}{2}}\big[U_{\neq,1}^{1,3}\cdot\nabla_{x,z}\Theta_{\neq,1}+U_{\neq,1}^2\partial_y^L\Theta_{\neq,1}\big]\rangle_{H^N}\big|{\rm d}\tau\\&\quad + \int_0^t\big|\langle\mathcal{A}Q_{\neq,1},\mathcal{A}|\nabla_L|^{\frac{3}{2}}|\nabla_{L,h}|^{-1}\big(U_{0,1}^1\partial_xU_{\neq,1}^3\big)\rangle_{H^N}\big|{\rm d}\tau\\&\quad + \int_0^t\big|\langle\mathcal{A}Q_{\neq,1},\mathcal{A}\partial_z|\nabla_L|^{-\frac{1}{2}}|\nabla_{L,h}|^{-1}\big[2\partial_z U^{1,3}_{0,1}\cdot\nabla_{x,z} U^3_{\neq,1}+2\partial_yU_{0,1}^1\partial_xU_{\neq,1}^2\big]\rangle_{H^N}\big|{\rm d}\tau\\ &\quad + \sqrt{\frac{\beta}{\beta-1}}\int_0^t\big|\langle\mathcal{A}K_{\neq,1},\mathcal{A}|\nabla_L|^{\frac{1}{2}}|\nabla_{L,h}|^{-1}\big[(\partial_xU_{\neq,1}\cdot\nabla U^2_{0,1}-\partial_y^LU_{\neq,1}^2\partial_y U_{0,1}^1-\partial_y U_{0,1}^{1,3}\cdot\nabla_{x,z} U_{\neq,1}^1)\\ &\quad +(U_{0,1}^1\partial_x W_{\neq,1}^3+U_{0,1}^3\partial_z W_{\neq,1}^3)\big]\rangle_{H^N}\big|{\rm d}\tau\\&\quad + \int_0^t\big|\langle\mathcal{A}H_{\neq,1},\mathcal{A}|\nabla_L|^{\frac{1}{2}}\big[U_{0,1}^{1,3}\nabla_{x,z}\Theta_{\neq,1}+U^3_{\neq,1}\partial_z\Theta_{0,1}\big]\rangle_{H^N}\big|{\rm d}\tau\\&=:\sum_{i=1}^4\left[I_i(\neq,\neq)+J_{i}(0,\neq)\right], \end{align}\] and \(\int_0^tG\mathcal{N}_{\mathcal{A}2}{\rm d}\tau\) is the associated nonlinear cross terms, \[\begin{align} \int_0^tG\mathcal{N}_{\mathcal{A}2}{\rm d}\tau=&\int_0^t\langle G\mathcal{A}K_{\neq,1},\text{Nonlinear terms in the equation of }Q_{\neq,1}\rangle_{H^N}{\rm d}\tau\\ &+\int_0^t\langle G\mathcal{A}Q_{\neq,1},\text{Nonlinear terms in the equation of }K_{\neq,1}\rangle_{H^N}{\rm d}\tau. \end{align}\]

Now we are ready to estimate each term on the right-hand side of ?? . We first focus on the interaction between two non-zero frequencies. For \(I_{1}\), using the Plancherel formula, we have \[\begin{align} I_1=&\int_0^t\big|\langle\mathcal{A}Q_{\neq,1},\mathcal{A}|\nabla_{L}|^{\frac{3}{2}}|\nabla_{L,h}|^{-1}(U^1_{\neq,1}\partial_xU^3_{\neq,1})\rangle_{H^N}\big|{\rm d}\tau\\ \leq&\int_0^t\sum_{\substack{k,l\\k',l'}}\iint_{\eta,\eta'}\widehat{\langle\nabla\rangle^N}|\mathcal{A}\widehat{Q_{\neq,1}}(k,\eta,l)|\mathcal{A}p^{\frac{3}{4}}p_h^{-\frac{1}{2}}(k,\eta,l)\widehat{\langle\nabla\rangle^N}|\widehat{U_{\neq,1}^1}(k-k',\eta-\eta',l-l')||k'\widehat{U^3_{\neq,1}}(k',\eta',l')|{\rm d}\eta{\rm d}\eta'{\rm d}\tau\\ \lesssim& \int_0^t\sum_{\substack{k,l\\k',l'}}\iint_{\eta,\eta'}\widehat{\langle\nabla\rangle^{N}}|\mathcal{A}\widehat{Q_{\neq,1}}(k,\eta,l)|e^{\lambda\nu^{\frac{1}{3}}\tau}\frac{|l|^{\frac{3}{2}}}{\sqrt{p_h(k,\eta)}}\widehat{\langle\nabla\rangle^N}|\widehat{U_{\neq,1}^1}(k-k',\eta-\eta',l-l')||k'\widehat{U^3_{\neq,1}}(k',\eta',l')|\\ &+ \nu^{-\frac{1}{6}}\widehat{\langle\nabla\rangle^N}|\mathcal{A}\widehat{Q_{\neq,1}}(k,\eta,l)|e^{\lambda\nu^{\frac{1}{3}}\tau}p^{\frac{1}{4}}(k,\eta)\widehat{\langle\nabla\rangle^N}|\widehat{U_{\neq,1}^1}(k-k',\eta-\eta',l-l')||\widehat{\partial_xU^3_{\neq,1}}(k',\eta',l')|{\rm d}\eta{\rm d}\eta'{\rm d}\tau\\ =:&I_{11}+I_{12}. \end{align}\] where in the second inequality we have used the following fact \[\label{multi1} mp^{\frac{3}{4}}p_h^{-\frac{1}{2}}\lesssim \frac{|l|^{\frac{3}{2}}}{\sqrt{p_h}}+\nu^{-\frac{1}{6}}p^{\frac{1}{4}},\tag{75}\] due to \[\label{6666}\begin{cases} mp^{\frac{3}{4}}p_h^{-\frac{1}{2}}\lesssim \sup(m)p_h^{\frac{1}{4}}\lesssim\min\{\nu^{-\frac{1}{6}},\frac{p_h^{\frac{1}{4}}}{(k^2+l^2)^{\frac{1}{4}}}\}p_h^{\frac{1}{4}}\lesssim p_h^{\frac{1}{2}}\lesssim p^{\frac{1}{2}}\langle l\rangle^{\frac{1}{2}},\;\text{if }l^2\leq p_h,\\ mp^{\frac{3}{4}}p_h^{-\frac{1}{2}}\lesssim\frac{|l|^{\frac{3}{2}}}{\sqrt{p_h}}\lesssim |l|^{\frac{3}{2}}\lesssim p^{\frac{1}{2}}\langle l\rangle^{\frac{1}{2}},\;\text{if }l^2\geq p_h. \end{cases}\tag{76}\] For \(I_{11}\), by the fact that \(\sqrt{|k'|}\leq \sqrt{|k-k'|}+\sqrt{|k|}\), we have \[\begin{align} I_{11}\leq&\int_0^t\sum_{\substack{k,l\\k',l'}}\iint_{\eta,\eta'}\widehat{\langle\nabla\rangle^{N}}|l||\widehat{\mathcal{A}Q_{\neq,1}}(k,\eta,l)|e^{\lambda\nu^{\frac{1}{3}}\tau}\widehat{\langle\nabla\rangle^{N}}\frac{|l|^{\frac{1}{2}}}{\sqrt{p_h(k,\eta)}}|\widehat{U_{\neq,1}^1}(k-k',\eta-\eta',l-l')|\\ &\times|k'||\widehat{U_{\neq,1}^3}(k',\eta',l')|{\rm d}\eta'{\rm d}\eta{\rm d}\tau \\ \leq& \int_0^t\sum_{\substack{k,l\\k',l'}}\iint_{\eta,\eta'}\widehat{\langle\nabla\rangle^{N}}|l||\widehat{\mathcal{A}Q_{\neq,1}}(k,\eta,l)|e^{\lambda\nu^{\frac{1}{3}}\tau}\widehat{\langle\nabla\rangle^{N}}\frac{|l|^{\frac{1}{2}}}{\sqrt{p_h(k,\eta)}}\sqrt{|k-k'|}|\widehat{U_{\neq,1}^1}(k-k',\eta-\eta',l-l')|\\ & \times\sqrt{|k'|}|\widehat{U_{\neq,1}^3}(k',\eta',l')|{\rm d}\eta'{\rm d}\eta{\rm d}\tau + \int_0^t\sum_{\substack{k,l\\k',l'}}\iint_{\eta,\eta'}\widehat{\langle\nabla\rangle^{N}}|l||\widehat{\mathcal{A}Q_{\neq,1}}(k,\eta,l)|e^{\lambda\nu^{\frac{1}{3}}\tau}\widehat{\langle\nabla\rangle^{N}}\frac{|l|^{\frac{1}{2}}\sqrt{|k|}}{\sqrt{p_h(k,\eta)}}\\ & \times|\widehat{U_{\neq,1}^1}(k-k',\eta-\eta',l-l')|\sqrt{|k'|}|\widehat{U_{\neq,1}^3}(k',\eta',l')|{\rm d}\eta'{\rm d}\eta{\rm d}\tau\\ \lesssim& \|\mathcal{A}\nabla_L Q_{\neq,1}\|_{L^2_tH^N}\left(\left\|e^{\lambda\nu^{\frac{1}{3}}\tau}|l|^{\frac{1}{2}}\big(\widehat{U_{\neq,1}^1}*\widehat{|\partial_x|^{\frac{1}{2}}U_{\neq,1}^3}\big)\right\|_{L^2_tH^N} + \left\|e^{\lambda\nu^{\frac{1}{3}}\tau}|l|^{\frac{1}{2}}\big(\widehat{|\partial_x|^{\frac{1}{2}}U^1_{\neq,1}}*\widehat{|\partial_x|^{\frac{1}{2}}U^3_{\neq,1}}\big)\right\|_{L^2_tH^N}\right)\\ \lesssim& \|\mathcal{A}\nabla_L Q_{\neq,1}\|_{L^2_tH^N}\|e^{\lambda\nu^{\frac{1}{3}}\tau}|\partial_z|^{\frac{1}{2}}(|\partial_x|^{\frac{1}{2}}U_{\neq,1}^1|\partial_x|^{\frac{1}{2}}U_{\neq,1}^3)\|_{L^2_tH^N}\\ \lesssim& \|\mathcal{A}\nabla_L Q_{\neq,1}\|_{L^2_tH^N}\|\mathcal{A}(Q_{\neq,1},K_{\neq,1})\|_{L^{\infty}_tH^N}\|\mathcal{A}|\nabla_L|^{\frac{1}{2}}(Q_{\neq,1},K_{\neq,1})\|_{L^2_tH^N}\\ \lesssim& \delta\nu^c\nu^{-\frac{1}{2}}\delta\nu^c\delta\nu^c\nu^{-\frac{1}{3}}=\delta^2\nu^{2c}\cdot\delta\nu^{c-\frac{5}{6}}, \end{align}\] where in the last inequality we have used ?? . For \(I_{12}\), we use ?? to obtain \[\begin{align} I_{12}\leq& \nu^{-\frac{1}{6}}\int_0^t\sum_{\substack{k,l\\k',l'}}\iint_{\eta,\eta'}\widehat{\langle\nabla\rangle^N} p^{\frac{1}{4}}(k,\eta,l)|\widehat{\mathcal{A}Q_{\neq,1}(k,\eta,l)}|\widehat{\langle\nabla\rangle^N}|\widehat{U_{\neq,1}^1}(k-k',\eta-\eta',l-l')||\widehat{\partial_xU_{\neq,1}^3}(k',\eta',l')|{\rm d}\eta'{\rm d}\eta{\rm d}\tau\\ \lesssim& \nu^{-\frac{1}{6}}\|\mathcal{A}|\nabla_L|^{\frac{1}{2}}Q_{\neq,1}\|_{L^2_tH^N}\big(\|\widehat{U^1_{\neq,1}}\|_{L^\infty_tL^1}\|\mathcal{A}|\nabla_L|^{\frac{1}{2}}Q_{\neq,1}\|_{L^2_tH^N}+\|\mathcal{A}(Q_{\neq,1},K_{\neq,1})\|_{L^2_tH^N}\|\widehat{\partial_xU^3_{\neq,1}}\|_{L^\infty_tL^1}\big)\\ \lesssim& \nu^{-\frac{1}{6}}\delta\nu^c\nu^{-\frac{1}{3}}(\delta^2\nu^{2c}\nu^{-\frac{1}{3}}+\delta\nu^c\nu^{-\frac{1}{6}}\delta\nu^c)\lesssim \delta^2\nu^{2c}\cdot\delta\nu^{c-\frac{5}{6}}. \end{align}\]

Now we focus on \(I_2\). Note that \[\label{multi4} m|l|p^{-\frac{1}{4}}p_h^{-\frac{1}{2}}\lesssim \frac{\sqrt{|l|}}{\sqrt {p_h}(k^2+l^2)^{\frac{1}{4}}}\cdot\sqrt{|l|},\tag{77}\] by definition of \(m\), using ?? and inviscid damping effect gives \[\begin{align} I_2=&\int_0^t\big|\langle\mathcal{A}Q_{\neq,1},\mathcal{A}\partial_z|\nabla_{L,h}|^{-1}|\nabla_L|^{-\frac{1}{2}}(\partial_{x,z} U^{1,3}_{\neq,1}\cdot\nabla_{x,z} U^{1,3}_{\neq,1}+2\nabla_{L,h}U_{\neq,1}^2\partial_y^LU_{\neq,1}^{1,2})\rangle_{H^N}\big|{\rm d}\tau\\ \leq& 2\int_0^t\sum_{\substack{k,l\\k',l'}}\iint_{\eta,\eta'}\widehat{\langle\nabla\rangle^N}\widehat{\mathcal{A}Q_{\neq,1}}(k,\eta,l)e^{\lambda\nu^{\frac{1}{3}}\tau}m|l|p_h^{-\frac{1}{2}}p^{-\frac{1}{4}}(k,\eta,l)\widehat{\langle\nabla\rangle^N}\big[|\widehat{\nabla_{x,z}U_{\neq,1}^{1,3}}(k-k',\eta-\eta',l-l')|\\ & \times|\widehat{\nabla_{x,z}U_{\neq,1}^{1,3}}(k',\eta',l')|+|\widehat{\nabla_{L,h}U_{\neq,1}^2}(k-k',\eta-\eta',l-l')||\widehat{\partial_y^LU_{\neq,1}^{1,2}}(k',\eta',l')|\big]{\rm d}\eta{\rm d}\eta'{\rm d}\tau\\ \leq& 2\int_0^t\sum_{\substack{k,l\\k',l'}}\iint_{\eta,\eta'}\widehat{\langle\nabla\rangle^N}\frac{|l|^{\frac{1}{2}}}{\sqrt{p_h}(k^2+l^2)^{\frac{1}{4}}}\widehat{\mathcal{A}Q_{\neq,1}}(k,\eta,l)e^{\lambda\nu^{\frac{1}{3}}\tau}\widehat{\langle\nabla\rangle^N}\sqrt{|l|}\big[|\widehat{\nabla_{x,z}U_{\neq,1}^{1,3}}(k-k',\eta-\eta',l-l')|\\ & \times|\widehat{\nabla_{x,z}U_{\neq,1}^{1,3}}(k',\eta',l')|+|\widehat{\nabla_{L,h}U_{\neq,1}^2}(k-k',\eta-\eta',l-l')||\widehat{\partial_y^LU_{\neq,1}^{1,2}}(k',\eta',l')|\big]{\rm d}\eta{\rm d}\eta'{\rm d}\tau\\ \lesssim& \left\|\mathcal{A}\sqrt{-\frac{\dot{M}}{M}}Q_{\neq,1}\right\|_{L^2_tH^N}\||l|^{\frac{1}{2}}\widehat{\nabla_LU_{\neq,1}}\|_{L^\infty_tL^1}\|\mathcal{A}\nabla_L(Q_{\neq,1},K_{\neq,1})\|_{L^2_tH^N} \\ \lesssim&\left\|\mathcal{A}\sqrt{-\frac{\dot{M}}{M}}Q_{\neq,1}\right\|_{L^2_tH^N}\|\mathcal{A}(Q_{\neq,1},K_{\neq,1})\|_{L^\infty_tL^N}\nu^{-\frac{1}{3}}\|\mathcal{A}\nabla_L(Q_{\neq,1},K_{\neq,1})\|_{L^2_tH^N}\\ \lesssim& \delta\nu^c\delta\nu^c\nu^{-\frac{1}{3}}\delta\nu^c\nu^{-\frac{1}{2}}=\delta^2\nu^{2c}\cdot\delta\nu^{c-\frac{5}{6}}. \end{align}\]

For the terms in the equation of \(K_{\neq,1}\), we decompose it into some parts as follows:

\[\begin{align} I_3=&\int_0^t\big|\langle\mathcal{A}K_{\neq,1},\mathcal{A}\sqrt{\frac{\beta}{\beta-1}}|\nabla_L|^{\frac{1}{2}}|\nabla_{L,h}|^{-1}\big[\partial_xU_{\neq,1}\cdot\nabla_L U^2_{\neq,1}-\partial_y^LU_{\neq,1}\cdot\nabla_LU^1_{\neq,1} \\ &+U_{\neq,1}^{1,3}\cdot\nabla_{x,z}W^3_{\neq,1}+U_{\neq,1}^{2}\partial_y^L W^3_{\neq,1}\big]\rangle_{H^N}\big|{\rm d}\tau =:I_{31}+I_{32}+I_{33}+I_{34} \end{align}\] For \(I_{31}\), we use ?? , ?? and the fact that \(\sqrt{|k-k'|}\lesssim \sqrt{|k'|}\sqrt{|k|}\) to obtain \[\begin{align} I_{31}=&\int_0^t\left|\langle\mathcal{A}K_{\neq,1},\mathcal{A}\sqrt{\frac{\beta}{\beta-1}}|\nabla_L|^{\frac{1}{2}}|\nabla_{L,h}|^{-1}\big[\partial_xU_{\neq,1}\cdot\nabla_{L} U^2_{\neq,1}\big]\rangle_{H^N}\right|{\rm d}\tau \\ \leq& \sqrt{\frac{\beta}{\beta-1}}\int_0^t\sum_{\substack{k,l\\k',l'}}\iint_{\eta,\eta'}\widehat{\langle\nabla\rangle^N}\mathcal{A}\widehat{K_{\neq,1}}(k,\eta,l)\widehat{\langle\nabla\rangle^N}\mathcal{A}p^{\frac{1}{4}}p_h^{-\frac{1}{2}}(k,\eta,l,t)|k-k'||\widehat{U_{\neq,1}}(k-k',\eta-\eta',l-l')|\\ & \times |\widehat{\nabla_L U_{\neq,1}^2}(k',\eta',l')|{\rm d}\eta{\rm d}\eta'{\rm d}\tau\\ \lesssim& \sqrt{\frac{\beta}{\beta-1}}\left[\int_0^t\sum_{\substack{k,l\\k',l'}}\iint_{\eta,\eta'}\widehat{\langle\nabla\rangle^N}\mathcal{A}\widehat{K_{\neq,1}}(k,\eta,l)\frac{e^{\lambda\nu^{\frac{1}{3}}\tau}}{(k^2+l^2)^{\frac{1}{4}}}\widehat{\langle\nabla\rangle^N}\left(\sqrt{|k|}\sqrt{|k'|}\sqrt{|k-k'|}\right.\right.\\ & \left.\times |\widehat{U_{\neq,1}}(k-k',\eta-\eta',l-l')||\widehat{\nabla_L U_{\neq,1}^2}(k',\eta',l')|\right){\rm d}\eta{\rm d}\eta'{\rm d}\tau + \int_0^t\sum_{\substack{k,l\\k',l'}}\iint_{\eta,\eta'}\widehat{\langle\nabla\rangle^N}\frac{\sqrt{|l|}}{\sqrt{p_h}(k^2+l^2)^{\frac{1}{4}}}\mathcal{A}\widehat{K_{\neq,1}}(k,\eta,l)\\ & \left.\times e^{\lambda\nu^{\frac{1}{3}}\tau}\widehat{\langle\nabla\rangle^N}\sqrt{|l|}\left(\sqrt{|k|}\sqrt{|k'|}\sqrt{|k-k'|}|\widehat{U_{\neq,1}}(k-k',\eta-\eta',l-l')| |\widehat{\nabla_L U_{\neq,1}^2}(k',\eta',l')|\right){\rm d}\eta{\rm d}\eta'{\rm d}\tau\right]\\ \lesssim& \sqrt{\frac{\beta}{\beta-1}} \left(\|\mathcal{A}K_{\neq,1}\|_{L^2_tH^N}+\left\|\mathcal{A}\sqrt{-\frac{\dot{M}}{M}}K_{\neq,1}\right\|_{L^2_tH^N}^{\frac{1}{2}}\|\mathcal{A}\nabla_L K_{\neq,1}\|_{L^2_tH^N}^{\frac{1}{2}}\right)\\ &\times\left( \|e^{\lambda\nu^{\frac{1}{3}}\tau}\widehat{|\partial_x|^{\frac{1}{2}}U_{\neq,1}}\|_{L^\infty_tL^1}\|\nabla_L|\partial_x|^{\frac{1}{2}} U_{\neq,1}^2\|_{L^2_tH^N}+\|e^{\lambda\nu^{\frac{1}{3}}\tau}\widehat{|\partial_x|^{\frac{1}{2}}U_{\neq,1}}\|_{L^\infty_tH^N}\|\nabla_L|\partial_x|^{\frac{1}{2}} U_{\neq,1}^2\|_{L^2_tL^1}\right)\\ \lesssim& \sqrt{\frac{\beta}{\beta-1}} \delta\nu^{c}\nu^{-\frac{1}{4}}\|\mathcal{A}(Q_{\neq,1},K_{\neq,1})\|_{L^\infty_tH^N}\|\mathcal{A}\nabla_L(Q_{\neq,1},K_{\neq,1})\|_{L^2_tH^N}\\ \lesssim &\sqrt{\frac{\beta}{\beta-1}} \delta\nu^{c}\nu^{-\frac{1}{4}}\delta\nu^c\delta\nu^c\nu^{-\frac{1}{2}}=\delta^2\nu^{2c}\cdot \delta\nu^{c-\frac{3}{4}}. \end{align}\] where we have used the following inequality \[mp^{\frac{1}{4}}p_h^{-\frac{1}{2}}\lesssim (k^2+l^2)^{-\frac{1}{4}}+\frac{\sqrt{|l|}\cdot\sqrt{|l|}}{\sqrt{p_h}(k^2+l^2)^{\frac{1}{4}}},\label{multi2}\tag{78}\] due to \[\label{555} \begin{cases} mp^{\frac{1}{4}}p_h^{-\frac{1}{2}}\lesssim \sup(m)p_h^{-\frac{1}{4}}\lesssim\min\{\nu^{-\frac{1}{6}},\frac{p_h^{\frac{1}{4}}}{(k^2+l^2)^{\frac{1}{4}}}\}p_h^{-\frac{1}{4}}\lesssim 1,\;\text{if }l^2\leq p_h,\\ mp^{\frac{1}{4}}p_h^{-\frac{1}{2}}\lesssim\sup(m)\frac{|l|^{\frac{1}{2}}}{\sqrt{p_h}} \lesssim\min\{\nu^{-\frac{1}{6}},\frac{|l|^{\frac{1}{2}}}{(k^2+l^2)^{\frac{1}{4}}}\}\frac{|l|^{\frac{1}{2}}}{\sqrt{p_h}} \lesssim \langle l\rangle^{\frac{1}{2}},\;\text{if }l^2\geq p_h. \end{cases}\tag{79}\] For \(I_{32}\), we use ?? , ?? , ?? and 78 to obtain \[\begin{align} I_{32}=&\int_0^t\left|\langle\mathcal{A}K_{\neq,1},\mathcal{A}\sqrt{\frac{\beta}{\beta-1}}|\nabla_L|^{\frac{1}{2}}|\nabla_{L,h}|^{-1}\big[\partial_y^LU_{\neq,1}\cdot\nabla_LU^1_{\neq,1}\big]\rangle_{H^N}\right|{\rm d}\tau\\ \leq& \sqrt{\frac{\beta}{\beta-1}}\int_0^t\sum_{\substack{k,l\\k',l'}}\iint_{\eta,\eta'}\widehat{\langle\nabla\rangle^N}\mathcal{A}\widehat{K_{\neq,1}}(k,\eta,l)\widehat{\langle\nabla\rangle^N}\mathcal{A}p^{\frac{1}{4}}p_h^{-\frac{1}{2}}(k,\eta,l)\\ &\times |\eta-\eta'-(k-k')t||\widehat{U_{\neq,1}}(k-k',\eta-\eta',l-l')|p^{\frac{1}{2}}(k',\eta',l')|\widehat{U_{\neq,1}^1}(k',\eta',l')|{\rm d}\eta{\rm d}\eta'{\rm d}\tau\\ \leq& \sqrt{\frac{\beta}{\beta-1}}\int_0^t\sum_{\substack{k,l\\k',l'}}\iint_{\eta,\eta'}\widehat{\langle\nabla\rangle^N}\mathcal{A}\widehat{K_{\neq,1}}(k,\eta,l)\widehat{\langle\nabla\rangle^N}\frac{e^{\lambda\nu^{\frac{1}{3}}\tau}}{(k^2+l^2)^{\frac{1}{4}}}\big[|\eta-\eta'-(k-k')t||\widehat{U_{\neq,1}}(k-k',\eta-\eta',l-l')|\\ &\times p^{\frac{1}{2}}(k',\eta',l')|\widehat{U_{\neq,1}^1}(k',\eta',l')|\big]{\rm d}\eta{\rm d}\eta'{\rm d}\tau + \sqrt{\frac{\beta}{\beta-1}}\int_0^t\sum_{\substack{k,l\\k',l'}}\iint_{\eta,\eta'}\widehat{\langle\nabla\rangle^N}\frac{\sqrt{|l|}}{\sqrt{p_h}(k^2+l^2)^{\frac{1}{4}}}\mathcal{A}\widehat{K_{\neq,1}}(k,\eta,l)\\ & \times e^{\lambda\nu^{\frac{1}{3}}\tau}\widehat{\langle\nabla\rangle^N}\sqrt{|l|}\big[|\eta-\eta'-(k-k')t||\widehat{U_{\neq,1}}(k-k',\eta-\eta',l-l')|p^{\frac{1}{2}}(k',\eta',l')|\widehat{U_{\neq,1}^1}(k',\eta',l')|\big]{\rm d}\eta{\rm d}\eta'{\rm d}\tau\\ \lesssim& \sqrt{\frac{\beta}{\beta-1}}\|\mathcal{A}K_{\neq,1}\|_{L^2_tH^N}\left(\|e^{\lambda\nu^{\frac{1}{3}}\tau}\partial_y^LU_{\neq,1}^{1,3} \nabla_{x,z}U_{\neq,1}^1\|_{L^2_tH^N}+\|e^{\lambda\nu^{\frac{1}{3}}\tau}\partial_y^LU_{\neq,1}^2\partial_y^LU_{\neq,1}^1\|_{L^2_tH^N}\right)\\ &+\left\|\mathcal{A}\sqrt{-\frac{\dot{M}}{M}}K_{\neq,1}\right\|_{L^2_tH^N}\left(\|e^{\lambda\nu^{\frac{1}{3}}\tau}|l|^{\frac{1}{2}}\widehat{\partial_y^LU_{\neq,1}}\|_{L^2_tH^N}\||l|^{\frac{1}{2}}\widehat{\nabla_L U_{\neq,1}^1}\|_{L^\infty_tL^1}\right.\\ & \left. +\|e^{\lambda\nu^{\frac{1}{3}}\tau}|l|^{\frac{1}{2}}\widehat{\partial_y^LU_{\neq,1}}\|_{L^\infty_tL^1}\||l|^{\frac{1}{2}}\widehat{\nabla_L U_{\neq,1}^1}\|_{L^2_tH^N}\right)\\ \lesssim&\sqrt{\frac{\beta}{\beta-1}}\|\mathcal{A}K_{\neq,1}\|_{L^2_tH^N}\left(\|e^{\lambda\nu^{\frac{1}{3}}\tau}\partial_y^LU_{\neq,1}^{1,3} \nabla_{x,z}U_{\neq,1}^1\|_{L^2_tH^N} + \|e^{\lambda\nu^{\frac{1}{3}}\tau}\partial_y^LU_{\neq,1}^2\partial_y^LU_{\neq,1}^1\|_{L^2_tH^N}\right)\\ &+\left\|\mathcal{A}\sqrt{-\frac{\dot{M}}{M}}K_{\neq,1}\right\|_{L^2_tH^N}\|\mathcal{A}\nabla_L(Q_{\neq,1},K_{\neq,1})\|_{L^2_tH^N}\|\widehat{\nabla_L U_{\neq,1}}\|_{L^\infty_tW^{\frac{1}{2},1}}\\ \lesssim&\sqrt{\frac{\beta}{\beta-1}}\|\mathcal{A}K_{\neq,1}\|_{L^2_tH^N}\left(\|e^{\lambda\nu^{\frac{1}{3}}\tau} \partial_y^LU_{\neq,1}^{1,3}\|_{L^2_tH^N}\|\nabla_{x,z}U_{\neq,1}^1\|_{L^\infty_tL^\infty}+\|\partial_y^LU_{\neq,1}^{1,3}\|_{L^\infty_tL^\infty}\|e^{\lambda\nu^{\frac{1}{3}}\tau}\nabla_{x,z}U_{\neq,1}^1\|_{L^2_tH^N}\right.\\ & \left. +\|\partial_y^LU_{\neq,1}^2\|_{L^\infty_tL^\infty}\|e^{\lambda\nu^{\frac{1}{3}}\tau}\partial_y^LU_{\neq,1}^1\|_{L^2_tH^N}+\|e^{\lambda\nu^{\frac{1}{3}}\tau}\partial_y^LU_{\neq,1}^2\|_{L^2_tH^N}\|\partial_y^LU_{\neq,1}^1\|_{L^\infty_tL^\infty}\right) \\ &+ \left\|\mathcal{A}\sqrt{-\frac{\dot{M}}{M}}K_{\neq,1}\right\|_{L^2_tH^N}\|\mathcal{A}\nabla_L(Q_{\neq,1},K_{\neq,1})\|_{L^2_tH^N}\|\widehat{\nabla_L U_{\neq,1}}\|_{L^\infty_tW^{\frac{1}{2},1}}\\ \lesssim&\sqrt{\frac{\beta}{\beta-1}}\|\mathcal{A}K_{\neq,1}\|_{L^2_tH^N}\left(\|\mathcal{A}\nabla_L(Q_{\neq,1},K_{\neq,1})\|_{L^2_tH^N}\|\nabla_{x,z}U_{\neq,1}^1\|_{L^\infty_tL^\infty}\right.\\ & \left.+\|\nabla_LU_{\neq,1}^{1,3}\|_{L^\infty_tL^\infty}\|\mathcal{A}|\nabla_L|^{\frac{1}{2}}(Q_{\neq,1},K_{\neq,1})\|_{L^2_tH^N}+\|\partial_y^LU_{\neq,1}^2\|_{L^\infty_tL^\infty}\|\mathcal{A}\nabla_L(Q_{\neq,1},K_{\neq,1})\|_{L^2_tH^N}\right) \\ &+ \left\|\mathcal{A}\sqrt{-\frac{\dot{M}}{M}}K_{\neq,1}\right\|_{L^2_tH^N}\|\mathcal{A}\nabla_L(Q_{\neq,1},K_{\neq,1})\|_{L^2_tH^N}\|\widehat{\nabla_L U_{\neq,1}}\|_{L^\infty_tW^{\frac{1}{2},1}}\\ \lesssim&\sqrt{\frac{\beta}{\beta-1}} \delta\nu^c\nu^{-\frac{1}{6}}(\delta\nu^c\nu^{-\frac{1}{2}}\delta\nu^c+\delta\nu^c\delta\nu^c\nu^{-\frac{1}{3}}+\delta\nu^c\delta\nu^c\nu^{-\frac{1}{2}})+\delta\nu^c(\delta\nu^c\nu^{-\frac{1}{2}}\delta\nu^c\nu^{-\frac{1}{3}})\\ \lesssim& \sqrt{\frac{\beta}{\beta-1}}\delta^2\nu^{2c}\cdot\delta\nu^{c-\frac{5}{6}}. \end{align}\] For \(I_{33}\), using ?? , ?? , 78 and \(|(k',\eta'-k't)|\leq |(k,\eta-kt)|+|(k-k',\eta-\eta'-(k-k')t)|\) gives \[\begin{align} I_{33}=&\int_0^t\left|\langle\mathcal{A}K_{\neq,1},\mathcal{A}\sqrt{\frac{\beta}{\beta-1}}|\nabla_L|^{\frac{1}{2}}|\nabla_{L,h}|^{-1}\big[U_{\neq,1}^{1,3}\cdot\nabla_{x,z}W^3_{\neq,1}\big]\rangle_{H^N}\right|{\rm d}\tau\\ \leq& \int_0^t\left|\langle\mathcal{A}K_{\neq,1},\mathcal{A}|\nabla_L|^{\frac{1}{2}}|\nabla_{L,h}|^{-1}\big[U_{\neq,1}^{1,3}|\nabla_{x,z}|^{\frac{1}{2}}|\nabla_{L,h}|mK_{\neq,1}\big]\rangle_{H^N}\right|{\rm d}\tau\\ \leq& \int_0^t\sum_{\substack{k,l\\k',l'}}\iint_{\eta,\eta'}\widehat{\langle\nabla\rangle^N}\mathcal{A}\widehat{K_{\neq,1}}(k,\eta,l)\widehat{\langle\nabla\rangle^N}\mathcal{A}p^{\frac{1}{4}}p_h^{-\frac{1}{2}}(k,\eta,l,t)|\widehat{U_{\neq,1}^{1,3}}(k-k',\eta-\eta',l-l')|\\ & \times(k'^2+l'^2)^{\frac{1}{4}}|(k',\eta'-k't)||\widehat{mK_{\neq,1}}(k',\eta',l')|{\rm d}\eta{\rm d}\eta'{\rm d}\tau\\ \leq& \int_0^t\sum_{\substack{k,l\\k',l'}}\iint_{\eta,\eta'}\widehat{\langle\nabla\rangle^N}\mathcal{A}\widehat{K_{\neq,1}}(k,\eta,l)e^{\lambda\nu^{\frac{1}{3}}\tau}\widehat{\langle\nabla\rangle^N}\left(\frac{1}{(k^2+l^2)^{\frac{1}{4}}}+\frac{\sqrt{|l|}}{\sqrt{p_h(k,\eta)}(k^2+l^2)^{\frac{1}{4}}}\sqrt{|l|}\right)\\ & \times|\widehat{U_{\neq,1}^{1,3}}(k-k',\eta-\eta',l-l')|(k'^2+l'^2)^{\frac{1}{4}}\big[|(k,\eta-kt)|\\ & +|(k-k',\eta-\eta'-(k-k')t)|\big]|\widehat{mK_{\neq,1}}(k',\eta',l')|{\rm d}\eta{\rm d}\eta'{\rm d}\tau\\ \lesssim&\|\mathcal{A}\nabla_LK_{\neq,1}\|_{L^2_tH^N}\|U_{\neq,1}^{1,3}\|_{L^\infty_tH^N}\|\mathcal{A}|\nabla_L|^{\frac{1}{2}}K_{\neq}\|_{L^2_tH^N} \\ &+ \|\mathcal{A}K_{\neq,1}\|_{L^2_tH^N}\left(\|e^{\lambda\nu^{\frac{1}{3}}\tau}\nabla_L U_{\neq,1}^{1,3}\|_{L^2_tH^N}\|\widehat{|\nabla_{x,z}|^{\frac{1}{2}}mK_{\neq,1}}\|_{L^\infty_tL^1}+\|e^{\lambda\nu^{\frac{1}{3}}\tau}\widehat{\nabla_L U_{\neq,1}^{1,3}}\|_{L^\infty_tL^1}\||\nabla_{x,z}|^{\frac{1}{2}}mK_{\neq,1}\|_{L^2_tH^N}\right)\\ &+ \left\|\mathcal{A}\sqrt{-\frac{\dot{M}}{M}}K_{\neq,1}\right\|_{L^2_tH^N}\left(\|e^{\lambda\nu^{\frac{1}{3}}\tau}|l|^{\frac{1}{2}}\widehat{\nabla_L U_{\neq,1}^{1,3}}\|_{L^2_tH^N}\|\widehat{|\nabla_{x,z}|^{\frac{1}{2}}mK_{\neq,1}}\|_{L^\infty_tL^1}\right.\\ &\left. +\|\widehat{\nabla_L U_{\neq,1}^{1,3}}\|_{L^\infty_tL^1}\|e^{\lambda\nu^{\frac{1}{3}}\tau}|\nabla_{x,z}|mK_{\neq,1}\|_{L^2_tH^N}\right)\\ &+ \|\mathcal{A}K_{\neq,1}\|_{L^2_tH^N}\left(\|e^{\lambda\nu^{\frac{1}{3}}\tau}|l|^{\frac{1}{2}}\widehat{U_{\neq,1}^{1,3}}\|_{L^2_tH^N}\|\widehat{|\nabla_{x,z}|^{\frac{1}{2}}mK_{\neq,1}}\|_{L^\infty_tL^1} + \|\widehat{U_{\neq,1}^{1,3}}\|_{L^\infty_tL^1}\|e^{\lambda\nu^{\frac{1}{3}}\tau}|\nabla_{x,z}|mK_{\neq,1}\|_{L^2_tH^N}\right)\\ \lesssim&\|\mathcal{A}\nabla_LK_{\neq,1}\|_{L^2_tH^N}\|\mathcal{A}(Q_{\neq,1},K_{\neq,1})\|_{L^\infty_tH^N}\|\mathcal{A}|\nabla_L|^{\frac{1}{2}}K_{\neq}\|_{L^2_tH^N} \\ &+ \left\|\mathcal{A}\sqrt{-\frac{\dot{M}}{M}}K_{\neq,1}\right\|_{L^2_tH^N}\left(\|\mathcal{A}\nabla_L(Q_{\neq,1},K_{\neq,1})\|_{L^2_tH^N}\|\widehat{\nabla_{x,z}mK_{\neq,1}}\|_{L^\infty_tL^1}\right.\\ &\left. +\|p_h^{\frac{1}{2}}\widehat{m(Q_{\neq,1},K_{\neq,1})}\|_{L^\infty_tL^1}\|\mathcal{A}\nabla_LK_{\neq,1}\|_{L^2_tH^N} \right)+\|\mathcal{A}K_{\neq,1}\|_{L^2_tH^N}\\ &\times\left(\|\mathcal{A}\nabla_L(Q_{\neq,1},K_{\neq,1})\|_{L^2_tH^N}\|\widehat{|\nabla_{x,z}|^{\frac{1}{2}}mK_{\neq,1}}\|_{L^\infty_tL^1} +\|p_h^{\frac{1}{2}}\widehat{m(Q_{\neq,1},K_{\neq,1})}\|_{L^\infty_tL^1}\|\mathcal{A}|\nabla_L|^{\frac{1}{2}}K_{\neq,1}\|_{L^2_tH^N} \right)\\ \lesssim& \delta\nu^c\nu^{-\frac{1}{2}}\delta\nu^c\delta\nu^{c-\frac{1}{3}}+\delta\nu^c\nu^{-\frac{1}{6}}(\delta\nu^c\nu^{-\frac{1}{2}}\delta\nu^c+\delta\nu^c\nu^{-\frac{1}{3}}\delta\nu^c\nu^{-\frac{1}{3}})\\ &\\ \lesssim& \delta^2\nu^{2c}\cdot\delta\nu^{c-\frac{5}{6}}. \end{align}\] Similar to the proof of 78 , since \[mp^{\frac{1}{4}}p_h^{-\frac{1}{2}}\lesssim\nu^{-\frac{1}{6}}p_h^{-\frac{1}{4}}+\frac{\sqrt{|l|}}{\sqrt{p_h}},\label{multi3}\tag{80}\] for the last term \(I_{34}\), combining 80 and \(\sqrt{p_h(k',\eta')}\leq\sqrt{p_h(k,\eta)}+\sqrt{p_h(k-k',\eta-\eta')}\) yields \[\begin{align} I_{34}=&\int_0^t\left|\langle\mathcal{A}K_{\neq,1},\mathcal{A}\sqrt{\frac{\beta}{\beta-1}}|\nabla_L|^{\frac{1}{2}}|\nabla_{L,h}|^{-1}\big[U_{\neq,1}^{2}\partial_y^L W^3_{\neq,1}\big]\rangle_{H^N}\right|{\rm d}\tau\\ \leq& \int_0^t\sum_{\substack{k,l\\k',l'}}\iint_{\eta,\eta'}\widehat{\langle\nabla\rangle^N}\mathcal{A}\widehat{K_{\neq,1}}(k,\eta,l)\widehat{\langle\nabla\rangle^N}\mathcal{A}p^{\frac{1}{4}}p_h^{-\frac{1}{2}}(k,\eta,l,t)\big[|\widehat{U_{\neq,1}^2}(k-k',\eta-\eta',l-l')|\\ & \times |\eta'-k't|p^{-\frac{1}{4}}(k',\eta',l',t)\sqrt{p_h(k',\eta')}|\widehat{K_{\neq,1}}(k',\eta',l')|\big]{\rm d}\eta{\rm d}\eta'{\rm d}\tau\\ \leq& \int_0^t\sum_{\substack{k,l\\k',l'}}\iint_{\eta,\eta'}\widehat{\langle\nabla\rangle^N}\mathcal{A}\widehat{K_{\neq,1}}(k,\eta,l)e^{\lambda\nu^{\frac{1}{3}}\tau}\widehat{\langle\nabla\rangle^N}\left(\nu^{-\frac{1}{6}}p_h^{-\frac{1}{4}}(k,\eta)+\frac{\sqrt{|l|}}{\sqrt{p_h(k,\eta)}}\right)\big[|\widehat{U_{\neq,1}^2}(k-k',\eta-\eta',l-l')|\\ &\times |\eta'-k't|p^{-\frac{1}{4}}(k',\eta',l')\big[\sqrt{p_h(k,\eta)}+\sqrt{p_h(k-k',\eta-\eta')}\big]|\widehat{K_{\neq,1}}(k',\eta',l')|\big]{\rm d}\eta{\rm d}\eta'{\rm d}\tau\\ \leq& \nu^{-\frac{1}{6}}\int_0^t\sum_{\substack{k,l\\k',l'}}\iint_{\eta,\eta'}\widehat{\langle\nabla\rangle^N}p_h^{\frac{1}{4}}(k,\eta)\mathcal{A}\widehat{K_{\neq,1}}(k,\eta,l)e^{\lambda\nu^{\frac{1}{3}}\tau}\widehat{\langle\nabla\rangle^N}\big[|\widehat{U_{\neq,1}^2}(k-k',\eta-\eta',l-l')|\\ &\times p_h^{\frac{1}{4}}(k',\eta',l')|\widehat{mK_{\neq,1}}(k',\eta',l')|\big]\nu^{-\frac{1}{6}}{\rm d}\eta{\rm d}\eta'{\rm d}\tau + \nu^{-\frac{1}{6}}\int_0^t\sum_{\substack{k,l\\k',l'}}\iint_{\eta,\eta'}\widehat{\langle\nabla\rangle^N}p_h^{-\frac{1}{4}}(k,\eta)\mathcal{A}\widehat{K_{\neq,1}}(k,\eta,l)\\ &\times e^{\lambda\nu^{\frac{1}{3}}\tau}\widehat{\langle\nabla\rangle^N}\big[\sqrt{p_h(k-k',\eta-\eta')}|\widehat{U_{\neq,1}^2}(k-k',\eta-\eta',l-l')|p_h^{\frac{1}{4}}(k',\eta',l')|\widehat{mK_{\neq,1}}(k',\eta',l')|\big]\nu^{-\frac{1}{6}}{\rm d}\eta{\rm d}\eta'{\rm d}\tau\\ &+\int_0^t\sum_{\substack{k,l\\k',l'}}\iint_{\eta,\eta'}\widehat{\langle\nabla\rangle^N}\sqrt{|l|}\mathcal{A}\widehat{K_{\neq,1}}(k,\eta,l)\widehat{\langle\nabla\rangle^N}\frac{e^{\lambda\nu^{\frac{1}{3}}\tau}}{\sqrt{p_h(k,\eta)}}\sqrt{p_h(k,\eta)}\big[|\widehat{U_{\neq,1}^2}(k-k',\eta-\eta',l-l')|\\ &\times p_h^{\frac{1}{4}}(k',\eta',l')|\widehat{mK_{\neq,1}}(k',\eta',l')|\big]\nu^{-\frac{1}{6}}{\rm d}\eta{\rm d}\eta'{\rm d}\tau + \int_0^t\sum_{\substack{k,l\\k',l'}}\iint_{\eta,\eta'}\widehat{\langle\nabla\rangle^N}p_h^{-\frac{1}{2}}(k,\eta)\mathcal{A}\widehat{K_{\neq,1}}(k,\eta,l)e^{\lambda\nu^{\frac{1}{3}}\tau}\\ &\times \widehat{\langle\nabla\rangle^N}|l|^{\frac{1}{2}}\big[\sqrt{p_h(k-k',\eta-\eta')}|\widehat{U_{\neq,1}^2}(k-k',\eta-\eta',l-l')|p_h^{\frac{1}{4}}(k',\eta',l')|\widehat{mK_{\neq,1}}(k',\eta',l')|\big]\nu^{-\frac{1}{6}}{\rm d}\eta{\rm d}\eta'{\rm d}\tau\\ \lesssim& \nu^{-\frac{1}{6}-\frac{1}{6}}\big(\|\mathcal{A}|\nabla_L|^{\frac{1}{2}}K_{\neq,1}\|_{L^4_tH^N}^2\|\widehat{U_{\neq,1}^2}\|_{L^2_tL^1}+\|\mathcal{A}|\nabla_L|^{\frac{1}{2}}K_{\neq,1}\|_{L^2_tH^N}\|e^{\lambda\nu^{\frac{1}{3}}\tau}U_{\neq,1}^2\|_{L^2_tH^N}\|\widehat{|\nabla_L|^{\frac{1}{2}}mK_{\neq,1}}\|_{L^\infty_tL^1}\big)\\ &+ \nu^{-\frac{1}{6}-\frac{1}{6}}\|\mathcal{A}|\nabla_L|^{-\frac{1}{2}}K_{\neq,1}\|_{L^2_tH^N}\big(\|\widehat{\nabla_{L,h}U_{\neq,1}^2}\|_{L^\infty_tL^1}\|\mathcal{A}|\nabla_L|^{\frac{1}{2}}K_{\neq,1}\|_{L^2_tH^N}\\ & +\|\mathcal{A}|\nabla_L|^{\frac{1}{2}}(Q_{\neq,1},K_{\neq,1})\|_{L^2_tH^N}\|\widehat{|\nabla_L|^{\frac{1}{2}}mK_{\neq,1}}\|_{L^\infty_tL^1}\big)+\nu^{-\frac{1}{6}}\left\|\mathcal{A}\sqrt{-\frac{\dot{M}}{M}}K_{\neq,1}\right\|_{L^2_tH^N}\\ &\times \left(\|\widehat{|\partial_z|^{\frac{1}{2}}\nabla_{L,h}U_{\neq,1}^2}\|_{L^\infty_tL^1}\|\mathcal{A}\nabla_L (Q_{\neq,1},K_{\neq,1})\|_{L^2_tH^N}+\|\mathcal{A}\nabla_L (Q_{\neq,1},K_{\neq,1})\|_{L^2_tH^N}\|\widehat{|\partial_z|^{\frac{1}{2}}|\nabla_L|^{\frac{1}{2}}mK_{\neq,1}}\|_{L^\infty_tL^1}\right)\\ \lesssim& \nu^{-\frac{1}{3}}(\delta^2\nu^{2c}\nu^{-\frac{1}{2}}\delta\nu^c+\delta\nu^c\nu^{-\frac{1}{3}}\delta\nu^c\nu^{-\frac{\kappa}{6(1+\kappa)}}\delta\nu^c\nu^{-\frac{1}{6}})\\&+\nu^{-\frac{1}{3}}\delta\nu^c\nu^{-\frac{\kappa}{6(1+\kappa)}}(\delta\nu^c\delta\nu^c\nu^{-\frac{1}{3}}+\delta\nu^c\nu^{-\frac{1}{3}}\delta\nu^{c-\frac{1}{6}})+\nu^{-\frac{1}{6}}\delta\nu^c(\delta^2\nu^c\nu^c\nu^{-\frac{1}{2}}+\delta^2\nu^c \nu^{-\frac{1}{2}}\nu^c\nu^{-\frac{1}{6}}) \\ \lesssim&\delta^2\nu^{2c}\cdot\delta\nu^{c-\frac{5}{6}-\frac{\kappa}{6(1+\kappa)}}, \end{align}\] here we use ?? , ?? , ?? and ?? .

For the terms in the equation of \(H_{\neq,1}\), using ?? , ?? , ?? and ?? yields that \[\begin{align} I_4=&\int_0^t\left|\langle\mathcal{A}H_{\neq,1},\mathcal{A}|\nabla_L|^{\frac{1}{2}}\big[U_{\neq,1}^{1,3}\cdot\nabla_{x,z}\Theta_{\neq,1}+U_{\neq,1}^2\partial_y^L\Theta_{\neq,1}\big]\rangle_{H^N}\right|{\rm d}\tau \\ \leq& \int_0^t\sum_{\substack{k,l\\k',l'}}\iint_{\eta,\eta'}\widehat{\langle\nabla\rangle^N}\mathcal{A}\widehat{H_{\neq,1}}(k,\eta,l)\widehat{\langle\nabla\rangle^N}\mathcal{A}p^{\frac{1}{4}}(k,\eta,l,t)\big[|\widehat{U_{\neq,1}^{1,3}}(k-k',\eta-\eta',l-l')|(k',l')||\widehat{\Theta_{\neq,1}}(k',\eta',l')|\\ & +|\widehat{U_{\neq,1}^2}(k-k',\eta-\eta',l-l')|\eta'-k't||\widehat{\Theta_{\neq,1}}(k',\eta',l')|\big]{\rm d}\eta{\rm d}\eta'{\rm d}\tau\\ \leq& \int_0^t\sum_{\substack{k,l\\k',l'}}\iint_{\eta,\eta'}\widehat{\langle\nabla\rangle^N}\mathcal{A}p^{\frac{1}{2}}(k,\eta,l)\widehat{H_{\neq,1}}(k,\eta,l)e^{\lambda\nu^{\frac{1}{3}}\tau}\widehat{\langle\nabla\rangle^N}\big[|\widehat{U_{\neq,1}^{1,3}}(k-k',\eta-\eta',l-l')(k'^2+l'^2)^{\frac{1}{2}}p^{-\frac{1}{4}}(k',\eta',l')\\ & \times|\widehat{mH_{\neq,1}}(k',\eta',l')|\frac{p^{\frac{1}{4}}(k',\eta',l')}{(k'^2+l'^2)^{\frac{1}{4}}}\big]{\rm d}\eta{\rm d}\eta'{\rm d}\tau + \int_0^t\sum_{\substack{k,l\\k',l'}}\iint_{\eta,\eta'}\widehat{\langle\nabla\rangle^N}\mathcal{A}p^{\frac{1}{4}}(k,\eta,l)\widehat{H_{\neq,1}}(k,\eta,l)e^{\lambda\nu^{\frac{1}{3}}\tau}\\ &\times\widehat{\langle\nabla\rangle^N}\nu^{-\frac{1}{6}}\big[|\widehat{U_{\neq,1}^2}(k-k',\eta-\eta',l-l')|\eta'-k't|p^{-\frac{1}{4}}(k',\eta',l')|\widehat{mH_{\neq,1}}(k',\eta',l')|\big]\nu^{-\frac{1}{6}}{\rm d}\eta{\rm d}\eta'{\rm d}\tau\\ \lesssim& \|\mathcal{A}|\nabla_L|^{\frac{1}{2}} H_{\neq,1}\|_{L^2_tH^N}\big(\|\mathcal{A}(Q_{\neq,1},K_{\neq,1})\|_{L^\infty_tH^N}\|\mathcal{A}\nabla_L H_{\neq,1}\|_{L^2_tH^N}\\ & +\nu^{-\frac{1}{6}}\|\mathcal{A}|\nabla_L|^{-\frac{1}{2}}(Q_{\neq,1},K_{\neq,1})\|_{L^2_tH^N}\|p^{\frac{1}{4}}\widehat{mH_{\neq,1}}\|_{L^\infty_tL^1}\nu^{-\frac{1}{6}}\big) + (\nu^{-\frac{1}{6}}\|\mathcal{A}|\nabla_L|^{\frac{1}{2}}H_{\neq,1}\|_{L^4_tH^N})^2\|\widehat{U^2_{\neq,1}}\|_{L^2_tL^1}\\ \lesssim& \delta\nu^c\nu^{-\frac{1}{3}}(\delta^2\nu^{2c}\nu^{-\frac{1}{2}}+\nu^{-\frac{1}{6}}\delta\nu^c\nu^{-\frac{\kappa}{6(1+\kappa)}}\delta\nu^c\nu^{-\frac{1}{6}}\delta\nu^c\nu^{-\frac{1}{6}})+(\nu^{-\frac{1}{6}}\delta\nu^c\nu^{-\frac{1}{4}})^2\delta\nu^c\\\lesssim& \delta^2\nu^{2c}\cdot\delta\nu^{c-\frac{5}{6}-\frac{\kappa}{6(1+\kappa)}}. \end{align}\]

We then handle the terms about nonzero mode interacts with simple zero modes in the equation of \(Q_{\neq,1}\). Similar to 75 , we have \(mp^{\frac{3}{4}}p_h^{-\frac{1}{2}}\lesssim \frac{|l|^{\frac{3}{2}}}{\sqrt{p_h}}+\min\{p^{\frac{1}{2}}(k^2+l^2)^{-\frac{1}{4}},\nu^{-\frac{1}{6}}p^{\frac{1}{4}}\}\) which is used to \[\begin{align} J_1=&\int_0^t\big|\langle\mathcal{A}Q_{\neq,1},\mathcal{A}|\nabla_L|^{\frac{3}{2}}|\nabla_{L,h}|^{-1}\big(U_{0,1}^1\partial_xU_{\neq,1}^3\big)\rangle_{H^N}\big|{\rm d}\tau\\ \lesssim& \int_0^t\sum_{\substack{k,l\\l'}}\iint_{\eta,\eta'}\widehat{\langle\nabla\rangle^N}|\widehat{\mathcal{A}Q_{\neq,1}}(k,\eta,l)|\widehat{\langle\nabla\rangle^N}e^{\lambda\nu^{\frac{1}{3}}\tau}mp_h^{-\frac{1}{2}}p^{\frac{3}{4}}(k,\eta,l) \big[|\widehat{U_{0,1}^1}(0,\eta-\eta',l-l')||k||\widehat{U_{\neq,1}^3}(k,\eta',l')|\big]{\rm d}\eta{\rm d}\eta'{\rm d}\tau\\ \lesssim& \int_0^t\sum_{\substack{k,l\\l'}}\iint_{\eta,\eta'}\widehat{\langle\nabla\rangle^N}|\widehat{\mathcal{A}Q_{\neq,1}}(k,\eta,l)|e^{\lambda\nu^{\frac{1}{3}}\tau}\widehat{\langle\nabla\rangle^N}\left(\frac{|l|^{\frac{3}{2}}}{\sqrt{p_h(k,\eta)}}+\frac{\sqrt{p(k,\eta,l)}}{(k^2+l^2)^{\frac{1}{4}}}\right)\\ &\times \big[|\widehat{U_{0,1}^1}(0,\eta-\eta',l-l')||k||\widehat{U_{\neq,1}^3}(k,\eta',l')|\big]{\rm d}\eta{\rm d}\eta'{\rm d}\tau\\ \lesssim& \int_0^t\sum_{\substack{k,l\\l'}}\iint_{\eta,\eta'}\Big\{\widehat{\langle\nabla\rangle^N}|l||\widehat{\mathcal{A}Q_{\neq,1}}(k,\eta,l)|e^{\lambda\nu^{\frac{1}{3}}\tau}\widehat{\langle\nabla\rangle^N}|l|^{\frac{1}{2}}\big[|\widehat{U_{0,1}}(0,\eta-\eta',l-l')|\frac{|k|}{\sqrt{p_h(k,\eta)}}|\widehat{U_{\neq,1}^3}(k,\eta',l')|\big]\\ &+\widehat{\langle\nabla\rangle^N}p^{\frac{1}{2}}(k,\eta,l)|\widehat{\mathcal{A}Q_{\neq,1}}(k,\eta,l)|e^{\lambda\nu^{\frac{1}{3}}\tau}\widehat{\langle\nabla\rangle^N}\big[|\widehat{U_{0,1}^1}(0,\eta-\eta',l-l')|\frac{|k|}{(k^2+l^2)^{\frac{1}{4}}}|\widehat{U_{\neq,1}^3}(k,\eta',l')|\big]\Big\}{\rm d}\eta{\rm d}\eta'{\rm d}\tau\\ \lesssim& \|\mathcal{A}\nabla_LQ_{\neq,1}\|_{L^2_tH^N}\big(\|\widehat{U^1_{0,1}}\|_{L^\infty_tW^{\frac{1}{2},1}}\|\mathcal{A}Q_{\neq,1}\|_{L^2_tH^N}+\|U_{0,1}^1\|_{L^\infty_tH^{N,N+\frac{1}{2}}}\|e^{\lambda\nu^{\frac{1}{3}}\tau}\widehat{|\nabla_{x,z}|^{\frac{1}{2}}U^3_{\neq,1}}\|_{L^2_tL^1}\big)\\ \lesssim& \delta\nu^c\nu^{-\frac{1}{2}}(\delta^2\nu^c\nu^c\nu^{-\frac{1}{6}}+\delta^2\nu^c\nu^c\nu^{-\frac{1}{6}})\\ \lesssim&\delta^2\nu^{2c}\cdot\delta\nu^{c-\frac{2}{3}}, \end{align}\] where we have used the fact that \(k'=k\), ?? and ?? . We also use ?? and \(m|l|p_h^{-\frac{1}{2}}p^{-\frac{1}{4}}\lesssim \frac{\sqrt{|l|}}{\sqrt{p_h}(k^2+l^2)^{\frac{1}{4}}}\cdot\sqrt{|l|}\) to obtain \[\begin{align} J_2=&\int_0^t\big|\langle\mathcal{A}Q_{\neq,1},\mathcal{A}\partial_z|\nabla_L|^{-\frac{1}{2}}|\nabla_{L,h}|^{-1}\big[2\partial_z U^{1,3}_{0,1}\cdot\nabla_{x,z} U^3_{\neq,1}+2\partial_yU_{0,1}^1\partial_xU_{\neq,1}^2\big]\rangle_{H^N}\big|{\rm d}\tau\\ \lesssim& \int_0^t\sum_{\substack{k,l\\l'}}\iint_{\eta,\eta'}\widehat{\langle\nabla\rangle^N}\widehat{\mathcal{A}Q_{\neq,1}}(k,\eta,l)e^{\lambda\nu^{\frac{1}{3}}\tau}m|l|p_h^{-\frac{1}{2}}p^{-\frac{1}{4}}(k,\eta,l)\widehat{\langle\nabla\rangle^N}\big[|l-l'||\widehat{U_{0,1}^{1,3}}(0,\eta-\eta',l-l')|\\ & \times|(k,l')||\widehat{U_{\neq,1}^{3}}(k,\eta',l')|+|k||\widehat{U_{\neq,1}^2}(k,\eta-\eta',l-l')||\eta'|\widehat{U_{0,1}^{1}}(0,\eta',l')\big]{\rm d}\eta{\rm d}\eta'{\rm d}\tau\\ \lesssim& \int_0^t\sum_{\substack{k,l\\l'}}\iint_{\eta,\eta'}\widehat{\langle\nabla\rangle^N}\sqrt{-\frac{\dot{M}}{M}}\widehat{\mathcal{A}Q_{\neq,1}}(k,\eta,l)|l|^{\frac{1}{2}}e^{\lambda\nu^{\frac{1}{3}}\tau}\widehat{\langle\nabla\rangle^N}\big[|l-l'||\widehat{U_{0,1}^{1,3}}(0,\eta-\eta',l-l')|\\ & \times|(k,l')||\widehat{U_{\neq,1}^{3}}(k,\eta',l')|+|k||\widehat{U_{\neq,1}^2}(k,\eta-\eta',l-l')||\eta'|\widehat{U_{0,1}^{1}}(0,\eta',l')\big]{\rm d}\eta{\rm d}\eta'{\rm d}\tau\\ \lesssim& \left\|\mathcal{A}\sqrt{-\frac{\dot{M}}{M}}Q_{\neq,1}\right\|_{L^2_tH^N}\big(\|\widehat{|\partial_z|^{\frac{1}{2}}\nabla U_{0,1}}\|_{L^\infty_tL^1}\|\mathcal{A}\nabla_L(Q_{\neq,1},K_{\neq,1})\|_{L^2_tH^N} \\ &+ \|\nabla U_{0,1}\|_{L^2_tH^{N,N+\frac{1}{2}}}\|e^{\lambda\nu^{\frac{1}{3}}\tau}\widehat{|\partial_z|^{\frac{1}{2}}\nabla_{x,z} U_{\neq,1}}\|_{L^\infty_tL^1}\big) \\ \lesssim& \delta\nu^c(\delta^2\nu^c\nu^c\nu^{-\frac{1}{2}}+\delta^2\nu^c\nu^{-\frac{1}{2}}\nu^c)\\ \lesssim& \delta^2\nu^{2c}\cdot\delta\nu^{c-\frac{1}{2}}. \end{align}\] For the terms about nonzero mode interacts with simple zero modes in the equation of \(K_{\neq,1}\), one has \[\begin{align} J_3=&\sqrt{\frac{\beta}{\beta-1}}\int_0^t\big|\langle\mathcal{A}K_{\neq,1},\mathcal{A}|\nabla_L|^{\frac{1}{2}}|\nabla_{L,h}|^{-1}\big[(\partial_xU_{\neq,1}\cdot\nabla U^2_{0,1}-\partial_y^LU_{\neq,1}^2\partial_y U_{0,1}^1-\partial_y U_{0,1}^{1,3}\cdot\nabla_{x,z} U_{\neq,1}^1)\\ & +(U_{0,1}^1\partial_x W_{\neq,1}^3+U_{0,1}^3\partial_z W_{\neq,1}^3)\big]\rangle_{H^N}\big|{\rm d}\tau=:J_{31}+J_{32} \end{align}\] For \(J_{31}\), we use ?? , ?? and 78 to obtain \[\begin{align} J_{31}=&\sqrt{\frac{\beta}{\beta-1}}\int_0^t\big|\langle\mathcal{A}K_{\neq,1},\mathcal{A}|\nabla_L|^{\frac{1}{2}}|\nabla_{L,h}|^{-1}\big[(\partial_xU_{\neq,1}\cdot\nabla U^2_{0,1}-\partial_y^LU_{\neq,1}^2\partial_y U_{0,1}^1-\partial_y U_{0,1}^{1,3}\cdot\nabla_{x,z} U_{\neq,1}^1)\big]\rangle_{H^N}\big|{\rm d}\tau\\ \leq& \sqrt{\frac{\beta}{\beta-1}}\int_0^t\sum_{\substack{k,l\\l'}}\iint_{\eta,\eta'}\widehat{\langle\nabla\rangle^N}\mathcal{A}\widehat{K_{\neq,1}}(k,\eta,l)\mathcal{A}p^{\frac{1}{4}}p_h^{-\frac{1}{2}}(k,\eta,l)\widehat{\langle\nabla\rangle^N}\\ &\times\big[|k||\widehat{U_{\neq,1}}(k,\eta-\eta',l-l')||(0,\eta',l')||\widehat{U_{0,1}^2}(0,\eta',l')|+|\eta-\eta'-kt||\widehat{U_{\neq,1}^2}(k,\eta-\eta',l-l')||\eta'||\widehat{U_{0,1}^1}(0,\eta',l')| \\ &+ |\eta-\eta'|\widehat{U_{0,1}^{1,3}}(0,\eta-\eta',l-l')||(k,l')||\widehat{U^1_{\neq,1}}(k,\eta',l')|\big]{\rm d}\eta{\rm d}\eta'{\rm d}\tau\\ \leq& \sqrt{\frac{\beta}{\beta-1}}\int_0^t\sum_{\substack{k,l\\l'}}\iint_{\eta,\eta'}\widehat{\langle\nabla\rangle^N}\left(\frac{1}{(k^2+l^2)^{\frac{1}{4}}}+\sqrt{-\frac{\dot{M}}{M}}\right)\mathcal{A}\widehat{K_{\neq,1}}(k,\eta,l)|l|^{\frac{1}{2}}e^{\lambda\nu^{\frac{1}{3}}\tau}\widehat{\langle\nabla\rangle^N}\\ &\times\big[|k||\widehat{U_{\neq,1}}(k,\eta-\eta',l-l')||(0,\eta',l')||\widehat{U_{0,1}^2}(0,\eta',l')|+|\eta-\eta'-kt||\widehat{U_{\neq,1}^2}(k,\eta-\eta',l-l')||\eta'||\widehat{U_{0,1}^1}(0,\eta',l')| \\ & + |\eta-\eta'|\widehat{U_{0,1}^{1,3}}(0,\eta-\eta',l-l')||(k,l')||\widehat{U^1_{\neq,1}}(k,\eta',l')|\big]{\rm d}\eta{\rm d}\eta'{\rm d}\tau\\ \lesssim&\sqrt{\frac{\beta}{\beta-1}} (\|\mathcal{A}K_{\neq,1}\|_{L^2_tH^N}+\left\|\mathcal{A}\sqrt{-\frac{\dot{M}}{M}}K_{\neq,1}\right\|_{L^2_tH^N})\big(\|\widehat{U_{0,1}}\|_{L^\infty_tW^{2,1}}\|\mathcal{A}\nabla_L(Q_{\neq,1},K_{\neq,1})\|_{L^2_tH^N}\\ & +\|\nabla U_{0,1}\|_{L^2_tH^{N,N+\frac{1}{2}}}\|e^{\lambda\nu^{\frac{1}{3}}\tau}\widehat{|\partial_z|^{\frac{1}{2}}\nabla_{x,z}U^1_{\neq,1}}\|_{L^\infty_tL^1}+\|\nabla U_{0,1}\|_{L^2_tH^{N,N+\frac{1}{2}}}\|e^{\lambda\nu^{\frac{1}{3}}\tau}\widehat{|\partial_z|^{\frac{1}{2}}\nabla_LU^2_{\neq,1}}\|_{L^\infty_tL^1}\big)\\ \lesssim&\sqrt{\frac{\beta}{\beta-1}} \delta(\nu^c\nu^{-\frac{1}{6}}+\nu^c)\delta^2(\nu^c\nu^c\nu^{-\frac{1}{2}}+\nu^c\nu^{-\frac{1}{2}}\nu^c+\nu^c\nu^{-\frac{1}{2}}\nu^c)\\ \lesssim&\sqrt{\frac{\beta}{\beta-1}} \delta^2\nu^{2c}\cdot\delta\nu^{c-\frac{2}{3}}. \end{align}\] For \(J_{32}\), we use ?? and ?? to obtain \[\begin{align} J_{32}=&\sqrt{\frac{\beta}{\beta-1}}\int_0^t\big|\langle\mathcal{A}K_{\neq,1},\mathcal{A}|\nabla_L|^{\frac{1}{2}}|\nabla_{L,h}|^{-1}\big[U_{0,1}^1\partial_x W_{\neq,1}^3+U_{0,1}^3\partial_z W_{\neq,1}^3\big]\rangle_{H^N}\big|{\rm d}\tau\\ \leq& \int_0^t\sum_{\substack{k,l\\l'}}\iint_{\eta,\eta'}\widehat{\langle\nabla\rangle^N}\widehat{\mathcal{A}K_{\neq,1}}(k,\eta,l)\mathcal{A}p^{\frac{1}{4}}p_h^{-\frac{1}{2}}(k,\eta,l,t)\widehat{\langle\nabla\rangle^N}\big[|\widehat{U_{0,1}}(0,\eta-\eta',l-l')||k|p^{-\frac{1}{4}}(k,\eta',l')p_h^{\frac{1}{2}}(k,\eta')\\ &\times |\widehat{K_{\neq,1}}(k,\eta',l')|+|\widehat{U_{0,1}^3}(0,\eta-\eta',l-l')||l'|p^{-\frac{1}{4}}(k,\eta',l')p_h^{\frac{1}{2}}(k,\eta')|\widehat{K_{\neq,1}}(k,\eta',l')| \big]{\rm d}\eta{\rm d}\eta'{\rm d}\tau\\ \leq& \int_0^t\sum_{\substack{k,l\\l'}}\iint_{\eta,\eta'}\widehat{\langle\nabla\rangle^N}\left(\frac{1}{(k^2+l^2)^{\frac{1}{4}}}+\frac{\sqrt{|l|}\cdot\sqrt{|l|}}{\sqrt{p_h(k,\eta)}(k^2+l^2)^{\frac{1}{4}}}\right)\widehat{\mathcal{A}K_{\neq,1}}(k,\eta,l)|e^{\lambda\nu^{\frac{1}{3}}\tau}|\widehat{\langle\nabla\rangle^N}\big[|\widehat{U_{0,1}}(0,\eta-\eta',l-l')|\\ &\times |k|p^{-\frac{1}{4}}(k,\eta',l')p_h^{\frac{1}{2}}(k,\eta')|\widehat{K_{\neq,1}}(k,\eta',l')|\\ &+|\widehat{U_{0,1}^3}(0,\eta-\eta',l-l')||l'|p^{-\frac{1}{4}}(k,\eta',l')p_h^{\frac{1}{2}}(k,\eta')|\widehat{K_{\neq,1}}(k,\eta',l')| \big]{\rm d}\eta{\rm d}\eta'{\rm d}\tau\\ \lesssim&\left(\|\mathcal{A}K_{\neq,1}\|_{L^2_tH^N}+\left\|\mathcal{A}\sqrt{-\frac{\dot{M}}{M}}K_{\neq,1}\right\|_{L^2_tH^N}\right)\big(\|U_{0,1}\|_{L^\infty_tH^{N,N+\frac{1}{2}}}\|e^{\lambda\nu^{\frac{1}{3}}\tau}|l|^{\frac{1}{2}}p_h^{\frac{1}{2}}\widehat{mK_{\neq,1}}\|_{L^2_tL^1}\\ &+\|U_{0,1}\|_{L^\infty_tH^{N,N+\frac{1}{2}}}\|e^{\lambda\nu^{\frac{1}{3}}\tau}|l|^{\frac{1}{2}}p_h^{\frac{1}{2}}\widehat{|\nabla_{x,z}|mK_{\neq,1}}\|_{L^2_tL^1}\big)+\|\mathcal{A}\nabla_LK_{\neq,1}\|_{L^2_tH^N}\|\widehat{U_{0,1}}\|_{L^\infty_tW^{2,1}}\|\mathcal{A}K_{\neq,1}\|_{L^2_tH^N}\\ \lesssim& \delta(\nu^c\nu^{-\frac{1}{6}}+\nu^c)\delta^2(\nu^{2c}\nu^{-\frac{1}{2}})+\delta\nu^c\nu^{-\frac{1}{2}}\delta^2\nu^c\nu^c\nu^{-\frac{1}{6}}\\ \lesssim&\delta^2\nu^{2c}\cdot\delta\nu^{c-\frac{2}{3}}. \end{align}\] For the terms about nonzero mode interacts with simple zero modes in the equation of \(H_{\neq,1}\), we get \[\begin{align} J_4=&\int_0^t\big|\langle\mathcal{A}H_{\neq,1},\mathcal{A}|\nabla_L|^{\frac{1}{2}}\big[U_{0,1}^{1,3}\nabla_{x,z}\Theta_{\neq,1}+U^3_{\neq,1}\partial_z\Theta_{0,1}\big]\rangle_{H^N}\big|{\rm d}\tau \\ \leq& \int_0^t\sum_{\substack{k,l\\l'}}\iint_{\eta,\eta'}\widehat{\langle\nabla\rangle^N}\widehat{\mathcal{A}H_{\neq,1}}(k,\eta,l)\mathcal{A}p^{\frac{1}{4}}(k,\eta,l)\widehat{\langle\nabla\rangle^N}\big[|\widehat{U_{0,1}^{1,3}}(0,\eta-\eta',l-l')||(k,l')||\widehat{\Theta_{\neq,1}}(k,\eta',l')|\\ & +|\widehat{U_{\neq,1}^3}(k,\eta-\eta',l-l')|l'||\widehat{\Theta_{0,1}}(0,\eta',l')|\big]{\rm d}\eta{\rm d}\eta'{\rm d}\tau\\ \leq& \int_0^t\sum_{\substack{k,l\\l'}}\iint_{\eta,\eta'}\widehat{\langle\nabla\rangle^N}p^{\frac{1}{2}}(k,\eta,l)|\widehat{\mathcal{A}H_{\neq,1}}(k,\eta,l)|(k^2+l^2)^{-\frac{1}{4}}\widehat{\langle\nabla\rangle^N}e^{\lambda\nu^{\frac{1}{3}}\tau}\\ &\times\big[|\widehat{U_{0,1}^{1,3}}(0,\eta-\eta',l-l')||(k,l')||\widehat{\Theta_{\neq,1}}(k,\eta',l')|+|\widehat{U_{\neq,1}^3}(k,\eta-\eta',l-l')|l'||\widehat{\Theta_{0,1}}(0,\eta',l')|\big]{\rm d}\eta{\rm d}\eta'{\rm d}\tau\\ \lesssim& \|\mathcal{A}\nabla_L H_{\neq,1}\|_{L^2_tH^N}\big(\|\widehat{U_{0,1}^{1,3}}\|_{L^\infty_tL^1}\|\mathcal{A}|\nabla_L|^{\frac{1}{2}}H_{\neq,1}\|_{L^2_tH^N}+\|U_{0,1}^{1,3}\|_{L^\infty_tH^N}\|e^{\lambda\nu^{\frac{1}{3}}\tau}\widehat{\nabla_{x,z}\Theta_{\neq,1}}\|_{L^2_tL^1}\\ &+\|\widehat{\partial_z\Theta_{0,1}}\|_{L^\infty_tL^1}\|\mathcal{A}Q_{\neq,1}\|_{L^2_tH^N}+\|e^{\lambda\nu^{\frac{1}{3}}\tau}\widehat{U^3_{\neq,1}}\|_{L^4_tL^1}\||\partial_z|^{\frac{1}{2}}\Theta_{0,1}\|_{L^4_tH^{N,N+\frac{1}{2}}}\big) \\ \lesssim& \delta\nu^c\nu^{-\frac{1}{2}}\delta^2(\nu^c\nu^c\nu^{-\frac{1}{3}}+\nu^c\nu^c\nu^{-\frac{1}{6}}+\nu^c\nu^c\nu^{-\frac{1}{6}}+\nu^c\nu^{-\frac{1}{12}}\nu^c\nu^{-\frac{1}{4}})\\ \lesssim& \delta^2\nu^{2c}\cdot\delta\nu^{c-\frac{5}{6}}, \end{align}\] here we used the estimate ?? .

For nonlinear cross terms, thanks to the fact \(|\hat{G}|\leq 1\), the proof process for nonlinear terms mentioned above is no different from that for cross-terms, so we omit its proof process. Thus, we finish the proof of proposition 22. \(\square\)

4.4 Energy estimates on \(U_{0,2}\) and \(\Theta_{0,2}\)↩︎

In this section, we prove that under the bootstrap hypotheses of Proposition 15, the estimates on \(U_{0,2}\), and \(\Theta_{0,2}\) hold (i.e., ?? ), with 100 replaced by 50 on the right-hand side.

Before proving this proposition, we present the following lemma 24. We first establish an improved estimation of the double-zero modes.

Proof.Now we consider the perturbation system 61 ,62 and 63 . The energy estimates tell us that \[\begin{align} &\|(\overline{U^1_{0,2}},\overline{U^3_{0,2}},\overline{\Theta_{0,2}})\|_{H^{s}}^2 + \nu\|\partial_y (\overline{U^1_{0,2}},\overline{U^3_{0,2}},\overline{\Theta_{0,2}})\|_{L^2_t H^{s}}^2 \\ & \quad\lesssim \nu^{-1}\int_0^{t} \|\overline{U_{0,1}^2\tilde{\Theta}_{0,1}}\|_{H^{s}}^2+\sum_{F\in U^1,U^3,\Theta}(\|\overline{U_{0,1}^2\tilde{F}_{0,2}}\|_{H^{s}}^2 + {\|\overline{\tilde{U}_{0,2}^{2,3}\cdot\nabla_{y,z}\tilde{F}_{0,1}}\|_{H^{s-1}}^2}+\|\overline{U_{0,2}^2\tilde{F}_{0,2}}\|_{H^{s}}^2) {\rm d}\tau\\ &\qquad+ \nu^{-1}\int_0^{t} \sum_{F\in U^1,U^3,\Theta}(\|(\overline{U_{\neq,1}^2F_{\neq,2}})_0\|_{H^{s}}^2 + \|(\overline{U_{\neq,2}^2F_{\neq,1}})_0\|_{H^{s}}^2 + \|(\overline{U_{\neq,2}^2F_{\neq,2}})_0\|_{H^{s}}^2) {\rm d}\tau. \end{align}\] For \(F\in\{U^1,U^3,\Theta\}\), using estimates of Lemma 17 to handle the zero modes contributions, which gives \[\begin{align} \nu^{-1}\int_0^{t} \|\overline{U_{0,1}^2\tilde{F}_0}\|_{H^{s}}^2{\rm d}\tau\lesssim& \nu^{-1}\int_0^{t} \|U_{0,1}^2\|_{W^{s,\infty}}^2\|(\tilde{F}_{0,1},\tilde{F}_{0,2})\|_{H^{s}}^2{\rm d}\tau\\ \lesssim& \nu^{-1}\int_0^tq^{-\frac{2}{3}}\delta^2\nu^{2c}(\tau^{-\frac{1}{3}}e^{-\nu \tau}+\delta\nu^{c-\frac{2}{3}})^2\|(\tilde{F}_{0,1},\tilde{F}_{0,2})\|_{H^s}^2{\rm d}\tau\\ \lesssim& q^{-\frac{2}{3}}\delta^2(\nu^{2c}+\nu^{2a})(\delta^2\nu^{2c-\frac{4}{3}}+\delta^4\nu^{4c-\frac{10}{3}}),\\ \nu^{-1}\int_0^{t} {\|\overline{\tilde{U}_{0,2}^{2,3}\cdot\nabla_{y,z}\tilde{F}_{0,1}}\|_{H^{s-1}_y}^2}{\rm d}\tau\lesssim& \nu^{-1}\int_0^{t} \|\tilde{U}_{0,2}^{2,3}\|_{W^{s-3,\infty}}\|\nabla\tilde{U}_{0,2}\|_{H^{s,s+\frac{1}{2}}}\|\nabla\tilde{F}_{0,1}\|_{H^{s+1}}^2{\rm d}\tau\\ \lesssim& q^{-\frac{1}{3}}\delta^5\nu^{4c+a-\frac{8}{3}},\\ \nu^{-1}\int_0^{t} \|\overline{U_{0,2}^2\tilde{F}_{0,2}}\|_{H^{s}}^2{\rm d}\tau\lesssim& \nu^{-1} \|U_{0,2}\|_{L^{\infty}_t H^s}^2\|F_{0,2}\|_{L^2_tH^s}^2\lesssim \delta^4\nu^{4a-2}. \end{align}\] For the nonzero modes contributions, we have \[\begin{align} &\nu^{-1}\int_0^{t} \sum_{F\in U^1,U^3,\Theta}(\|(\overline{U_{\neq,1}^2F_{\neq,2}})_0\|_{H^{s}}^2 + \|(\overline{U_{\neq,2}^2F_{\neq,1}})_0\|_{H^{s}}^2 + \|(\overline{U_{\neq,2}^2F_{\neq,2}})_0\|_{H^{s}}^2) {\rm d}\tau\\ & \quad\lesssim\nu^{-1}\int_0^t (\|\mathcal{A}(Q_{\neq,1},K_{\neq,1},H_{\neq,1})\|_{H^s}^2+\|\mathcal{B}(Q_{\neq,2},K_{\neq,2},H_{\neq,2})\|_{H^s}^2)\|\mathcal{B}(Q_{\neq,2},K_{\neq,2},H_{\neq,2})\|_{H^s}^2{\rm d}\tau\\ &\quad\lesssim \delta^2(\nu^{2c}+\nu^{2b})\delta^2\nu^{2b-\frac{4}{3}}. \end{align}\] The proof of ?? is similar to that of ?? , it suffice to note that \[\begin{align} \nu^{-1}\int_0^{t} {\|\overline{\tilde{U}_{0,2}^{2,3}\cdot\nabla_{y,z}\tilde{F}_{0,1}}\|_{H^{s-4}_y}^2}{\rm d}\tau\lesssim& \nu^{-1}\int_0^{t} \|\tilde{U}_{0,2}^{2,3}\|_{W^{s-4,\infty}}^2\|\nabla\tilde{F}_{0,1}\|_{H^s}^2{\rm d}\tau \lesssim q^{-\frac{2}{3}}\delta^6\nu^{6c-\frac{10}{3}},\\ \nu^{-1}\int_0^{t} {\|\overline{U_{0,2}^2\tilde{F}_{0,2}}\|_{H^{s-3}}^2}{\rm d}\tau\lesssim& \nu^{-1} \|U_{0,2}\|_{L^\infty_t W^{s-3,\infty}}^2\|\nabla\tilde{F}_{0,2}\|_{L^2_tH^s}^2\lesssim \delta^6\nu^{4c+2a-\frac{10}{3}}. \end{align}\] where we need \(c\leq a\) and \(c \leq b\). Thus we finish the proof.\(\square\)

Proof of Proposition 23. In the following, We will use the above lemma to prove Proposition 23. Energy estimates yield that \[\begin{align} &\; \frac{1}{2} \frac{{\rm d}}{{\rm d}t}(\|U^2_{0,2}\|_{H^{s,s+\frac{1}{2}}}^2+\|V^2_{0,2}\|_{H^{s,s+\frac{1}{2}}}^2) + \nu\|\nabla(U^2_{0,2},V^2_{0,2})\|_{H^{s,s+\frac{1}{2}}}^2 \\&\quad = \langle\mathcal{N}_{\neq}(U_{0,2}^2),U_{0,2}^2\rangle_{H^{s,s+\frac{1}{2}}}+\langle\mathcal{N}_{\neq}(V_{0,2}^2),V_{0,2}^2\rangle_{H^{s,s+\frac{1}{2}}}-\langle U_0\cdot\nabla U_0^2,U^2_{0,2}\rangle_{H^{s,s+\frac{1}{2}}} + \langle U_{0,1}\cdot\nabla U_{0,1}^2,U^2_{0,2}\rangle_{H^{s,s+\frac{1}{2}}} \\ &\qquad+ 2 \langle \partial_{y}\Delta_{L,0}^{-1}(\partial_yU_0^2\partial_yU_0^2+\partial_yU_0^3 \partial_zU^2_0),U^2_{0,2}\rangle_{H^{s,s+\frac{1}{2}}}-2 \langle \partial_{y}\Delta_{L,0}^{-1}(\partial_yU_{0,1}^2\partial_yU_{0,1}^2+\partial_y\tilde{U}_{0,1}^3 \partial_zU^2_{0,1}),U^2_{0,2}\rangle_{H^{s,s+\frac{1}{2}}} \\ &\qquad- \langle G_1\widetilde{(U_0\cdot\nabla U^1_0)},V^2_{0,2}\rangle_{H^{s,s+\frac{1}{2}}} - \langle G_2\widetilde{(U_0\cdot\nabla\Theta_0)},V^2_{0,2}\rangle_{H^{s,s+\frac{1}{2}}} + \langle \tilde{U}_{0,1}\cdot\nabla V^2_{0,1},V^2_{0,2}\rangle_{H^{s,s+\frac{1}{2}}}, \end{align}\] and \[\begin{align} & \frac{1}{2} \frac{{\rm d}}{{\rm d}t}(\|\tilde{U}^3_{0,2}\|_{H^{s,s+\frac{1}{2}}}^2+\| \tilde{V}^3_{0,2}\|_{H^{s,s+\frac{1}{2}}}^2) + \nu\|\nabla(\tilde{U}^3_{0,2}, \tilde{V}^3_{0,2})\|_{H^{s,s+\frac{1}{2}}}^2\\& =\langle\mathcal{N}_{\neq}(U_{0,2}^2),U_{0,2}^2\rangle_{H^{s,s+\frac{1}{2}}}+\langle\mathcal{N}_{\neq}(V_{0,2}^2),V_{0,2}^2\rangle_{H^{s,s+\frac{1}{2}}}-\langle \widetilde{U_0\cdot\nabla U_0^3},\tilde{U}^3_{0,2}\rangle_{H^{s,s+\frac{1}{2}}} + \langle \widetilde{\tilde{U}_{0,1}\cdot\nabla\tilde{U}_{0,1}^3},\tilde{U}^3_{0,2}\rangle_{H^{s,s+\frac{1}{2}}} \\ &\quad+ 2\langle \partial_{z}\Delta_{L,0}^{-1}(\partial_yU_0^2\partial_yU^2_0+\partial_{y}U_0^3 \partial_zU^2_0),\tilde{U}^3_{0,2}\rangle_{H^{s,s+\frac{1}{2}}}- 2\langle \partial_{z}\Delta_{L,0}^{-1}(\partial_yU_{0,1}^2\partial_yU^2_{0,1} + \partial_{y}\tilde{U}_{0,1}^3 \partial_zU^2_{0,1}),\tilde{U}^3_{0,2}\rangle_{H^{s,s+\frac{1}{2}}}\\ &\quad -\langle \widetilde{G_1'(U_0\cdot\nabla U^1_0)}, \tilde{V}^3_{0,2}\rangle_{H^{s,s+\frac{1}{2}}} - \langle \widetilde{G_2'(U_0\cdot\nabla\Theta_0)}, \tilde{V}^3_{0,2}\rangle_{H^{s,s+\frac{1}{2}}} - {\langle \partial_z^{-1}\partial_y(\tilde{U}_{0,1}\cdot\nabla V_{0,1}^2), \tilde{V}^3_{0,2}\rangle_{H^{s,s+\frac{1}{2}}}}, \end{align}\] and \[\begin{align} &\frac{1}{2} \frac{{\rm d}}{{\rm d}t}\|\Lambda_{0,2}\|_{H^{s,s+\frac{1}{2}}}^2+\nu\|\nabla\Lambda_{0,2}\|_{H^{s,s+\frac{1}{2}}}^2 \\ &\quad= \langle\mathcal{N}_{\neq}(\Lambda_{0,2}),\Lambda_{0,2}\rangle_{H^{s,s+\frac{1}{2}}}+\langle {G_3(\tilde{U}_{0,1}\cdot\nabla U^1_{0,1})+G_4(\tilde{U}_{0,1}\cdot\nabla\Theta_{0,1})},\Lambda_{0,2}\rangle_{H^{s,s+\frac{1}{2}}}\\ &\qquad- \langle G_3\widetilde{(U_0\cdot\nabla U_0^1)},\Lambda_{0,2}\rangle_{H^{s,s+\frac{1}{2}}} - \langle G_4\widetilde{(U_0\cdot\nabla\Theta_0)},\Lambda_{0,2}\rangle_{H^{s,s+\frac{1}{2}}}. \end{align}\] Then adding all the above terms together, we have \[\begin{align} \frac{{\rm d}}{{\rm d}t}&E_{0,2}(t) + 2F_{0,2}(t)\\ & \lesssim \left|\langle \nabla(U_{0,1}\cdot\nabla(\tilde{U}_{0,1},\tilde{\Theta}_{0,1})), (U_{0,2}^2,\tilde{U}_{0,2}^3,V_{0,2}^2,\tilde{V}_{0,2}^3)\rangle_{H^{s,s+\frac{1}{2}}}\right|+ \left|\langle \overline{U_{0,1}^3}\partial_z(U_{0,1},\Theta_{0,1}), (\tilde{U}_{0,2},\tilde{\Theta}_{0,2})\rangle_{H^{s,s+\frac{1}{2}}}\right| \\ &\quad+ \left|\langle U_{0,1}\cdot\nabla(U_{0,2},\Theta_{0,2}), (\tilde{U}_{0,2},\tilde{\Theta}_{0,2})\rangle_{H^{s,s+\frac{1}{2}}}\right| + \left|\langle U_{0,2}\cdot\nabla(U_{0,1},\Theta_{0,1}), (\tilde{U}_{0,2},\tilde{\Theta}_{0,2})\rangle_{H^{s,s+\frac{1}{2}}}\right| \\ &\quad+ \left|\langle U_{0,2}\cdot\nabla(U_{0,2},\Theta_{0,2}), (\tilde{U}_{0,2},\tilde{\Theta}_{0,2})\rangle_{H^{s,s+\frac{1}{2}}}\right| + \left|\langle U_{\neq,1}\cdot\nabla_L (U_{\neq,2},\Theta_{\neq,2})_0, (\tilde{U}_{0,2},\tilde{\Theta}_{0,2})\rangle_{H^{s,s+\frac{1}{2}}}\right| \\ &\quad+ \left|\langle U_{\neq,2}\cdot\nabla_L (U_{\neq,1},\Theta_{\neq,1})_0, (\tilde{U}_{0,2},\tilde{\Theta}_{0,2})\rangle_{H^{s,s+\frac{1}{2}}}\right| + \left|\langle U_{\neq,2}\cdot\nabla_L (U_{\neq,2},\Theta_{\neq,2})_0, (\tilde{U}_{0,2},\tilde{\Theta}_{0,2})\rangle_{H^{s,s+\frac{1}{2}}}\right|\\ &\lesssim F^{\frac{1}{2}}_{0,1}(t)\nu^{-\frac{1}{2}}\left(F_{0,1}^{\frac{1}{2}}(t)\nu^{-\frac{1}{2}}\|(U_{0,2}^2,\tilde{U}_{0,2}^3,V_{0,2}^2,\tilde{V}_{0,2}^3)\|_{W^{s-3}} + F_{0,2}^{\frac{1}{2}}(t)\nu^{-\frac{1}{2}}\|\overline{U_{0,1}^3}\|_{H^{s,s+\frac{1}{2}}}\right) \\ &\quad+ F_{0,2}(t)\nu^{-1} \|(\overline{U_{0,1}^3},\overline{\Theta_{0,1}})\|_{H^{s+1}} + \|\langle\partial_z\rangle^{\frac{1}{2}}\tilde{U}_{0,1}^{2,3}\|_{W^{s,\infty}}F_{0,2}^{\frac{1}{2}}\nu^{-\frac{1}{2}}F_{0,2}^{\frac{1}{4}}(t)\nu^{-\frac{1}{4}}\|(\tilde{U}_{0,2},\tilde{\Theta}_{0,2})\|_{H^{s+1}}^{\frac{1}{2}} \\ &\quad+ { \|\overline{U_{0,2}^3}\|_{H^{s-3}}^{\frac{1}{2}}\|\partial_y\overline{U_{0,2}^3}\|_{H^s}^{\frac{1}{2}}E_{0,1}^{\frac{1}{4}}(t)F_{0,1}^{\frac{1}{4}}(t)\nu^{-\frac{1}{4}}F_{0,2}^{\frac{1}{2}}(t)\nu^{-\frac{1}{2}}} + {F_{0,2}^{\frac{3}{4}}(t)\nu^{-\frac{3}{4}}q^{-\frac{1}{6}}\delta\nu^{c-\frac{1}{3}}F_{0,1}^{\frac{1}{4}}(t)\nu^{-\frac{1}{4}}E_{0,1}^{\frac{1}{4}}(t)}\\ &\quad+ \delta\nu^a F_{0,2}(t)\nu^{-1} + \delta\nu^a\cdot F^{\frac{1}{2}}_{\neq,1}(t)F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{6}-\frac{1}{2}} + \delta\nu^a F_{\neq,2}(t)\nu^{-\frac{1}{6}}\nu^{-\frac{1}{2}}\\ &\lesssim \left({\delta^8\nu^{2a+6c-\frac{22}{3}} + \delta^6\nu^{6c-\frac{16}{3}}} + \delta^4\nu^{4c-\frac{10}{3}} + q^{-\frac{2}{3}}(\delta^2\nu^{2a-\frac{5}{3}} + {\delta^6\nu^{6c-\frac{16}{3}}} + {\delta^8\nu^{8c-\frac{22}{3}}} )\right)F_{0,1}(t) \\ &\quad+F_{0,2}(t)+ (q^{-\frac{1}{3}}{\delta^3\nu^{3c-\frac{8}{3}}} + q^{-\frac{1}{3}}\delta^2\nu^{2c-\frac{5}{3}} + q^{-\frac{4}{3}}\delta^4\nu^{4c-3} + \delta\nu^{a-1})F_{0,2}(t) \\ &\quad+ \delta\nu^{a-\frac{2}{3}}F^{\frac{1}{2}}_{\neq,1}(t)F^{\frac{1}{2}}_{\neq,2}(t) + \delta\nu^{a-\frac{2}{3}} F_{\neq,2}(t), \end{align}\] where in the fifth line from the bottom, we have used the following fact for \(G\in\{U,\Theta\}\) \[\begin{align} \langle \tilde{U}_{0,2}^{2,3}\cdot\nabla_{y,z}G_{0,1},\tilde{G}_{0,2}\rangle_{H^{s,s+\frac{1}{2}}}\lesssim& F_{0,2}^{\frac{1}{2}}(t)\nu^{-\frac{1}{2}}\|\langle\partial_z\rangle^{\frac{1}{2}}\left(\tilde{U}_{0,2}^{2,3}\cdot\nabla_{y,z}G_{0,1}\right)\|_{H^{s-1}}\\ \lesssim& F_{0,2}^{\frac{1}{2}}(t)\nu^{-\frac{1}{2}}\|\langle\partial_z\rangle^{\frac{1}{2}}\tilde{U}_{0,2}^{2,3}\|_{W^{s-3,\infty}}^{\frac{1}{2}}\|\nabla\tilde{U}_{0,2}\|_{H^{s,s+\frac{1}{2}}}^{\frac{1}{2}}\|\nabla G_{0,1}\|_{H^{s+\frac{3}{2}}}\\ \lesssim& F_{0,2}^{\frac{3}{4}}(t)\nu^{-\frac{3}{4}}q^{-\frac{1}{6}}\delta\nu^{c-\frac{1}{3}}F_{0,1}^{\frac{1}{4}}(t)\nu^{-\frac{1}{4}}E_{0,1}^{\frac{1}{4}}(t). \end{align}\] By using 50 , we can return to the estimates of \(\tilde{U}^1_{0,2}\) and \(\tilde{\Theta}_{0,2}\). Integrating the above inequality respect to \(t\) immediately gives ?? . Thus, the proof of Proposition 23 is finished.\(\square\)

4.5 Energy estimates on \(Q_{\neq,2}, K_{\neq,2}\) and \(H_{\neq,2}\)↩︎

In this section, we aim to prove that under the bootstrap hypotheses of Proposition 15, the estimates on \(U_{\neq,2}\), and \(\Theta_{\neq,2}\) hold (i.e., ?? ), with 100 replaced by 50 on the right-hand side.

Proof.According to the linear analysis, an energy estimate yields \[\begin{align}\label{LLL} &\|\mathcal{B}(Q_{\neq,2},K_{\neq,2},H_{\neq,2})\|_{H^s}^2 + \nu\|\mathcal{B}\nabla_L(Q_{\neq,2},K_{\neq,2},H_{\neq,2})\|_{L^2_tH^s}^2 + \left\|\mathcal{B}\sqrt{-\frac{\dot{M}}{M}}(Q_{\neq,2},K_{\neq,2},H_{\neq,2})\right\|_{L^2_tH^s}^2\\ &\quad\lesssim \frac{\nu^{\frac{1}{3}}}{32}\|\mathcal{B}(Q_{\neq,2},K_{\neq,2},H_{\neq,2})\|_{L^2_tH^s}^2+\frac{4\sqrt{\beta(\beta-1)}}{2\sqrt{\beta(\beta-1)}-1}\int_0^t\mathcal{N}_{\mathcal{B}1}{\rm d}\tau+\frac{4\sqrt{\beta(\beta-1)}}{2\sqrt{\beta(\beta-1)}-1}\int_0^tG\mathcal{N}_{\mathcal{B}2}{\rm d}\tau, \end{align}\qquad{(29)}\] where \[\begin{align} \int_0^t\mathcal{N}_{\mathcal{B}1}{\rm d}\tau=&\int_0^t\langle \mathcal{B}Q_{\neq,2},\text{Nonlinear terms in the equation of }Q_{\neq,2}\rangle_{H^s}{\rm d}\tau\\ &+\int_0^t\langle \mathcal{B}K_{\neq,2},\text{Nonlinear terms in the equation of }K_{\neq,2}\rangle_{H^s}{\rm d}\tau,\\ &+\int_0^t\langle \mathcal{B}H_{\neq,2},\text{Nonlinear terms in the equation of }H_{\neq,2}\rangle_{H^s}{\rm d}\tau, \end{align}\] and \(\int_0^tG\mathcal{N}_{\mathcal{B}2}{\rm d}\tau\) is the associated nonlinear cross terms \[\begin{align} \int_0^tG\mathcal{N}_{\mathcal{B}2}{\rm d}\tau=&\int_0^t\langle G\mathcal{B}K_{\neq,2},\text{Nonlinear terms in the equation of }Q_{\neq,2}\rangle_{H^s}{\rm d}\tau\\ &+\int_0^t\langle G\mathcal{B}Q_{\neq,2},\text{Nonlinear terms in the equation of }K_{\neq,2}\rangle_{H^s}{\rm d}\tau. \end{align}\]

We are ready to estimate each term on the right-hand side ?? as follows.

4.5.1 Contribution from two external force terms: Using 38 , 44 and 75 , we have \[\begin{align} &\big|\langle\mathcal{B}Q_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{3}{2}}|\nabla_{L,h}|^{-1}\big[U_{\neq,1}^2\partial_y^L U_{\neq,1}^3+U_{\neq,1}^3\partial_zU_{\neq,1}^3\big]\rangle_{H^s}\big|\\ &\quad \lesssim\|\mathcal{B}Q_{\neq,2}\|_{H^{s}}\nu^{-\frac{1}{6}}\langle t\rangle^{\frac{1}{2}}e^{-\frac{\lambda}{2}\nu^{\frac{1}{3}}t}\left\| |\nabla|^{\frac{5}{2}}\left(|\nabla|^{\frac{1}{2}}|\nabla|\langle t\rangle^{-1}e^{\lambda\nu^{\frac{1}{3}}t}m(K_{\neq,1},Q_{\neq,1})|\nabla|\langle t\rangle mQ_{\neq,1} \right) \right\|_{H^{s}}\\&\qquad+\|BQ_{\neq,2}\|_{H^{s}}\nu^{-\frac{1}{6}}\langle t\rangle^{\frac{1}{2}}e^{-\frac{\lambda}{2}\nu^{\frac{1}{3}}t}\left\||\nabla|^{\frac{5}{2}}\left(mQ_{\neq,1}e^{\lambda\nu^{\frac{1}{3}}t}|\nabla|mQ_{\neq,1}\right)\right\|_{H^{s}}\\&\quad\lesssim \|BQ_{\neq,2}\|_{H^{s}}\nu^{-\frac{1}{6}}\nu^{-\frac{1}{6}}\|\mathcal{A}(K_{\neq,1},Q_{\neq,1})\|_{H^{s+4}}\|\mathcal{A}Q_{\neq,1}\|_{H^{s+\frac{7}{2}}} \\&\quad\lesssim (\delta \nu^{b})\nu^{-\frac{1}{6}}\nu^{-\frac{1}{6}}F_{\neq,1}\nu^{-\frac{1}{6}}F_{\neq,1}\nu^{-\frac{1}{6}}\\&\quad\lesssim\delta\nu^{b-\frac{2}{3}}F_{\neq,1}, \end{align}\] where we have used \(mp^{\frac{3}{4}}p_h^{-\frac{1}{2}}\lesssim\nu^{-\frac{1}{6}}|k, \eta, l|^{\frac{5}{2}}\langle t\rangle^{\frac{1}{2}}\) due to the fact \(p\lesssim \langle t\rangle^{2}|k,\eta,l|^{2}\) and \(p^{-1}\lesssim \langle t\rangle^{-2}|k,\eta,l|^{2}\). For the term \(U_{0,1}^2\partial_y^LU_{\neq,1}^3\), when \(|l|\leq\sqrt{p_h}\), we use the fact that \(p^{\frac{3}{4}}p_h^{-\frac{1}{2}}\lesssim p_h^{\frac{1}{4}}\) and ?? to obtain \[\begin{align} &\big|\langle\mathcal{B}Q_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{3}{2}}|\nabla_{L,h}|^{-1}(U_{0,1}^2\partial_y^LU_{\neq,1}^3)\rangle_{H^s}\big|\\ &\quad\leq \sum_{\substack{k,l\\k',l'}}\iint_{\eta,\eta'}\widehat{\langle\nabla\rangle^s}\widehat{\mathcal{B}Q_{\neq,2}}(k,\eta,l)e^{\frac{\lambda}{2}\nu^{\frac{1}{3}}\tau}mp^{\frac{3}{4}}p_h^{-\frac{1}{2}}(t,k,\eta,l)\widehat{\langle\nabla\rangle^s}\big[\widehat{U_{0,1}^2}(0,\eta-\eta',l-l') |\widehat{|\nabla_L|^{\frac{1}{2}}Q_{\neq,1}}(k,\eta',l')|\big]{\rm d}\eta{\rm d}\eta'\\ &\quad\leq \sum_{\substack{k,l\\k',l'}}\iint_{\eta,\eta'}\widehat{\langle\nabla\rangle^s}\widehat{\mathcal{B}|\nabla_L|^{\frac{1}{2}}Q_{\neq,2}}(k,\eta,l)e^{\frac{\lambda}{2}\nu^{\frac{1}{3}}\tau}\widehat{\langle\nabla\rangle^s}\big[\widehat{|\nabla|^{\frac{1}{2}}U_{0,1}^2}(0,\eta-\eta',l-l') |\widehat{|\nabla_L|^{\frac{1}{2}}mQ_{\neq,1}}(k,\eta',l')|\big]{\rm d}\eta{\rm d}\eta'\\ &\quad\lesssim F_{\neq,2}^{\frac{1}{2}}(t)\nu^{-\frac{1}{3}}\|U_{0,1}^2\|_{H^{s+\frac{1}{2}}}F_{\neq,1}^{\frac{1}{2}}(t)\nu^{-\frac{1}{3}} \\ &\quad\lesssim \delta F_{\neq,2}(t) + \delta\nu^{2c-\frac{4}{3}} F_{\neq,1}(t), \end{align}\] when \(|l|>\sqrt{p_h}\), by using ?? and 38 we have \[\begin{align} &\big|\langle\mathcal{B}Q_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{3}{2}}|\nabla_{L,h}|^{-1}(U_{0,1}^2\partial_y^LU_{\neq,1}^3)\rangle_{H^s}\big|\\ &\quad\lesssim \big|\langle\mathcal{B}|\nabla_{L,h}|^{-1}Q_{\neq,2},e^{\frac{\lambda}{2}\nu^{\frac{1}{3}}t}\big[|\partial_z|^{\frac{3}{2}}U_{0,1}^2|\partial_z|^{2}m|\nabla_L|^{\frac{1}{2}}Q_{\neq,1}\big]\rangle_{H^s}\big| \\ &\quad\lesssim F_{\neq,2}^{\frac{1}{2}}(t)\|U_{0,1}^2\|_{H^{s+2}}F_{\neq,1}^{\frac{1}{2}}(t)\nu^{-\frac{1}{3}}\\&\quad\leq \delta F_{\neq,2}(t)+ \delta\nu^{2c-\frac{2}{3}}F_{\neq,1}(t). \end{align}\]

Due to 76 , 38 we have \[\begin{align} &\big|\langle\mathcal{B}Q_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{3}{2}}|\nabla_{L,h}|^{-1}(U_{0,1}^3\partial_zU_{\neq,1}^3)\rangle_{H^s}\big|\\ &\quad\lesssim \big|\langle\mathcal{B}\nabla_{L}Q_{\neq,2},e^{\frac{\lambda}{2}\nu^{\frac{1}{3}}t}\big[\langle\partial_z\rangle^{\frac{1}{2}}U_{0,1}^2\langle\partial_z\rangle mQ_{\neq,1}\big]\rangle_{H^s}\big|\\ &\quad\lesssim F_{\neq,2}^{\frac{1}{2}}(t)\nu^{-\frac{1}{2}}E_{0,1}^{\frac{1}{2}}(t)F_{\neq,1}^{\frac{1}{2}}(t)\nu^{-\frac{1}{6}}\\&\quad\leq \delta F_{\neq,2}(t) + \delta\nu^{2c-\frac{4}{3}}F_{\neq,1}(t). \end{align}\] Direct calculation shows \[\begin{align} &\big|\langle\mathcal{B}Q_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{3}{2}}|\nabla_{L,h}|^{-1}\big[U_{\neq,1}^{2,3}\cdot\nabla_{y,z}U_{0,1}^3\big]\rangle_{H^s}\big|\\ &\quad\lesssim F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{6}}\cdot\nu^{-\frac{1}{6}}\cdot\langle t\rangle^{\frac{1}{2}}e^{-\frac{\lambda}{2}\nu^{\frac{1}{3}}t}F^{\frac{1}{2}}_{\neq,1}(t)\nu^{-\frac{1}{6}}(\delta\nu^c)\\ &\quad\lesssim \delta F_{\neq,2}(t) + \delta\nu^{2c-\frac{4}{3}}F_{\neq,1}(t), \end{align}\] and \[\begin{align} &\big|\langle\mathcal{B}Q_{\neq,2}, \mathcal{B}\partial_z|\nabla_{L,h}|^{-1}|\nabla_L|^{-\frac{1}{2}}\big[\partial_zU_{\neq,1}^2\partial_y^LU_{\neq,1}^3+\partial_yU_{0,1}^3\partial_zU_{\neq,1}^2+\partial_iU_{0,1}^2\partial_y^LU_{\neq,1}^i\big]\rangle_{H^s}\big|\\ &\quad\lesssim F^{\frac{1}{2}}_{\neq,2}(t)(\delta\nu^c)F^{\frac{1}{2}}_{\neq,1}(t)\nu^{-\frac{1}{2}}\\ &\quad\lesssim \delta F_{\neq,2}(t) + \delta\nu^{2c-1}F_{\neq,1}(t). \end{align}\] where we have used the fact \(mp^{\frac{3}{4}}p_{h}^{-\frac{1}{2}}\lesssim m|\hat{\nabla}|^{\frac{5}{2}}\langle t \rangle^{\frac{1}{2}}\) and \(m|l|p_h^{-\frac{1}{2}}p^{-\frac{1}{4}}\lesssim \frac{\sqrt{|l|}}{\sqrt{p_{h}}}.\)

We now focus on the equation of \(K_{\neq,2}\). For the term \(U_{0,1}^2\partial_y^LW_{\neq,1}^3\), if \(|l|\leq\sqrt{p_h}\), using 23 , ?? and \(p^{\frac{1}{4}}p_h^{-\frac{1}{2}}\lesssim p_h^{-\frac{1}{4}}\) and integrating by parts, one has \[\begin{align} &\sqrt{\frac{\beta}{\beta-1}}\big|\langle\mathcal{B}K_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{1}{2}}|\nabla_{L,h}|^{-1}(U_{0,1}^2\partial_y^LW_{\neq,1}^3)\rangle_{H^s}\big|\\ &\leq \sqrt{\frac{\beta}{\beta-1}}(\big|\langle\mathcal{B}K_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{1}{2}}(U_{0,1}^2W_{\neq,1}^3)\rangle_{H^s}\big| + \big|\langle\mathcal{B}K_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{1}{2}}|\nabla_{L,h}|^{-1}(\partial_yU_{0,1}^2W_{\neq,1}^3)\rangle_{H^s}\big|)\\ &\leq \sum_{\substack{k,l\\k',l'}}\iint_{\eta,\eta'}\widehat{\langle\nabla\rangle^s}\widehat{\mathcal{B}K_{\neq,2}}(k,\eta,l)e^{\lambda\nu^{\frac{1}{3}}\tau}mp^{\frac{1}{4}}(t,k,\eta,l)\widehat{\langle\nabla\rangle^s}\big[\widehat{U_{0,1}^2}(0,\eta-\eta',l-l') |\widehat{|\nabla_L|^{\frac{1}{2}}K_{\neq,1}}(k,\eta',l')|\big]{\rm d}\eta{\rm d}\eta'\\ &\quad+ \sum_{\substack{k,l\\k',l'}}\iint_{\eta,\eta'}\widehat{\langle\nabla\rangle^s}\widehat{\mathcal{B}K_{\neq,2}}(k,\eta,l)e^{\lambda\nu^{\frac{1}{3}}\tau}mp^{\frac{1}{4}}p_h^{-\frac{1}{2}}(t,k,\eta,l)\widehat{\langle\nabla\rangle^s}\big[\widehat{\partial_yU_{0,1}^2}(0,\eta-\eta',l-l') |\widehat{|\nabla_L|^{\frac{1}{2}}K_{\neq,1}}(k,\eta',l')|\big]{\rm d}\eta{\rm d}\eta'\\ &\leq \sum_{\substack{k,l\\k',l'}}\iint_{\eta,\eta'}\widehat{\langle\nabla\rangle^s}\widehat{\mathcal{B}|\nabla_L|^{\frac{1}{2}}K_{\neq,2}}(k,\eta,l)e^{\lambda\nu^{\frac{1}{3}}\tau}\widehat{\langle\nabla\rangle^s}\big[\widehat{|\nabla_{y,z}|^{\frac{1}{2}}U_{0,1}^2}(0,\eta-\eta',l-l') |\widehat{|\nabla_L|^{\frac{1}{2}}mK_{\neq,1}}(k,\eta',l')|\big]{\rm d}\eta{\rm d}\eta'\\ &\quad + \sum_{\substack{k,l\\k',l'}}\iint_{\eta,\eta'}\widehat{\langle\nabla\rangle^s}\widehat{\mathcal{B}|\nabla_{L,h}|^{-\frac{1}{2}}K_{\neq,2}}(k,\eta,l)e^{\lambda\nu^{\frac{1}{3}}\tau}\widehat{\langle\nabla\rangle^s}\big[\widehat{|\nabla_{y,z}|^{\frac{1}{2}}\partial_yU_{0,1}^2}(0,\eta-\eta',l-l') |\widehat{|\nabla_L|^{\frac{1}{2}}mK_{\neq,1}}(k,\eta',l')|\big]{\rm d}\eta{\rm d}\eta'\\ &\lesssim F_{\neq,2}^{\frac{1}{2}}(t)\nu^{-\frac{1}{3}}\|U_{0,1}^2\|_{H^{s+\frac{1}{2}}}F_{\neq,1}^{\frac{1}{2}}(t)\nu^{-\frac{1}{3}} + F_{\neq,2}^{\frac{1}{2}}(t)\nu^{-\frac{\kappa}{6(1+\kappa)}}\|U_{0,1}^2\|_{H^{s+\frac{3}{2}}}F_{\neq,1}^{\frac{1}{2}}(t)\nu^{-\frac{1}{3}}\\ &\lesssim \delta F_{\neq,2}(t) + \delta(\nu^{2c-\frac{4}{3}}+\nu^{2c-\frac{2}{3}-\frac{\kappa}{3(1+\kappa)}}) F_{\neq,1}(t), \end{align}\] if \(|l|>\sqrt{p_h}\), using the fact that \(mp^{\frac{1}{4}}p_h^{-\frac{1}{2}}\lesssim \frac{|l|^{\frac{1}{2}}}{\sqrt{p_h}}\) and 23 and integrating by parts yields that \[\begin{align} &{\sqrt{\frac{\beta}{\beta-1}}\big|\langle \mathcal{B}K_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{1}{2}}|\nabla_{L,h}|^{-1}\big[U_{0,1}^2\partial_y^LW_{\neq,1}^3\big]\rangle_{H^s}\big|}\\ &\quad\leq\big|\langle \mathcal{B}|\nabla_{L,h}|^{-1}K_{\neq,2},e^{\frac{\lambda}{2}\nu^{\frac{1}{3}}t}|\partial_z|^{\frac{1}{2}}\big[U_{0,1}^2\partial_y^L|\nabla_L|^{-\frac{1}{2}}|\nabla_{L,h}|mK_{\neq,1}\big]\rangle_{H^s}\big|\nu^{-\frac{1}{6}}\\ &\quad\leq e^{\frac{\lambda}{2}\nu^{\frac{1}{3}}t}\nu^{-\frac{1}{6}}\big(\big|\langle \mathcal{B}K_{\neq,2},\big[|\partial_z|^{\frac{1}{2}}U_{0,1}^2|\partial_z|^{\frac{1}{2}}m|\nabla_L|^{\frac{1}{2}}K_{\neq,1}\big]\rangle_{H^s}\big| \\&\qquad + \big|\langle \mathcal{B}|\nabla_{L,h}|^{-1}K_{\neq,2},\big[|\partial_z|^{\frac{1}{2}}\partial_yU_{0,1}^2|\partial_z|^{\frac{1}{2}}m|\nabla_L|^{\frac{1}{2}}K_{\neq,1}\big]\rangle_{H^s}\big|\big)\\& \quad\lesssim \nu^{-\frac{1}{6}}F_{\neq,2}^{\frac{1}{2}}(t)\nu^{-\frac{1}{6}}\|U_{0,1}^2\|_{H^{s+\frac{1}{2}}}F_{\neq,1}^{\frac{1}{2}}(t)\nu^{-\frac{1}{3}} + \nu^{-\frac{1}{6}}F_{\neq,2}^{\frac{1}{2}}(t)\|U_{0,1}^2\|_{H^{s+\frac{3}{2}}}F_{\neq,1}^{\frac{1}{2}}(t)\nu^{-\frac{1}{3}}\\& \quad\lesssim \delta F_{\neq,2}(t) + \delta F_{\neq,1}(t)\nu^{2c-\frac{4}{3}}. \end{align}\]

By using \(mp^{\frac{1}{4}}p_h^{-\frac{1}{2}}\lesssim p^\frac{1}{2}p_{h}^{-\frac{1}{2}} \lesssim|\hat{\nabla}|^2\), one has \[\begin{align} &\sqrt{\frac{\beta}{\beta-1}}\big|\langle \mathcal{B}K_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{1}{2}}|\nabla_{L,h}|^{-1}\big[U_{\neq,1}^{2,3}\cdot\nabla_{y,z}\partial_yU_{0,1}^1\big]\rangle_{H^s}\big|\\&\quad \lesssim \sqrt{\frac{\beta}{\beta-1}}F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{6}}F^{\frac{1}{2}}_{\neq,1}(t)\nu^{-\frac{1}{6}}(\delta\nu^c)\\&\quad \lesssim \delta F_{\neq,2}(t)+ \delta\frac{\beta}{\beta-1}\nu^{2c-\frac{2}{3}}F_{\neq,1}(t), \end{align}\] and \[\begin{align} &\sqrt{\frac{\beta}{\beta-1}}\big|\langle \mathcal{B}K_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{1}{2}}|\nabla_{L,h}|^{-1}\big[\nabla_{y,z}U_{0,1}\partial_y^L U_{\neq,1}\big]\rangle_{H^s}\big|\\&\quad \lesssim \sqrt{\frac{\beta}{\beta-1}}F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{6}}(\delta\nu^c)F^{\frac{1}{2}}_{\neq,1}(t)\nu^{-\frac{1}{2}}\\&\quad\lesssim \delta F_{\neq,2}(t) + \delta\frac{\beta}{\beta-1}\nu^{2c-\frac{4}{3}}F_{\neq,1}(t). \end{align}\]

We now focus on the equation of \(H_{\neq,2}\). For the term \(U_{0,1}^2\partial_y^L\Theta_{\neq,1}\), we use 23 , ?? to obtain \[\begin{align} &\big|\langle\mathcal{B}H_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{1}{2}}(U_{0,1}^2\partial_y^L\Theta_{\neq,1})\rangle_{H^s}\big|\\ &\quad\leq \sum_{\substack{k,l\\k',l'}}\iint_{\eta,\eta'}\widehat{\langle\nabla\rangle^s}\widehat{\mathcal{B}H_{\neq,2}}(k,\eta,l)e^{\lambda\nu^{\frac{1}{3}}\tau}mp^{\frac{1}{4}}(t,k,\eta,l)\widehat{\langle\nabla\rangle^s}\big[\widehat{U_{0,1}^2}(0,\eta-\eta',l-l') |\widehat{|\nabla_L|^{\frac{1}{2}}H_{\neq,1}}(k,\eta',l')|\big]{\rm d}\eta{\rm d}\eta'\\ &\quad\leq \sum_{\substack{k,l\\k',l'}}\iint_{\eta,\eta'}\widehat{\langle\nabla\rangle^s}\widehat{\mathcal{B}|\nabla_L|^{\frac{1}{2}}H_{\neq,2}}(k,\eta,l)e^{\lambda\nu^{\frac{1}{3}}\tau}\widehat{\langle\nabla\rangle^s}\big[\widehat{|\nabla_{y,z}|^{\frac{1}{2}}U_{0,1}^2}(0,\eta-\eta',l-l') |\widehat{|\nabla_L|^{\frac{1}{2}}mH_{\neq,1}}(k,\eta',l')|\big]{\rm d}\eta{\rm d}\eta'\\ &\quad\lesssim F_{\neq,2}^{\frac{1}{2}}(t)\nu^{-\frac{1}{3}}\|U_{0,1}^2\|_{H^{s+\frac{1}{2}}}F_{\neq,1}^{\frac{1}{2}}(t)\nu^{-\frac{1}{3}}\\&\quad \lesssim \delta F_{\neq,2}(t) + \delta\nu^{2c-\frac{4}{3}} F_{\neq,1}(t). \end{align}\]

Due to \(mp^{\frac{1}{4}}\lesssim p^{\frac{1}{2}}\) and 44 , direct calculation yields \[\begin{align} \big|\langle \mathcal{B}H_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{1}{2}}\big[U_{\neq,1}^2\partial_y\Theta_{0,1}\big]\rangle_{H^s}\big|\lesssim F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{2}}\|U_{\neq,1}^2\|_{H^s}(\delta\nu^c) \lesssim \delta F_{\neq,2}(t) + \delta F_{\neq,1}(t)\nu^{2c-1}. \end{align}\]

Recalling the system 66 , we have already completed the estimation of the nonlinear terms for the two external force terms.

4.5.2 Contribution with only one external force term We consider the contribution of only one external force term for the second system. Focus on the equation of \(Q_{\neq,2}\). Thanks to the incompressible condiction, the fact \(p_h(k',\eta')^{\frac{1}{4}}\lesssim p_h(k,\eta)^{\frac{1}{4}}p_h(k-k',\eta-\eta')^{\frac{1}{4}}\), 75 and ?? , we have \[\begin{align} &\left|\langle\mathcal{B}Q_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{3}{2}}|\nabla_{L,h}|^{-1}\big[U_{\neq,1}^3\partial_zU_{\neq,2}^3\big]\rangle_{H^s}\right|\\ &\quad\leq \sum_{k,k',l,l'}\int_{\eta,\eta'}\langle\hat{\nabla}\rangle^s|\widehat{\mathcal{B}Q_{\neq,2}}(k,\eta,l)|e^{\frac{\lambda}{2}\nu^{\frac{1}{3}}t}\frac{|l|^{\frac{3}{2}}}{\sqrt{p_h}}\langle\hat{\nabla}\rangle^s(|\widehat{U_{\neq,1}^3}(k-k',\eta-\eta',l-l')||\widehat{\nabla_{L,h}\cdot U_{\neq,2}^{1,2}}(k',\eta',l')|){\rm d}\eta{\rm d}\eta'\\ &\qquad+ \sum_{k,k',l,l'}\int_{\eta,\eta'}\langle\hat{\nabla}\rangle^s|\widehat{\mathcal{B}Q_{\neq,2}}(k,\eta,l)|e^{\frac{\lambda}{2}\nu^{\frac{1}{3}}t}\frac{\sqrt{p_h}}{\sqrt{|k,l|}}\langle\hat{\nabla}\rangle^s(|\widehat{U_{\neq,1}^3}(k-k',\eta-\eta',l-l')||l'||\widehat{U_{\neq,2}^3}(k',\eta',l')|){\rm d}\eta{\rm d}\eta'\\ &\quad\lesssim \sum_{k,k',l,l'}\int_{\eta,\eta'}\langle\hat{\nabla}\rangle^s|l||\widehat{\mathcal{B}Q_{\neq,2}}(k,\eta,l)|\langle\hat{\nabla}\rangle^s\frac{|l|^{\frac{1}{2}}e^{\frac{\lambda}{2}\nu^{\frac{1}{3}}t}}{p_h^{\frac{1}{4}}(k,\eta)}(|\widehat{|\nabla_{L,h}|^{\frac{1}{2}}U_{\neq,1}^3}(k-k',\eta-\eta',l-l')|(|\widehat{|\partial_x|^{\frac{1}{2}}U_{\neq,2}^1}| \\ &\qquad + |\widehat{|\partial_y^L|^{\frac{1}{2}}U_{\neq,2}^2}|)(k',\eta',l')){\rm d}\eta{\rm d}\eta' + \sum_{k,k',l,l'}\int_{\eta,\eta'}\langle\hat{\nabla}\rangle^s|\widehat{\mathcal{B}Q_{\neq,2}}(k,\eta,l)|e^{\frac{\lambda}{2}\nu^{\frac{1}{3}}t}\frac{\sqrt{p_h}}{\sqrt{|k,l|}}\langle\hat{\nabla}\rangle^s\\ &\qquad\times(|\widehat{U_{\neq,1}^3}(k-k',\eta-\eta',l-l')||l'||\widehat{U_{\neq,2}^3}(k',\eta',l')|){\rm d}\eta{\rm d}\eta'\\ &\quad\lesssim F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{2}}(\delta\nu^{\frac{5}{6}+\frac{\kappa}{6(1+\kappa)}})F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{4}} + F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{2}}\langle t\rangle^{\frac{1}{2}} e^{-\frac{\lambda}{2}\nu^{\frac{1}{3}}t}(\delta\nu^{\frac{5}{6}+\frac{\kappa}{6(1+\kappa)}})F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{4}} \\ &\qquad+ F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{2}}(\delta\nu^{\frac{5}{6}+\frac{\kappa}{6(1+\kappa)}})F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{3}}\\ &\quad\lesssim \delta F_{\neq,2}(t)[\nu^{\frac{\kappa}{6(1+\kappa)}-\frac{1}{12}} + \nu^{\frac{\kappa}{6(1+\kappa)}}], \end{align}\] and \[\begin{align} &\left|\langle\mathcal{B}Q_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{3}{2}}|\nabla_{L,h}|^{-1}\big[U_{\neq,1}^h\cdot\nabla_{L,h}U_{\neq,2}^3\big]\rangle_{H^s}\right|\\ &\leq \sum_{k,k',l,l'}\int_{\eta,\eta'}\langle\hat{\nabla}\rangle^s|\widehat{\mathcal{B}Q_{\neq,2}}(k,\eta,l)|e^{\frac{\lambda}{2}\nu^{\frac{1}{3}}t}\frac{l^{\frac{3}{2}}}{\sqrt{p_h}}\langle\hat{\nabla}\rangle^s(|\widehat{U_{\neq,1}^h}(k-k',\eta-\eta',l-l')|p_h^{\frac{1}{2}}(k',\eta')|\widehat{U_{\neq,2}^3}(k',\eta',l')|){\rm d}\eta{\rm d}\eta'\\ &\quad+ \sum_{k,k',l,l'}\int_{\eta,\eta'}\langle\hat{\nabla}\rangle^s|\widehat{\mathcal{B}Q_{\neq,2}}(k,\eta,l)|e^{\frac{\lambda}{2}\nu^{\frac{1}{3}}t}\nu^{-\frac{1}{6}}p_h^{\frac{1}{4}}(k,\eta)\langle\hat{\nabla}\rangle^s(|\widehat{U_{\neq,1}^1}(k-k',\eta-\eta',l-l')|k'|\widehat{U_{\neq,2}^3}(k',\eta',l')|){\rm d}\eta{\rm d}\eta'\\ &\quad+ \sum_{k,k',l,l'}\int_{\eta,\eta'}\langle\hat{\nabla}\rangle^s|\widehat{\mathcal{B}Q_{\neq,2}}(k,\eta,l)|e^{\frac{\lambda}{2}\nu^{\frac{1}{3}}t}\nu^{-\frac{1}{6}}p_h^{\frac{1}{4}}(k,\eta)\langle\hat{\nabla}\rangle^s(|\widehat{U_{\neq,1}^2}(k-k',\eta-\eta',l-l')|\eta'-k't|\widehat{U_{\neq,2}^3}(k',\eta',l')|){\rm d}\eta{\rm d}\eta'\\ &\lesssim F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{2}}(\delta\nu^{\frac{5}{6}+\frac{\kappa}{6(1+\kappa)}})F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{6}} + F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{\kappa}{6(1+\kappa)}}\langle t\rangle e^{-\frac{\lambda}{2}\nu^{\frac{1}{3}}t}(\delta\nu^{\frac{5}{6}+\frac{\kappa}{6(1+\kappa)}})F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{2}} \\ &\quad+ F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{2}}(\delta\nu^{\frac{5}{6}+\frac{\kappa}{6(1+\kappa)}})F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{3}} + F^{\frac{1}{4}}_{\neq,2}(t)\nu^{-\frac{1}{4}}\nu^{-\frac{1}{6}}(\delta\nu^b)^{\frac{1}{2}}F^{\frac{1}{2}}_{\neq,1}(t)\nu^{-\frac{\kappa}{6(1+\kappa)}}(\delta\nu^b)^{\frac{1}{2}}F^{\frac{1}{4}}_{\neq,2}(t)\nu^{-\frac{1}{4}}\cdot\nu^{-\frac{1}{6}}\\& \lesssim \delta F_{\neq,2}(t) + \delta\nu^{2b-\frac{5}{3}-\frac{\kappa}{3(1+\kappa)}}F_{\neq,1}(t), \end{align}\] where we have used the following inequality \[\begin{align} |\widehat{\partial_{y}U_{\neq,2}^{3}}|\lesssim |\hat{|\partial_{y}|}\widehat{|\nabla_{L}|}^{-\frac{3}{2}}\widehat{|\nabla_{L,h}|}\widehat{|mQ_{\neq,2}|}|m^{-1}\lesssim |\widehat{m|\nabla_{L}|^{\frac{1}{2}}Q_{\neq,2}}|\nu^{-\frac{1}{6}}. \end{align}\]

Similarly, by 76 we have \[\begin{align} &\big|\langle\mathcal{B}Q_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{3}{2}}|\nabla_{L,h}|^{-1}\big[U_{\neq,2}^3\partial_zU_{\neq,1}^3\big]\rangle_{H^s}\big|\\ &\quad\lesssim F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{2}}F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{6}}(\delta\nu^{\frac{5}{6}+\frac{\kappa}{6(1+\kappa)}})\\&\quad=\delta F_{\neq,2}(t)\nu^{\frac{1}{6}+\frac{\kappa}{6(1+\kappa)}}, \end{align}\] Similarly, since \(\sqrt{|\eta'-k't|}\lesssim \sqrt{|\eta-\eta'-(k-k')t|}p_h^{\frac{1}{4}}(k,\eta)\), we have \[\begin{align} & \big|\langle\mathcal{B}Q_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{3}{2}}|\nabla_{L,h}|^{-1}\big[U_{\neq,2}^1\partial_xU_{\neq,1}^3+U_{\neq,2}^2\partial_y^LU_{\neq,1}^3\big]\rangle_{H^s}\big|\\ &\quad\lesssim F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{2}}F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{6}}(\delta\nu^{\frac{5}{6}+\frac{\kappa}{6(1+\kappa)}})\\ &\qquad+ \sum_{k,k',l,l'}\int_{\eta,\eta'}\Big[ \langle\hat{\nabla}\rangle^s|\widehat{\mathcal{B}Q_{\neq,2}}(k,\eta,l)|\frac{|l|^{\frac{3}{2}}e^{\frac{\lambda}{2}\nu^{\frac{1}{3}}t}}{p_h^{\frac{1}{4}}(k,\eta)}\langle\hat{\nabla}\rangle^s(|\widehat{|\partial_y^L|^{\frac{1}{2}}U_{\neq,2}^2}(k-k',\eta-\eta',l-l')|\eta'-k't|^{\frac{1}{2}}\widehat{U_{\neq,1}^3}(k',\eta',l')|)\\ &\qquad+ \langle\hat{\nabla}\rangle^s|\widehat{\mathcal{B}Q_{\neq,2}}(k,\eta,l)|e^{\frac{\lambda}{2}\nu^{\frac{1}{3}}t}\frac{\sqrt{p_h}}{\sqrt{|k,l|}}(k,\eta)\langle\hat{\nabla}\rangle^s(|\widehat{U_{\neq,2}^2}(k-k',\eta-\eta',l-l')|\eta'-k't|\widehat{U_{\neq,1}^3}(k',\eta',l')|)\Big]{\rm d}\eta{\rm d}\eta'\\ &\quad\lesssim \delta\nu^{\frac{1}{6}+\frac{\kappa}{6(1+\kappa)}}F_{\neq,2}(t) + F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{2}}\left(F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{4}}\delta\nu^{\frac{5}{6}+\frac{\kappa}{6(1+\kappa)}}\langle t\rangle^{\frac{1}{2}}e^{-\frac{\lambda}{2}\nu^{\frac{1}{3}}t} + F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{\kappa}{6(1+\kappa)}}\nu^{-\frac{1}{3}}(\delta\nu^{\frac{5}{6}+\frac{\kappa}{6(1+\kappa)}})\right)\\ &\quad\lesssim F_{\neq,2}(t)\delta[\nu^{\frac{\kappa}{6(1+\kappa)}-\frac{1}{12}} + \nu^{\frac{\kappa}{6(1+\kappa)}} + 1], \end{align}\] and \[\begin{align} \big|\langle\mathcal{B}Q_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{3}{2}}|\nabla_{L,h}|^{-1}\big[U_{0,2}^{1,3}\cdot\nabla_{x,z}U_{\neq,1}^3\big]\rangle_{H^s}&\lesssim F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{2}}\cdot(\delta\nu^a)F^{\frac{1}{2}}_{\neq,1}(t)\nu^{-\frac{1}{6}}\\&\lesssim \delta F_{\neq,2}(t) + \delta\nu^{2a-\frac{4}{3}}F_{\neq,1}(t). \end{align}\]

For term \(U_{0,2}^2\partial_y^LU_{\neq,1}^3\), if \(|l|\leq\sqrt{p_h}\), we have \[\begin{align} \big|\langle\mathcal{B}Q_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{3}{2}}|\nabla_{L,h}|^{-1}\big[U_{0,2}^2\partial_y^LU_{\neq,1}^3\big]\rangle_{H^s}\big|&\lesssim \big|\langle\mathcal{B}|\nabla_L|^{\frac{1}{2}}Q_{\neq,2},\big[|\nabla_{y,z}|^{\frac{1}{2}}U_{0,2}^2\partial_y^LmU_{\neq,1}^3\big]\rangle_{H^s}\big|\\ &\lesssim F_{\neq,2}^{\frac{1}{2}}(t)\nu^{-\frac{1}{3}}F_{0,2}^{\frac{1}{4}}(t)\nu^{-\frac{1}{4}}E_{0,2}^{\frac{1}{4}}(t)F_{\neq,1}^{\frac{1}{4}}(t)\nu^{-\frac{1}{4}}E_{\neq,1}^{\frac{1}{4}}(t)\\&\lesssim \delta F_{\neq,2}(t) + \delta\nu^{a-\frac{5}{6} + \frac{\kappa}{6(1+\kappa)}}F_{0,2}^{\frac{1}{2}}(t)F_{\neq,1}^{\frac{1}{2}}(t). \end{align}\] If \(|l|>\sqrt{p_h}\), then \(m\leq 1\). One gets \[\begin{align} &\big|\langle\mathcal{B}Q_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{3}{2}}|\nabla_{L,h}|^{-1}\big[U_{0,2}^2\partial_y^LU_{\neq,1}^3\big]\rangle_{H^s}\big|\\ &\lesssim \sum_{k,l,l'}\int_{\eta,\eta'}\langle\hat{\nabla}\rangle^s|\left(\frac{p}{p_h}\right)^{\frac{1}{4}}\widehat{\mathcal{B}Q_{\neq,2}}(k,\eta,l)|e^{\frac{\lambda}{2}\nu^{\frac{1}{3}}t}\frac{|l||(k,\eta)|^{\frac{1}{2}}}{\langle\tau\rangle^{\frac{1}{2}}}\\&\quad \times \langle\hat{\nabla}\rangle^s(|\widehat{U_{0,2}^2}(0,\eta-\eta',l-l')\nu^{-\frac{1}{6}}|\eta'-kt|\widehat{mU_{\neq,1}^3}(k,\eta',l')|){\rm d}\eta{\rm d}\eta'\\ & \lesssim F_{\neq,2}^{\frac{1}{2}}\nu^{-\frac{1}{4}}\langle t\rangle^{-\frac{1}{2}}\cdot F_{0,2}^{\frac{1}{2}}\nu^{-\frac{1}{2}}\nu^{-\frac{1}{6}}\|\mathcal{A}Q_{\neq,1}\|_{H^{s+2}}\langle t\rangle^{\frac{1}{2}}\\ &\lesssim \delta F_{\neq,2}(t) +\delta \nu^{\frac{\kappa}{3(1+\kappa)}-\frac{1}{6}}F_{0,2}(t), \end{align}\] where we have used \[\begin{align}\label{LL88} m p^{\frac{3}{4}}p_h^{-\frac{1}{2}}\lesssim p^{\frac{1}{4}}\cdot |l|\cdot p_h^{-\frac{1}{4}}p_{h}^{-\frac{1}{4}}\lesssim p^{\frac{1}{4}}p_h^{-\frac{1}{4}}\frac{|l|\cdot |k,\eta|^{\frac{1}{2}}}{\langle \tau\rangle^{\frac{1}{2}}},\end{align}\tag{81}\] and \[\left\|\left(\frac{p}{p_{h}}\right)^{\frac{1}{4}}\widehat{\mathcal{B}Q_{\neq,2}}\right\|_{L^{2}} \leq \|\widehat{\mathcal{B}\nabla_{L}Q_{\neq,2}}\|_{L^{2}}^{\frac{1}{2}}\|\widehat{\mathcal{B}\nabla_{L}^{-1}Q_{\neq,2}}\|_{L^{2}}^{\frac{1}{2}}.\]

The estimate of \(U_{0,1}\cdot\nabla_L U_{\neq,2}^3\) is similar to the term \(U_{0,1}^2\partial_y^L U_{\neq,1}^3\), we have \[\begin{align} \big|\langle\mathcal{B}Q_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{3}{2}}|\nabla_{L,h}|^{-1}\big[U_{0,1}\cdot\nabla_L U_{\neq,2}^3\big]\rangle_{H^s}\big|&\lesssim F_{\neq,2}^{\frac{1}{2}}\nu^{-\frac{1}{3}}\|U_{0,1}\|_{H^{s+\frac{3}{2}}}F_{\neq,2}^{\frac{1}{2}}\nu^{-\frac{1}{3}}\\&\lesssim \delta \nu^{c-\frac{2}{3}}F_{\neq,2}(t). \end{align}\] Integrating by part and using 76 we have \[\begin{align} &\big|\langle\mathcal{B}Q_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{3}{2}}|\nabla_{L,h}|^{-1}\big[U_{\neq,1}^2\partial_yU_{0,2}^3\big]\rangle_{H^s}\big|\\ &\quad\lesssim \big|\langle\mathcal{B}Q_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{3}{2}}\big[U_{\neq,1}^2U_{0,2}^3\big]\rangle_{H^s}\big| + \big|\langle\mathcal{B}Q_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{3}{2}}|\nabla_{L,h}|^{-1}\big[\partial_y^LU_{\neq,1}^2U_{0,2}^3\big]\rangle_{H^s}\big|\\ &\quad\lesssim \nu^{-\frac{1}{6}}F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{2}}[F^{\frac{1}{4}}_{\neq,1}(t)(\delta\nu^{\frac{5}{6}+\frac{\kappa}{6(1+\kappa)}})^{\frac{1}{2}}][F^{\frac{1}{4}}_{0,2}(t)\nu^{-\frac{1}{4}}(\delta\nu^a)^{\frac{1}{2}}] + F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{2}}F^{\frac{1}{2}}_{\neq,1}(t)\nu^{-\frac{1}{6}}(\delta\nu^a)\\ &\quad\lesssim 2\delta F_{\neq,2}(t) + \delta\nu^{a-1+\frac{\kappa}{6(1+\kappa)}}F^{\frac{1}{2}}_{\neq,1}(t)F^{\frac{1}{2}}_{0,2}(t) + \delta\nu^{2a-\frac{4}{3}}F_{\neq,1}(t), \end{align}\] and \[\begin{align} &\big|\langle\mathcal{B}Q_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{3}{2}}|\nabla_{L,h}|^{-1}\big[U_{\neq,2}^2\partial_yU_{0,1}^3\big]\rangle_{H^s}\big|+\big|\langle\mathcal{B}Q_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{3}{2}}|\nabla_{L,h}|^{-1}\big[U_{\neq,2}^3\partial_zU_{0,1}^3\big]\rangle_{H^s}\big|\\ &\quad\lesssim F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{2}}F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{4}}(\delta\nu^c)\\ &\quad=\delta \nu^{c-\frac{3}{4}}F_{\neq,2}(t). \end{align}\] The estimate of term \(U_{\neq,1}^3\partial_zU_{0,2}^3\) is similar to \(U_{0,2}^2\partial_y^LU_{\neq,1}^3\). Integrating by parts give \[\begin{align} &\big|\langle\mathcal{B}Q_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{3}{2}}|\nabla_{L,h}|^{-1}\big[U_{\neq,1}^3\partial_zU_{0,2}^3\big]\rangle_{H^s}\big|\\ &\quad\lesssim \big|\langle\mathcal{B}Q_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{3}{2}}|\nabla_{L,h}|^{-1}\big[\partial_y^LU_{\neq,1}^3U_{0,2}^2\big]\rangle_{H^s}\big| + \big|\langle\mathcal{B}Q_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{3}{2}}\big[U_{\neq,1}^3U_{0,2}^2\big]\rangle_{H^s}\big|\\ & \quad\lesssim\delta F_{\neq,2}(t) + \delta\nu^{a-\frac{5}{6} + \frac{\kappa}{6(1+\kappa)}}F_{0,2}^{\frac{1}{2}}(t)F_{\neq,1}^{\frac{1}{2}}(t) +\delta \nu^{\frac{\kappa}{3(1+\kappa)}-\frac{1}{6}}F_{0,2}(t)\\&\qquad + \big|\langle\mathcal{B}\nabla_L Q_{\neq,2},e^{\frac{\lambda}{2}\nu^{\frac{1}{3}}t}\big[|\nabla_L|^{\frac{1}{2}}mU_{\neq,1}^3|\partial_z|^{\frac{1}{2}}|\nabla|^{\frac{1}{2}}U_{0,2}\big]\rangle_{H^s}\big|\\ &\quad\lesssim \delta F_{\neq,2}(t) + \delta\nu^{a-\frac{5}{6} + \frac{\kappa}{6(1+\kappa)}}F_{0,2}^{\frac{1}{2}}(t)F_{\neq,1}^{\frac{1}{2}}(t) + \delta\nu^{\frac{\kappa}{3(1+\kappa)}-\frac{1}{6}}F_{0,2}(t)\\&\qquad +F_{\neq,2}^{\frac{1}{2}}(t)\nu^{-\frac{1}{2}}F_{\neq,1}^{\frac{1}{4}}\nu^{-\frac{1}{12}}E_{\neq,1}^{\frac{1}{4}}(t)F_{0,2}^{\frac{1}{4}}\nu^{-\frac{1}{4}}E_{0,2}^{\frac{1}{4}}(t)\\ &\quad\lesssim \delta F_{\neq,2}(t) + \delta\nu^{a-\frac{5}{6} + \frac{\kappa}{6(1+\kappa)}}F_{0,2}^{\frac{1}{2}}(t)F_{\neq,1}^{\frac{1}{2}}(t) + \delta\nu^{\frac{\kappa}{3(1+\kappa)}-\frac{1}{6}}F_{0,2}(t), \end{align}\] where we have used ?? and the incompressible condition and \[|m|\nabla_{L}|^{\frac{1}{2}} U_{\neq,1}^{3}|\leq |mQ_{\neq,1}|\leq|mQ_{\neq,1}|^{\frac{1}{2}} |mQ_{\neq,1}|^{\frac{1}{2}} .\]

For the pressure terms, using the fact that \(\sqrt{|\eta'-k't|}\lesssim \sqrt{|\eta-kt|}p_h^{\frac{1}{4}}(k',\eta')\), 77 and the incompressible condition, we have \[\begin{align} &\big|\langle\mathcal{B}Q_{\neq,2},\mathcal{B}\partial_z|\nabla_L|^{-\frac{1}{2}}|\nabla_{L,h}|^{-1}\big[\partial_{x,z}U_{\neq,1}\cdot\nabla_L U_{\neq,2}^{1,3}+\partial_y^LU_{\neq,1}^{1,2}\cdot\nabla_{L,h}U_{\neq,2}^2+\partial_y^LU_{\neq,1}^3\partial_zU_{\neq,2}^2\big]\rangle_{H^s}\big|\\ &\quad\lesssim F^{\frac{1}{2}}_{\neq,2}(t)(1+\langle t\rangle e^{-\frac{\lambda}{2}\nu^{\frac{1}{3}}t})(\delta\nu^{\frac{5}{6}+\frac{\kappa}{6(1+\kappa)}})F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{2}} + \big|\langle \mathcal{B}Q_{\neq,2},\mathcal{B}\partial_z|\nabla_L|^{-\frac{1}{2}}|\nabla_{L,h}|^{-\frac{1}{2}}\big[|\partial_y^L|^{\frac{1}{2}}U_{\neq,1}^3\partial_z|\nabla_{L,h}|^{\frac{1}{2}} U_{\neq,2}^2\big] \rangle_{H^s}\big|\\ &\quad\lesssim \delta\nu^{\frac{\kappa}{6(1+\kappa)}}F_{\neq,2}(t)+F_{\neq,2}^{\frac{1}{2}}(t)\nu^{-\frac{1}{4}}\langle t\rangle^{\frac{1}{2}}e^{-\frac{\lambda}{2}\nu^{\frac{1}{3}}t}(\delta\nu^{\frac{5}{6}+\frac{\kappa}{6(1+\kappa)}})F_{\neq,2}^{\frac{1}{2}}(t)\nu^{-\frac{1}{2}}\\ &\quad \lesssim \delta F_{\neq,2}(t)\nu^{\frac{\kappa}{6(1+\kappa)}-\frac{1}{12}}, \end{align}\] \[\begin{align} &\big|\langle\mathcal{B}Q_{\neq,2},\mathcal{B}\partial_z|\nabla_L|^{-\frac{1}{2}}|\nabla_{L,h}|^{-1}\big[\partial_iU_{0,1}^j\partial_j^LU_{\neq,2}^i\big]\rangle_{H^s}\big|\\&\quad\lesssim F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{4}}(\delta\nu^c)F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{2}}=\delta\nu^{c-\frac{3}{4}}F_{\neq,2}(t), \end{align}\] and \[\begin{align} &\big|\langle\mathcal{B}Q_{\neq,2},\mathcal{B}\partial_z|\nabla_L|^{-\frac{1}{2}}|\nabla_{L,h}|^{-1}\big[\partial_iU_{0,2}^j\partial_j^LU_{\neq,1}^i\big]\rangle_{H^s}\big|\\&\quad \lesssim F^{\frac{1}{2}}_{\neq,2}(t)F^{\frac{1}{2}}_{0,2}(t)\nu^{-\frac{1}{2}}\nu^{-\frac{1}{3}}(\delta\nu^{\frac{5}{6}+\frac{\kappa}{6(1+\kappa)}})\\&\quad\lesssim\delta F_{\neq,2}(t) + \delta\nu^{\frac{\kappa}{3(1+\kappa)}}F_{0,2}(t). \end{align}\]

Now we focus on the equation of \(K_{\neq,2}\). 79 and \(|k-k',l-l'|^{\frac{1}{2}}\lesssim |k',l'|^{\frac{1}{4}}|k,l|^{\frac{1}{4}}\) yields that \[\begin{align} &\sqrt{\frac{\beta}{\beta-1}}\big|\langle \mathcal{B}K_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{1}{2}}|\nabla_{L,h}|^{-1}\big[\partial_xU_{\neq,1}^{1,3}\cdot\nabla_{x,z} U^2_{\neq,2}+\partial_xU_{\neq,1}^2\partial_y^L U^2_{\neq,2}+\partial_xU_{\neq,2}\cdot\nabla_L U^2_{\neq,1}\big]\rangle_{H^s}\big|\\ &\quad\lesssim \sqrt{\frac{\beta}{\beta-1}}\left(\big|\langle \mathcal{B}K_{\neq,2},e^{\frac{\lambda}{2}\nu^{\frac{1}{3}}t}|\nabla_L||\nabla_{L,h}|^{-1}\big[\partial_x|\nabla_{x,z}|^{\frac{1}{2}}U_{\neq,1}^{1,3}\cdot|\nabla_{x,z}|^{\frac{1}{2}} U^2_{\neq,2}\big]\rangle_{H^s}\big|\right.\\ &\qquad \left.+F_{\neq,2}^{\frac{1}{2}}(t)\nu^{-\frac{1}{2}}(\delta\nu^{\frac{5}{6}+\frac{\kappa}{6(1+\kappa)}})F_{\neq,2}^{\frac{1}{2}}(t)\nu^{-\frac{1}{3}}+\nu^{-\frac{1}{6}}(F_{\neq,2}^{\frac{1}{4}}(t)\nu^{-\frac{1}{4}}(\delta\nu^b)^{\frac{1}{2}})^2F_{\neq,1}^{\frac{1}{2}}(t)\right)\\ &\quad \lesssim \sqrt{\frac{\beta}{\beta-1}}\left((1+\delta\nu^{\frac{\kappa}{6(1+\kappa)}})F_{\neq,2}(t) + \delta^2\nu^{2b-\frac{4}{3}}F_{\neq,1}(t)\right),\\ &\sqrt{\frac{\beta}{\beta-1}}\big|\langle \mathcal{B}K_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{1}{2}}|\nabla_{L,h}|^{-1}\big[\partial_xU_{\neq,2}\cdot\nabla U_{0,1}^2-\partial_y^LU_{\neq,2}\cdot\nabla U_{0,1}^1-\partial_yU_{0,1}\cdot\nabla_LU^1_{\neq,2}\big]\rangle_{H^s}\big|\\ &\quad\lesssim \sqrt{\frac{\beta}{\beta-1}}F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{3}}F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{2}}(\delta\nu^c)\\&\quad=\sqrt{\frac{\beta}{\beta-1}}\delta\nu^{c-\frac{5}{6}} F_{\neq,2}(t), \end{align}\] and \[\begin{align} &\sqrt{\frac{\beta}{\beta-1}}\big|\langle \mathcal{B}K_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{1}{2}}|\nabla_{L,h}|^{-1}\big[\partial_xU_{\neq,1}\cdot\nabla U_{0,2}^2-U_{\neq,2}\cdot\nabla\partial_yU^1_{0,1}\big]\rangle_{H^s}\big|\\ &\quad\lesssim \sqrt{\frac{\beta}{\beta-1}}F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{6}}(\delta\nu^{\frac{5}{6}+\frac{\kappa}{6(1+\kappa)}})F^{\frac{1}{2}}_{0,2}(t)\nu^{-\frac{1}{2}} + \sqrt{\frac{\beta}{\beta-1}}F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{6}}F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{3}}(\delta\nu^c)\\ & \quad\lesssim\delta F_{\neq,2}(t) + \frac{\beta}{\beta-1}\delta\nu^{\frac{1}{3}+\frac{\kappa}{3(1+\kappa)}}F_{0,2}(t) + \sqrt{\frac{\beta}{\beta-1}}\delta\nu^{c-\frac{1}{2}} F_{\neq,2}(t). \end{align}\]

Now we divide the term \(\partial_y^LU_{\neq,1}\cdot\nabla U_{0,2}^1+\partial_yU_{0,2}\cdot\nabla_LU^1_{\neq,1}=(\partial_yU_{0,2}^2\partial_y^LU_{\neq,1}^1 + \partial_y^LU_{\neq,1}^3\partial_zU_{0,2}^1) + (\partial_y^LU_{\neq,1}^2\partial_yU_{0,2}^1+\partial_yU_{0,2}^{1,3}\cdot\nabla_{x,z}U_{\neq,1}^1)\). The estimate of first one is similar to \(U_{0,2}^2\partial_y^LU_{\neq,1}^3\), we discuss two case \(|l|\leq\sqrt{p_h}\) and \(|l|>\sqrt{p_h}\) using 81 gives \[\begin{align} &\sqrt{\frac{\beta}{\beta-1}}\big|\langle \mathcal{B}K_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{1}{2}}|\nabla_{L,h}|^{-1}\big[\partial_yU_{0,2}^2\partial_y^LU_{\neq,1}^1 + \partial_y^LU_{\neq,1}^3\partial_zU_{0,2}^1\big]\rangle_{H^s}\big|\\ &\quad\lesssim \sqrt{\frac{\beta}{\beta-1}}F_{\neq,2}^{\frac{1}{2}}\nu^{-\frac{\kappa}{6(1+\kappa)}}E_{\neq,1}^{\frac{1}{2}}\nu^{-\frac{1}{6}}F_{0,2}^{\frac{1}{2}}\nu^{-\frac{1}{2}} \\ &\qquad+ \sqrt{\frac{\beta}{\beta-1}}\big|\langle \mathcal{B}|\nabla_L|^{\frac{1}{2}}|\nabla_{L,h}|^{-\frac{1}{2}}K_{\neq,2},e^{\frac{\lambda}{2}\nu^{\frac{1}{3}}t}\frac{|\nabla_{x,y}|^{\frac{1}{2}}}{\langle t\rangle^{\frac{1}{2}}}\big[\partial_zU_{0,2}\nu^{-\frac{1}{6}}|\nabla_L|^{\frac{1}{2}}m(Q_{\neq,1},K_{\neq,1})\big]\rangle_{H^s}\big|\\ &\quad\lesssim \sqrt{\frac{\beta}{\beta-1}}\left(F_{\neq,2}^{\frac{1}{2}}\nu^{-\frac{\kappa}{6(1+\kappa)}}E_{\neq,1}^{\frac{1}{2}}\nu^{-\frac{1}{6}}F_{0,2}^{\frac{1}{2}}\nu^{-\frac{1}{2}}+F_{\neq,2}^{\frac{1}{2}}(t)\nu^{-\frac{1}{4}}\nu^{-\frac{1}{6}}E_{\neq,1}^{\frac{1}{2}}F_{0,2}^{\frac{1}{2}}\nu^{-\frac{1}{2}}\right)\\ &\quad\lesssim \delta F_{\neq,2}(t) + \frac{\beta}{\beta-1}[\delta\nu^{2c-\frac{11}{6}}F_{0,2}(t)+\delta \nu^{\frac{1}{3}}F_{0,2}(t)], \end{align}\] Similarly, we have \[\begin{align} &\sqrt{\frac{\beta}{\beta-1}}\big|\langle \mathcal{B}K_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{1}{2}}|\nabla_{L,h}|^{-1}\big[\partial_y^LU_{\neq,1}^2\partial_yU_{0,2}^1+\partial_yU_{0,2}^{1,3}\cdot\nabla_{x,z}U_{\neq,1}^1\big]\rangle_{H^s}\big|\\ &\quad\lesssim \sqrt{\frac{\beta}{\beta-1}}\left(F_{\neq,2}^{\frac{1}{2}}(t)\nu^{-\frac{\kappa}{6(1+\kappa)}}\nu^{-\frac{1}{6}}E_{\neq,1}^{\frac{1}{2}}F_{0,2}^{\frac{1}{2}}(t)\nu^{-\frac{1}{2}} + F_{\neq,2}^{\frac{1}{2}}(t)F_{0,2}^{\frac{1}{2}}\nu^{-\frac{1}{2}}E_{\neq,1}^{\frac{1}{2}}\right)\\ &\quad\lesssim\delta F_{\neq,2}(t) + \frac{\beta}{\beta-1}[\delta\nu^{2c-1}F_{0,2}(t) + \delta\nu^{\frac{1}{3}}F_{0,2}(t)], \end{align}\] where we have used the fact \(mp^{\frac{1}{4}}p_{h}^{-\frac{1}{2}}\lesssim \frac{\langle l \rangle^{\frac{1}{2}}}{p_{h}^{\frac{1}{2}}},\) if \(|l|\geq \sqrt{p_{h}}\) and \(mp^{\frac{1}{4}}p_{h}^{-\frac{1}{2}}\lesssim m|\nabla_{L}|^{-\frac{1}{2}},\) if \(|l|\leq \sqrt{p_{h}}\).

For \(\partial_y^LU_{\neq,1}\cdot\nabla_LU^1_{\neq,2}+\partial_y^LU_{\neq,2}\cdot\nabla_LU^1_{\neq,1}\), we use 78 , ?? , the facts \(|\hat{\nabla}\lesssim |\hat{\nabla}|\langle t \rangle\) and \(\sqrt{|k',l'|}\leq C\sqrt{|k-k',l-l'|}+C\sqrt{|k,l|}\) to obtain \[\begin{align} &\sqrt{\frac{\beta}{\beta-1}}\big|\langle \mathcal{B}K_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{1}{2}}|\nabla_{L,h}|^{-1}\big[\partial_y^LU_{\neq,1}\cdot\nabla_LU^1_{\neq,2}+\partial_y^LU_{\neq,2}\cdot\nabla_LU^1_{\neq,1}\big]\rangle_{H^s}\big|\\ &\quad\lesssim \sqrt{\frac{\beta}{\beta-1}}\sum_{k,k',l,l'}\int_{\eta,\eta'}\langle\hat{\nabla}\rangle^s|\widehat{\mathcal{B}K_{\neq,2}}(k,\eta,l)|\frac{e^{\frac{\lambda}{2}\nu^{\frac{1}{3}}t}\sqrt{l}}{\sqrt{p_h}(k,\eta)}\langle\hat{\nabla}\rangle^s\left[|\widehat{\partial_y^LU_{\neq,1}}(k-k',\eta-\eta',l-l')||(k',\eta'-k't,l')|\right.\\ &\qquad\left.\times |\widehat{U_{\neq,2}^1}(k',\eta',l')| + |\widehat{\partial_y^LU_{\neq,2}}(k-k',\eta-\eta',l-l')||(k',\eta'-k't,l')||\widehat{U_{\neq,1}^1}(k',\eta',l')|\right]{\rm d}\eta{\rm d}\eta'\\ &\qquad+ \sqrt{\frac{\beta}{\beta-1}}\sum_{k,k',l,l'}\int_{\eta,\eta'}\langle\hat{\nabla}\rangle^s|\widehat{\mathcal{B}K_{\neq,2}}(k,\eta,l)|\frac{e^{\frac{\lambda}{2}\nu^{\frac{1}{3}}t}}{\sqrt{|k,l|}}\langle\hat{\nabla}\rangle^s\left[|\widehat{\partial_y^LU_{\neq,1}^{1,3}}(k-k',\eta-\eta',l-l')|\right.\\ &\qquad \times|\widehat{\nabla_{x,z}U_{\neq,2}^1}(k',\eta',l')|+\widehat{\partial_y^LU_{\neq,2}^{1,3}}(k-k',\eta-\eta',l-l')||\widehat{\nabla_{x,z}U_{\neq,1}^1}(k',\eta',l')| + |\widehat{\partial_y^LU_{\neq,1}^2}(k-k',\eta-\eta',l-l')|\\ &\qquad \times|\widehat{\partial_y^L U_{\neq,2}^1}(k',\eta',l')|\left.+ |\widehat{\partial_y^LU_{\neq,2}^2}(k-k',\eta-\eta',l-l')||\widehat{\partial_y^L U_{\neq,1}^1}(k',\eta',l')|\right]{\rm d}\eta{\rm d}\eta'\\ &\quad\lesssim \sqrt{\frac{\beta}{\beta-1}}\left(F^{\frac{1}{2}}_{\neq,2}(t)(\delta\nu^{\frac{5}{6}+\frac{\kappa}{6(1+\kappa)}})\nu^{-\frac{1}{3}}F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{2}} + F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{6}}(\delta\nu^{\frac{5}{6}+\frac{\kappa}{6(1+\kappa)}})\nu^{-\frac{1}{3}}F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{6}}\right.\\ &\qquad\left.+ F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{6}}F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{2}}(\delta\nu^{\frac{5}{6}+\frac{\kappa}{6(1+\kappa)}}) + F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{6}}F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{3}}(\delta\nu^{\frac{5}{6}+\frac{\kappa}{6(1+\kappa)}})\nu^{-\frac{1}{3}}\right)\\&\quad \lesssim \sqrt{\frac{\beta}{\beta-1}}\delta\nu^{\frac{\kappa}{6(1+\kappa)}}F_{\neq,2}(t). \end{align}\]

Using \(mp^{\frac{1}{4}}\lesssim \min\{\nu^{-\frac{1}{6}}p^{\frac{1}{4}},p^{\frac{1}{2}}\}\), 44 , 78 and recalling \(W_{\neq}^3=i\sqrt{\frac{\beta-1}{\beta}}|\nabla_L|^{-\frac{1}{2}}|\nabla_{L,h}|K_{\neq}\), and integration by parts, we have \[\begin{align} &\sqrt{\frac{\beta}{\beta-1}}\big|\langle \mathcal{B}K_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{1}{2}}|\nabla_{L,h}|^{-1}\big[U_{\neq,1}^{1,3}\cdot\nabla_{x,z}W^3_{\neq,2}+U_{\neq,1}^2\partial_y^L W^3_{\neq,2}\big]\rangle_{H^s}\big|\\ & \quad\lesssim\big|\langle \mathcal{B}K_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{1}{2}}\big[U_{\neq,1}^{1,3}\cdot\nabla_{x,z}|\nabla_L|^{-\frac{1}{2}}K_{\neq,2}\big]\rangle_{H^s}\big| + \big|\langle \mathcal{B}K_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{1}{2}}\big[U_{\neq,1}^2\partial_y^L|\nabla_L|^{-\frac{1}{2}}K_{\neq,2}\big]\rangle_{H^s}\big|\\ &\qquad+ \big|\langle \mathcal{B}K_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{1}{2}}|\nabla_{L,h}|^{-1}\big[|\nabla_{L,h}|U_{\neq,1}^{1,3}\cdot\nabla_{x,z}|\nabla_L|^{-\frac{1}{2}}K_{\neq,2}\big]\rangle_{H^s}\big|\\ &\qquad+ \big|\langle \mathcal{B}K_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{1}{2}}|\nabla_{L,h}|^{-1}\big[|\nabla_{L,h}|U_{\neq,1}^2\partial_y^L|\nabla_L|^{-\frac{1}{2}}K_{\neq,2}\big]\rangle_{H^s}\big|\\ &\quad\lesssim F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{2}}(\delta\nu^{\frac{5}{6}+\frac{\kappa}{6(1+\kappa)}})F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{3}} + \nu^{-\frac{1}{6}}F^{\frac{1}{4}}_{\neq,2}(t)\nu^{-\frac{1}{4}}(\delta\nu^b)^{\frac{1}{2}}\|U_{\neq,1}^2\|_{H^s}F^{\frac{1}{4}}_{\neq,2}(t)\nu^{-\frac{1}{4}}(\delta\nu^b)^{\frac{1}{2}}\nu^{-\frac{1}{6}} \\ &\qquad+ \sum_{k,k',l,l'}\int_{\eta,\eta'}\langle\hat{\nabla}\rangle^s|\widehat{\mathcal{B}K_{\neq,2}}(k,\eta,l)|\frac{e^{\frac{\lambda}{2}\nu^{\frac{1}{3}}t}\sqrt{l}}{\sqrt{p_h}(k,\eta)}\langle\hat{\nabla}\rangle^s|\widehat{|\nabla_{L,h}|U_{\neq,1}^{1,3}}(k-k',\eta-\eta',l-l')\\ &\qquad\times\widehat{\nabla_{x,z}|\nabla_{L}|^{-\frac{1}{2}}K_{\neq,2}}(k',\eta',l')|{\rm d}\eta{\rm d}\eta'+ \sum_{k,k',l,l'}\int_{\eta,\eta'}\langle\hat{\nabla}\rangle^s|\widehat{\mathcal{B}K_{\neq,2}}(k,\eta,l)|\frac{e^{\frac{\lambda}{2}\nu^{\frac{1}{3}}t}}{\sqrt{|k,l|}}\langle\hat{\nabla}\rangle^s\\ &\qquad\times|\widehat{|\nabla_{L,h}|U_{\neq,1}^{1,3}}(k-k',\eta-\eta',l-l')\widehat{\nabla_{x,z}|\nabla_{L}|^{-\frac{1}{2}}K_{\neq,2}}(k',\eta',l')|{\rm d}\eta{\rm d}\eta'+ (\delta\nu^b)F^{\frac{1}{2}}_{\neq,1}(t)\nu^{-\frac{1}{6}}F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{2}}\nu^{-\frac{1}{6}}\\ &\quad\lesssim \delta\nu^{\frac{\kappa}{6(1+\kappa)}}F_{\neq,2}(t) + \delta\nu^{b-\frac{5}{6}}F^{\frac{1}{2}}_{\neq,2}(t)F_{\neq,1}^{\frac{1}{2}}(t)+ F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{\kappa}{6(1+\kappa)}}\nu^{-\frac{1}{6}}(\delta\nu^{\frac{5}{6}+\frac{\kappa}{6(1+\kappa)}})F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{2}}\nu^{-\frac{1}{6}} \\ &\qquad+ F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{6}}\nu^{-\frac{1}{3}}(\delta\nu^{\frac{5}{6}+\frac{\kappa}{6(1+\kappa)}})F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{6}}\nu^{-\frac{1}{6}}+ \delta F_{\neq,2}(t) + \delta\nu^{2b-\frac{5}{3}}F_{\neq,1}(t)\\ &\quad \lesssim 5\delta F_{\neq,2}(t) + \delta\nu^{2b-\frac{5}{3}}F_{\neq,1}(t). \end{align}\] The estimate of \(U_{0,1}\cdot\nabla_LW^3_{\neq,2}\) is similar to \(U_{0,1}^2\partial_y^L W^3_{\neq,1}\), via the integration by part, we obtain \[\begin{align} &\sqrt{\frac{\beta}{\beta-1}}\big|\langle \mathcal{B}K_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{1}{2}}|\nabla_{L,h}|^{-1}\big[U_{0,1}\cdot\nabla_LW^3_{\neq,2}\big]\rangle_{H^s}\big|\\ &\quad\leq \big|\langle \mathcal{B}K_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{1}{2}}|\nabla_{L,h}|^{-1}\big[U_{0,1}\cdot\nabla_L|\nabla_L|^{-\frac{1}{2}}|\nabla_{L,h}|K_{\neq,2}\big]\rangle_{H^s}\big| \\ &\quad\lesssim \big|\langle \mathcal{B}K_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{1}{2}}\big[U_{0,1}|\nabla_L|^{\frac{1}{2}}K_{\neq,2}\big]\rangle_{H^s}\big| + \big|\langle \mathcal{B}K_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{1}{2}}|\nabla_{L,h}|^{-1}\big[|\partial_y|U_{0,1}\|\nabla_L|^{\frac{1}{2}}K_{\neq,2}\big]\rangle_{H^s}\big|\\ &\quad\lesssim F_{\neq,2}^{\frac{1}{2}}(t)\nu^{-\frac{1}{3}}(\|U_{0,1}\|_{H^{s+\frac{1}{2}}}+\|U_{0,1}\|_{H^{s+\frac{3}{2}}})F_{\neq,2}^{\frac{1}{2}}(t)\nu^{-\frac{1}{3}}\\&\quad\lesssim \delta\nu^{c-\frac{2}{3}}F_{\neq,2}(t). \end{align}\]

Using 78 , we have \[\begin{align} &\sqrt{\frac{\beta}{\beta-1}}\big|\langle \mathcal{B}K_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{1}{2}}|\nabla_{L,h}|^{-1}\big[U_{\neq,2}\cdot\nabla_LW^3_{\neq,1}\big]\rangle_{H^s}\big|\\ &\quad=\big|\langle \mathcal{B}K_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{1}{2}}|\nabla_{L,h}|^{-1}\big[U_{\neq,2}\cdot\nabla_L|\nabla_L|^{-\frac{1}{2}}|\nabla_{L,h}|K_{\neq,1}\big]\rangle_{H^s}\big|\\ &\quad\lesssim \sum_{k,k',l,l'}\int_{\eta,\eta'}\Big[\langle\hat{\nabla}\rangle^s|\widehat{\mathcal{B}K_{\neq,2}}(k,\eta,l)|\frac{e^{\frac{\lambda}{2}\nu^{\frac{1}{3}}t}\sqrt{l}}{p_h^{\frac{1}{4}}(k,\eta)}\langle\hat{\nabla}\rangle^s(|\widehat{U_{\neq,2}^{2}}(k-k',\eta-\eta',l-l')\widehat{|\nabla_{L}|^{\frac{1}{2}}|\nabla_{L,h}|^{\frac{1}{2}}K_{\neq,1}}(k',\eta',l')|) \\ & \qquad+ \langle\hat{\nabla}\rangle^s|\widehat{\mathcal{B}K_{\neq,2}}(k,\eta,l)|\frac{e^{\frac{\lambda}{2}\nu^{\frac{1}{3}}t}\sqrt{l}}{\sqrt{p_h}(k,\eta)}\langle\hat{\nabla}\rangle^s(|\widehat{|\nabla_{L,h}|^{\frac{1}{2}}U_{\neq,2}^{2}}(k-k',\eta-\eta',l-l')\widehat{|\nabla_{L}|^{\frac{1}{2}}|\nabla_{L,h}|^{\frac{1}{2}}K_{\neq,1}}(k',\eta',l')|) \\ & \qquad+ \langle\hat{\nabla}\rangle^s|\widehat{\mathcal{B}K_{\neq,2}}(k,\eta,l)|\frac{e^{\frac{\lambda}{2}\nu^{\frac{1}{3}}t}\sqrt{l}}{\sqrt{p_h}(k,\eta)}\langle\hat{\nabla}\rangle^s(|\widehat{|U_{\neq,2}^{1,3}}(k-k',\eta-\eta',l-l')\widehat{|\nabla_{L}|^{\frac{1}{2}}|\nabla_{L,h}|K_{\neq,1}}(k',\eta',l')|) \\ &\qquad+ \langle\hat{\nabla}\rangle^s|\widehat{\mathcal{B}K_{\neq,2}}(k,\eta,l)|\frac{e^{\frac{\lambda}{2}\nu^{\frac{1}{3}}t}}{\sqrt{|k,l|}}\langle\hat{\nabla}\rangle^s(|\widehat{U_{\neq,2}^{1,3}}(k-k',\eta-\eta',l-l')\widehat{\nabla_{x,z}|\nabla_{L}|^{-\frac{1}{2}}|\nabla_{L,h}|K_{\neq,1}}(k',\eta',l')|) \\ &\qquad+ \langle\hat{\nabla}\rangle^s|\widehat{\mathcal{B}K_{\neq,2}}(k,\eta,l)|\frac{e^{\frac{\lambda}{2}\nu^{\frac{1}{3}}t}}{\sqrt{|k,l|}}\langle\hat{\nabla}\rangle^s(|\widehat{U_{\neq,2}^2}(k-k',\eta-\eta',l-l')\widehat{\partial_y^L|\nabla_{L}|^{-\frac{1}{2}}|\nabla_{L,h}|K_{\neq,1}}(k',\eta',l')|)\Big]{\rm d}\eta{\rm d}\eta'\\ &\quad\lesssim F_{\neq,2}^{\frac{1}{2}}\nu^{-\frac{\kappa}{6(1+\kappa)}}F_{\neq,2}^{\frac{1}{2}}\nu^{-\frac{1}{3}}\langle t\rangle^{\frac{3}{2}} e^{-\frac{\lambda}{2}\nu^{\frac{1}{3}}t}\delta\nu^{\frac{5}{6}+\frac{\kappa}{6(1+\kappa)}}+ F^{\frac{1}{2}}_{\neq,2}(t)F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{6}}\langle t\rangle^2e^{-\frac{\lambda}{2}\nu^{\frac{1}{3}}t}\delta\nu^{\frac{5}{6}+\frac{\kappa}{6(1+\kappa)}} \\ &\qquad+F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{6}}F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{6}}\langle t\rangle e^{-\frac{\lambda}{2}\nu^{\frac{1}{3}}t}\delta\nu^{\frac{5}{6}+\frac{\kappa}{6(1+\kappa)}}+ F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{6}}F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{\kappa}{6(1+\kappa)}}\langle t\rangle^2 e^{-\frac{\lambda}{2}\nu^{\frac{1}{3}}t}\delta\nu^{\frac{5}{6}+\frac{\kappa}{6(1+\kappa)}}\\ &\quad\lesssim \delta F_{\neq,2}(t), \end{align}\] where we have using the following inequality \[\begin{align} \langle l\rangle^{\frac{1}{2}}p^{\frac{1}{4}}p_h^{\frac{1}{4}}=\langle l\rangle^{\frac{1}{2}}p^{\frac{1}{4}}p_h^{\frac{1}{4}}m \frac{p^{\frac{1}{4}}}{|k,l|^{\frac{1}{2}}}\lesssim p^{\frac{1}{2}}p_{h}^{\frac{1}{4}}m\lesssim|\hat{\nabla}|^{\frac{3}{2}}\langle t\rangle^{\frac{3}{2}}m, \end{align}\] and \[|k,l|p^{-\frac{1}{4}}p_{h}^{\frac{1}{2}}\lesssim \sqrt{|k,l|}|\nabla|^{\frac{1}{2}} \langle t\rangle^{2}m.\]

Similarly, one has \[\begin{align} &\sqrt{\frac{\beta}{\beta-1}}\big|\langle \mathcal{B}K_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{1}{2}}|\nabla_{L,h}|^{-1}\big[U_{0,2}^{1,3}\cdot\nabla_{x,z}W^3_{\neq,1}\big]\rangle_{H^s}\big|\\ &=\big|\langle \mathcal{B}K_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{1}{2}}|\nabla_{L,h}|^{-1}\big[U_{0,2}^{1,3}\cdot\nabla_{x,z}|\nabla_L|^{-\frac{1}{2}}|\nabla_{L,h}|K_{\neq,1}\big]\rangle_{H^s}\big|\\ &\quad\lesssim F^{\frac{1}{2}}_{\neq,2}(t)(\delta\nu^a)\langle t\rangle e^{-\frac{\lambda}{2}\nu^{\frac{1}{3}}t}F^{\frac{1}{2}}_{\neq,1}(t)\nu^{-\frac{1}{6}} + F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{6}}(\delta\nu^a)\langle t\rangle e^{-\frac{\lambda}{2}\nu^{\frac{1}{3}}t}F^{\frac{1}{2}}_{\neq,1}(t)\nu^{-\frac{1}{6}}\\& \quad \lesssim\delta F_{\neq,2}(t) + \delta\nu^{2a-\frac{4}{3}}F_{\neq,1}(t). \end{align}\] The estimate of \(U_{0,2}^2\partial_y^L W^3_{\neq,1}\) is similar to the \(U_{0,2}^2\partial_y^LU_{\neq,1}^3\) in the equation of \(Q_{\neq,2}\), it obtains that \[\begin{align} &\sqrt{\frac{\beta}{\beta-1}}\big|\langle \mathcal{B}K_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{1}{2}}|\nabla_{L,h}|^{-1}\big[U_{0,2}^2\partial_y^LW^3_{\neq,1}\big]\rangle_{H^s}\big|\\ &\quad=\big|\langle \mathcal{B}K_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{1}{2}}|\nabla_{L,h}|^{-1}\big[U_{0,2}^2\partial_y^L|\nabla_L|^{-\frac{1}{2}}|\nabla_{L,h}|K_{\neq,1}\big]\rangle_{H^s}\big|\\&\quad \lesssim \big|\langle \mathcal{B}|\nabla_L|^{\frac{1}{2}}|\nabla_{L,h}|^{-\frac{1}{2}}K_{\neq,2},e^{\frac{\lambda}{2}\nu^{\frac{1}{3}}t}\frac{|\nabla_{x,y}|^{\frac{1}{2}}}{\langle t\rangle^{\frac{1}{2}}}\big[|\nabla_{y,z}|^{\frac{1}{2}}U_{0,2}^2\partial_y^L|\nabla|^{\frac{1}{2}}\langle t\rangle^{\frac{1}{2}}mK_{\neq,1}\big]\rangle_{H^s}\big|\\ &\quad\lesssim \big|\langle \mathcal{B}|\nabla_L|^{\frac{1}{2}}|\nabla_{L,h}|^{-\frac{1}{2}}K_{\neq,2},e^{\frac{\lambda}{2}\nu^{\frac{1}{3}}t}\big[|\partial_z|^{\frac{1}{2}}|\nabla|^{\frac{1}{2}}U_{0,2}|\nabla|^{\frac{3}{2}}\langle t\rangle^{\frac{1}{2}}m|\nabla_L|^{\frac{1}{2}}K_{\neq,1}\big]\rangle_{H^s}\big|\\ &\quad\lesssim F_{\neq,2}^{\frac{1}{2}}(t)\nu^{-\frac{1}{4}}F_{0,2}^{\frac{1}{4}}(t)\nu^{-\frac{1}{4}}E_{0,2}^{\frac{1}{4}}(t)F_{\neq,1}^{\frac{1}{4}}(t)\nu^{-\frac{1}{4}}E_{\neq,1}^{\frac{1}{4}}(t)\nu^{-\frac{1}{6}}\\&\quad\lesssim \delta F_{\neq,2}(t) + \nu^{a-1+\frac{\kappa}{6(1+\kappa)}}F_{0,2}^{\frac{1}{2}}(t)F_{\neq,1}^{\frac{1}{2}}(t). \end{align}\]

Finally, the estimate of \(U_{\neq,1}^2\partial_{yy}U^1_{0,2}+U_{\neq,1}^3\partial_z\partial_yU^1_{0,2}\) is similar to \(\partial_y^LU_{\neq,1}\cdot\nabla U_{0,2}^1+\partial_yU_{0,2}\cdot\nabla_LU^1_{\neq,1}\). Integrating by part gives \[\begin{align} &\sqrt{\frac{\beta}{\beta-1}}\big|\langle \mathcal{B}K_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{1}{2}}|\nabla_{L,h}|^{-1}\big[U_{\neq,1}^2\partial_{yy}U^1_{0,2}+U_{\neq,1}^3\partial_z\partial_yU^1_{0,2}\big]\rangle_{H^s}\big|\\&\quad \lesssim \sqrt{\frac{\beta}{\beta-1}}\big|\langle \mathcal{B}K_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{1}{2}}\big[U_{\neq,1}^2\partial_{y}U^1_{0,2}+U_{\neq,1}^3\partial_zU^1_{0,2}\big]\rangle_{H^s}\big|\\ &\qquad+ \sqrt{\frac{\beta}{\beta-1}}\big|\langle \mathcal{B}K_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{1}{2}}|\nabla_{L,h}|^{-1}\big[\partial_y^LU_{\neq,1}^2\partial_{y}U^1_{0,2}+\partial_y^LU_{\neq,1}^3\partial_zU^1_{0,2}\big]\rangle_{H^s}\big|\\ &\quad \lesssim\sqrt{\frac{\beta}{\beta-1}}\big|\langle \mathcal{B}K_{\neq,2},\mathcal{B}|\nabla_L|\big[|\partial_y^L|^{\frac{1}{2}}U_{\neq,1}^2|\partial_y|^{\frac{1}{2}}U^1_{0,2}\big]\rangle_{H^s}\big|+ \sqrt{\frac{\beta}{\beta-1}}F_{\neq,2}^{\frac{1}{2}}(t)\nu^{-\frac{1}{3}}E_{\neq,1}^{\frac{1}{2}}(t)F_{0,2}^{\frac{1}{2}}(t)\nu^{-\frac{1}{2}} \\ &\qquad+ \delta F_{\neq,2}(t) + \frac{\beta}{\beta-1}[\delta\nu^{2c-\frac{11}{6} }F_{0,2}(t)+ \nu^{\frac{1}{3}}F_{0,2}(t)] \\&\quad \lesssim \sqrt{\frac{\beta}{\beta-1}}F_{\neq,2}^{\frac{1}{2}}(t)\nu^{-\frac{1}{2}}\nu^{-\frac{1}{6}}F_{\neq,1}^{\frac{1}{4}}(t)E_{\neq,1}^{\frac{1}{4}}(t)F_{0,2}^{\frac{1}{4}}(t)\nu^{-\frac{1}{4}}E_{0,2}^{\frac{1}{4}}(t) \\ &\qquad+ \delta F_{\neq,2}(t) + \frac{\beta}{\beta-1}[\delta\nu^{2c-\frac{11}{6} }F_{0,2}(t) + \nu^{\frac{1}{3}}F_{0,2}(t) + \nu^{\frac{\kappa}{3(1+\kappa)}}F_{0,2}(t)]\\ &\quad\lesssim \delta F_{\neq,2}(t) + \frac{\beta}{\beta-1}[\delta\nu^{a- 1 + \frac{\kappa}{6(1+\kappa)}}F_{0,2}^{\frac{1}{2}}(t)F_{\neq,1}^{\frac{1}{2}}(t)+\delta\nu^{2c-\frac{11}{6} }F_{0,2}(t) + \nu^{\frac{1}{3}}F_{0,2}(t) + \nu^{\frac{\kappa}{3(1+\kappa)}}F_{0,2}(t)], \end{align}\] where in the third inequality we have used 44 and \[\begin{align} \widehat{{|\partial_{y}^{L}}|^{\frac{1}{2}}|U_{\neq,1}^{2}|}&\lesssim m p_{h}^{\frac{1}{4}}\frac{p^{\frac{1}{4}}}{|k,l|^{\frac{1}{2}}}\frac{1}{p_{h}^{\frac{1}{2}}}\left(\frac{|k|}{p^{\frac{1}{4}}}|\widehat{K_{\neq,1}}|+\frac{|l|}{p^{\frac{1}{4}}}|\widehat{Q_{\neq,1}}|\right)\\&\lesssim mp_{h}^{-\frac{1}{4}}\left(|k|^{\frac{1}{2}}|\widehat{K_{\neq,1}}|+|l|^{\frac{1}{2}}|\widehat{Q_{\neq,1}}|\right). \end{align}\]

Now we focus on the equation of \(H_{\neq,2}\). We use the fact that to obtain \[\begin{align} &\big|\langle\mathcal{B}H_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{1}{2}}\big[U_{0,2}^2\partial_y^L\Theta_{\neq,1}+U_{\neq,2}^{1,3}\cdot\nabla_{x,z}\Theta_{\neq,1}+U_{\neq,2}^2\partial_y^L\Theta_{\neq,1}\big]\rangle_{H^s}\big|\\&\quad \lesssim F_{\neq,2}^{\frac{1}{2}}(t)\nu^{-\frac{1}{3}}\Big(F_{0,2}^{\frac{1}{4}}(t)\nu^{-\frac{1}{4}}E_{0,2}^{\frac{1}{4}}F_{\neq,1}^{\frac{1}{4}}(t)\nu^{-\frac{1}{4}}E_{\neq,1}^{\frac{1}{4}}(t) +\nu^{-\frac{1}{6}} F_{\neq,2}^{\frac{1}{2}}(t)\nu^{-\frac{1}{6}}E_{\neq,1}^{\frac{1}{2}}(t)\\ &\qquad+\nu^{-\frac{1}{6}}F_{\neq,2}^{\frac{1}{2}}(t)\nu^{-\frac{\kappa}{6(1+\kappa)}}E_{\neq,1}^{\frac{1}{2}}(t)\nu^{-\frac{1}{3}}\Big)\\ &\quad \lesssim \delta F_{\neq,2}(t) + \delta\nu^{a-\frac{5}{6} + \frac{\kappa}{6(1+\kappa)}}F_{0,2}^{\frac{1}{2}}(t)F_{\neq,1}^{\frac{1}{2}}(t) + \nu^{c-\frac{2}{3}}F_{\neq,2}(t), \end{align}\] and \[\begin{align} &\big|\langle\mathcal{B}H_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{1}{2}}\big[U_{\neq,2}\cdot\nabla\Theta_{0,1} + U_{0,2}^{1,3}\cdot\nabla_{x,z}\Theta_{\neq,1}\big]\rangle_{H^s}\big|\\&\quad \lesssim F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{2}}F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{6}}(\delta\nu^c) + F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{2}}(\delta\nu^a)F^{\frac{1}{2}}_{\neq,1}(t)\nu^{-\frac{1}{6}}\\&\quad\lesssim \delta\nu^{c-\frac{2}{3}}F_{\neq,2}(t) + \delta F_{\neq,2}(t) + \delta\nu^{2a-\frac{4}{3}}F_{\neq,1}(t). \end{align}\]

For the term \(U^2_{\neq,1}\partial_y\Theta_{0,2}+U_{\neq,1}^3\partial_z\Theta_{0,2}\), whose estimate is similar to \(U_{\neq,1}^2\partial_{yy}U^1_{0,2}+U_{\neq,1}^3\partial_z\partial_yU^1_{0,2}\) in \(K_{\neq,2}\) equation as follows \[\begin{align} &\big|\langle\mathcal{B}H_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{1}{2}}\big[U^2_{\neq,1}\partial_y\Theta_{0,2}+U_{\neq,1}^3\partial_z\Theta_{0,2}\big]\rangle_{H^s}\big|\\ &\quad\lesssim \big|\langle\mathcal{B}|\nabla_L|H_{\neq,2},\mathcal{B}\big[|\partial_y^L|^{\frac{1}{2}}U^2_{\neq,1}|\partial_y|^{\frac{1}{2}}\Theta_{0,2}\big]\rangle_{H^s}\big| + F_{\neq,2}^{\frac{1}{2}}(t)\nu^{-\frac{1}{3}}E_{\neq,1}^{\frac{1}{2}}(t)F_{0,2}^{\frac{1}{2}}(t)\nu^{-\frac{1}{2}}\\ &\quad\lesssim F_{\neq,2}^{\frac{1}{2}}(t)\nu^{-\frac{1}{2}}\nu^{-\frac{1}{6}}F_{\neq,1}^{\frac{1}{4}}(t)E_{\neq,1}^{\frac{1}{4}}(t)F_{0,2}^{\frac{1}{4}}(t)\nu^{-\frac{1}{4}}E_{0,2}^{\frac{1}{4}}(t) + \delta F_{\neq,2}(t) + \delta\nu^{\frac{\kappa}{3(1+\kappa)}} F_{0,2}(t)\\ &\quad\lesssim \delta F_{\neq,2}(t) + \delta\nu^{\frac{\kappa}{3(1+\kappa)}} F_{0,2}(t) + \delta\nu^{a- 1 + \frac{\kappa}{6(1+\kappa)}}F_{0,2}^{\frac{1}{2}}(t)F_{\neq,1}^{\frac{1}{2}}(t). \end{align}\] The estimate of term \(U_{0,1}\cdot\nabla_L \Theta_{\neq,2}\) is similar to \(U_{0,1}^2\partial_y^L\Theta_{\neq,1}\), which gives \[\begin{align} \big|\langle\mathcal{B}H_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{1}{2}}\big[U_{0,1}\cdot\nabla_L\Theta_{\neq,2}\big]\rangle_{H^s}\big|&\lesssim \big|\langle\mathcal{B}|\nabla_L|^{\frac{1}{2}}H_{\neq,2},\big[|\nabla_{y,z}|^{\frac{1}{2}}U_{0,1}|\nabla_L|^{\frac{1}{2}}mH_{\neq,2}\big]\rangle_{H^s}\big|\\ &\lesssim F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{3}}\|U_{0,1}\|_{H^{s+\frac{1}{2}}}F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{3}}\\&\lesssim \delta\nu^{c-\frac{2}{3}} F_{\neq,2}(t). \end{align}\]

Due to the fact that \(|k,l||\widehat{\Theta_{\neq,2}}|\lesssim \sqrt{|k,l|}|\widehat{mH_{\neq,2}}|,\;\sqrt{|k',l'|}\leq C\sqrt{|k,l|}+ C\sqrt{|k-k',l-l'|}\) and \(mp^{\frac{1}{4}}\lesssim \min\{\nu^{-\frac{1}{6}}p^{\frac{1}{4}},p^{\frac{1}{2}}\}\), we have \[\begin{align} &\big|\langle\mathcal{B}H_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{1}{2}}\big[U_{\neq,1}^{1,3}\cdot\nabla_{x,z}\Theta_{\neq,2}+U_{\neq,1}^2\partial_y^L\Theta_{\neq,2}\big]\rangle_{H^s}\big|\\ &\quad\lesssim \big|\langle\mathcal{B}H_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{1}{2}}|\nabla_{x,z}|^{\frac{1}{2}}\big[U_{\neq,1}^{1,3}\cdot mH_{\neq,2}\big]\rangle_{H^s}\big| + \big|\langle\mathcal{B}H_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{1}{2}}\big[|\nabla_{x,z}|^{\frac{1}{2}}U_{\neq,1}^{1,3}\cdot mH_{\neq,2}\big]\rangle_{H^s}\big| \\ &\qquad+ \big|\langle\mathcal{B}H_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{1}{2}}\big[U_{\neq,1}^2\partial_y^L\Theta_{\neq,2}\big]\rangle_{H^s}\big|\\ &\quad\lesssim F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{2}}\nu^{-\frac{1}{6}}(\delta\nu^{\frac{5}{6}+\frac{\kappa}{6(1+\kappa)}})F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{6}} + \nu^{-\frac{1}{6}}F^{\frac{1}{4}}_{\neq,2}(t)\nu^{-\frac{1}{4}}(\delta\nu^b)^{\frac{1}{2}}\|U_{\neq,1}^2\|_{H^s}F^{\frac{1}{4}}_{\neq,2}(t)\nu^{-\frac{1}{4}}\nu^{-\frac{1}{6}}(\delta\nu^b)^{\frac{1}{2}}\\ & \quad\lesssim\delta\nu^{\frac{\kappa}{6(1+\kappa)}} F_{\neq,2}(t) + \delta\nu^b F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{5}{6}}F^{\frac{1}{2}}_{\neq,1}(t)\\&\quad\lesssim \delta\nu^{\frac{\kappa}{6(1+\kappa)}} F_{\neq,2}(t) + \delta F_{\neq,2}(t) + \delta\nu^{2b-\frac{5}{3}}F_{\neq,1}(t). \end{align}\]

4.5.3 Contribution without any external force term We now consider the contribution of the second system without any force term. We focus on the equation of \(Q_{\neq,2}\). Due to \(mp^{\frac{1}{4}}\lesssim \min\{\nu^{-\frac{1}{6}}p^{\frac{1}{4}},p^{\frac{1}{2}}\}\) and \(\sqrt{\eta-\eta'-(k-k')t}\lesssim \sqrt{|\eta'-k't|}p_h^{\frac{1}{4}}(k,\eta)\) , one gets \[\begin{align} &\big|\langle\mathcal{B}Q_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{3}{2}}|\nabla_{L,h}|^{-1}\big[U_{\neq,2}^{1,3}\cdot\nabla_{x,z}U^3_{\neq,2} + U_{0,2}\cdot\nabla_LU^3_{\neq,2}+U_{\neq,2}^2\partial_y^LU_{\neq,2}^3\big]\rangle_{H^s}\big|\\ &\quad \lesssim F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{2}}F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{2}}[\delta\nu^b+\delta\nu^a] + \big|\langle\mathcal{B}Q_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{3}{2}}|\nabla_{L,h}|^{-\frac{1}{2}}\big[|\partial_y^L|^{\frac{1}{2}}U_{\neq,2}^2|\partial_y^L|^{\frac{1}{2}}U_{\neq,2}^3\big]\rangle_{H^s}\big|\\ &\quad \lesssim\delta[\nu^{b-1}+\nu^{a-1}]F_{\neq,2}(t) + F_{\neq,2}^{\frac{1}{2}}(t)\nu^{-\frac{1}{2}}(F_{\neq,2}^{\frac{1}{4}}(t)\nu^{-\frac{1}{4}})^2(\delta\nu^b)\\ &\quad \lesssim \delta[\nu^{b-1}+\nu^{a-1}]F_{\neq,2}(t), \end{align}\] and since \(\sqrt{|\eta-\eta'|}\lesssim \sqrt{|\eta'-k't|}p_h^{\frac{1}{4}}(k,\eta)\), \[\begin{align} &\big|\langle\mathcal{B}Q_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{3}{2}}|\nabla_{L,h}|^{-1}\big[U_{\neq,2}^3\partial_z U_{0,2}^3+U_{\neq,2}^2\partial_y U_{0,2}^3\big]\rangle_{H^s}\big|\\ &\quad\lesssim F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{2}}(\delta\nu^b)F^{\frac{1}{2}}_{0,2}(t)\nu^{-\frac{1}{2}} + \big|\langle\mathcal{B}Q_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{3}{2}}|\nabla_{L,h}|^{-\frac{1}{2}}\big[|\partial_y^L|^{\frac{1}{2}}U_{\neq,2}^2|\partial_y|^{\frac{1}{2}} U_{0,2}^3\big]\rangle_{H^s}\big|\\ &\quad\lesssim \delta F_{\neq,2}(t) + \delta^2\nu^{2b-2}F_{0,2}(t) + F_{\neq,2}^{\frac{1}{2}}(t)\nu^{-\frac{1}{2}}F_{\neq,2}^{\frac{1}{4}}(t)\nu^{-\frac{1}{4}}F_{0,2}^{\frac{1}{4}}(t)\nu^{-\frac{1}{4}}\delta\nu^{\frac{a+b}{2}}\\ &\quad\lesssim \delta F_{\neq,2}(t) + (\delta^2\nu^{2b-2}+\delta^4\nu^{2a+2b-4})F_{0,2}(t). \end{align}\] For the pressure terms, due to the fact that \(\sqrt{|\eta-\eta'-(k-k')t|}\lesssim \sqrt{|\eta'-k't|}p_h^{\frac{1}{4}}(k,\eta)\) and \(\sqrt{|\eta-\eta'|}\lesssim \sqrt{|\eta'-kt|}p_h^{\frac{1}{4}}(k,\eta)\), we have \[\begin{align} &\big|\langle\mathcal{B}Q_{\neq,2},\mathcal{B}\partial_z|\nabla_L|^{-\frac{1}{2}}|\nabla_{L,h}|^{-1}\big[\partial_{x,z}U_{\neq,2}\cdot\nabla_L U_{\neq,2}^{1,3}+\partial_y^LU_{\neq,2}^{1,2}\cdot\nabla_{L,h}U_{\neq,2}^2+\partial_y^LU_{\neq,2}^3\partial_zU_{\neq,2}^2\big]\rangle_{H^s}\big|\\ &\quad +\big|\langle\mathcal{B}Q_{\neq,2},\mathcal{B}\partial_z|\nabla_L|^{-\frac{1}{2}}|\nabla_{L,h}|^{-1}\big[\partial_zU_{0,2}\cdot\nabla_LU_{\neq,2}^3+\partial_yU_{0,2}^{1,2}\cdot\nabla_{L,h}U_{\neq,2}^2 + \partial_yU_{0,2}^3\partial_zU_{\neq,2}^2\big]\rangle_{H^s}\big|\\ &\quad\lesssim (\delta\nu^b)(F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{2}} + F^{\frac{1}{2}}_{0,2}(t)\nu^{-\frac{1}{2}})F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{2}} \\ &\qquad+ \big|\langle\mathcal{B}Q_{\neq,2},\mathcal{B}\partial_z|\nabla_L|^{-\frac{1}{2}}|\nabla_{L,h}|^{-\frac{1}{2}}\big[(|\partial_y^L|^{\frac{1}{2}}U_{\neq,2}^3+|\partial_y|^{\frac{1}{2}}U_{0,2}^3)\partial_z|\partial_y^L|^{\frac{1}{2}}U_{\neq,2}^2\big]\rangle_{H^s}\big|\\ &\quad \lesssim \delta\nu^{b-1}F_{\neq,2}(t) + \delta F_{\neq,2}(t) + \delta^2\nu^{2b-2}F_{0,2}(t) + F_{\neq,2}^{\frac{1}{4}}(t)\nu^{-\frac{1}{4}}(\delta\nu^b)^{\frac{1}{2}}(F_{\neq,2}^{\frac{1}{4}}\nu^{-\frac{1}{4}}(\delta\nu^b)^{\frac{1}{2}}+F_{0,2}^{\frac{1}{4}}\nu^{-\frac{1}{4}}(\delta\nu^a)^{\frac{1}{2}})F_{\neq,2}^{\frac{1}{2}}(t)\nu^{-\frac{1}{2}}\\ &\quad \lesssim \delta\nu^{b-1}F_{\neq,2}(t) + \delta F_{\neq,2}(t) + (\delta^2\nu^{2b-2}+\delta^4\nu^{2a+2b-4})F_{0,2}(t) \end{align}\]

Next we focus on the equation of \(K_{\neq,2}\). Since \(\sqrt{|k'|}\lesssim \sqrt{|k-k'|}p_h^{\frac{1}{4}}(k,\eta)\), we have \[\begin{align} &\sqrt{\frac{\beta}{\beta-1}}\big|\langle\mathcal{B}K_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{1}{2}}|\nabla_{L,h}|^{-1}\big[\partial_xU_{\neq,2}^{1,2}\cdot\nabla_{L,h}U_{\neq,2}^2-\partial_y^LU_{\neq,2}\cdot\nabla_LU_{\neq,2}^1+\partial_xU_{\neq,2}^3\partial_z U_{\neq,2}^2\big]\rangle_{H^s}\big|\\ &\quad\lesssim \sqrt{\frac{\beta}{\beta-1}}(\delta\nu^b)F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{2}}F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{2}} + \sqrt{\frac{\beta}{\beta-1}}\big|\langle\mathcal{B}K_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{1}{2}}|\nabla_{L,h}|^{-\frac{1}{2}}\big[|\partial_x|^{\frac{1}{2}}U_{\neq,2}^3\partial_z|\partial_x|^{\frac{1}{2}} U_{\neq,2}^2\big]\rangle_{H^s}\big|\\ &\quad\lesssim \sqrt{\frac{\beta}{\beta-1}}\delta\nu^{b-1}F_{\neq,2}(t) + \sqrt{\frac{\beta}{\beta-1}}F_{\neq,2}^{\frac{1}{2}}(t)\nu^{-\frac{1}{4}}(\delta\nu^b)F_{\neq,2}^{\frac{1}{2}}(t)\nu^{-\frac{1}{2}}\\ &\quad\lesssim \sqrt{\frac{\beta}{\beta-1}}\delta\nu^{b-1}F_{\neq,2}(t), \end{align}\] and \[\begin{align} &\sqrt{\frac{\beta}{\beta-1}}\big|\langle\mathcal{B}K_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{1}{2}}|\nabla_{L,h}|^{-1}\big[\partial_xU_{\neq,2}\cdot\nabla U_{0,2}^2-\partial_y^LU_{\neq,2}\cdot\nabla U_{0,2}^1-\partial_yU_{0,2}\cdot\nabla_LU_{\neq,2}^1\big]\rangle_{H^s}\big|\\ &\quad\lesssim \sqrt{\frac{\beta}{\beta-1}}(\delta\nu^b)F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{2}}F^{\frac{1}{2}}_{0,2}(t)\nu^{-\frac{1}{2}}\\&\quad \lesssim \delta F_{\neq,2}(t) + \frac{\beta}{\beta-1}\delta\nu^{2b-2}F_{0,2}(t). \end{align}\] Integration by part yields that \[\begin{align} &\sqrt{\frac{\beta}{\beta-1}}\big|\langle\mathcal{B}K_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{1}{2}}|\nabla_{L,h}|^{-1}\big[U_{\neq,2}\cdot\nabla_LW_{\neq,2}^3\big]\rangle_{H^s}\big|\\ &\quad\lesssim\big|\langle\mathcal{B}K_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{1}{2}}\big[U_{\neq,2}\cdot\nabla_L|\nabla_L|^{-\frac{1}{2}}K_{\neq,2}\big]\rangle_{H^s}\big| \\ &\qquad+ \big|\langle\mathcal{B}K_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{1}{2}}|\nabla_{L,h}|^{-1}\big[|\nabla_{L,h}|U_{\neq,2}\cdot\nabla_L|\nabla_L|^{-\frac{1}{2}}K_{\neq,2}\big]\rangle_{H^s}\big|\\ &\quad\lesssim F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{2}}(\delta\nu^b)F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{2}} + (\delta\nu^b)F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{2}}F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{2}}\\ &\quad\lesssim \delta\nu^{b-1}F_{\neq,2}(t), \end{align}\] and \[\begin{align} &\sqrt{\frac{\beta}{\beta-1}}\big|\langle\mathcal{B}K_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{1}{2}}|\nabla_{L,h}|^{-1}\big[U_{0,2}\cdot\nabla_LW_{\neq,2}^3\big]\rangle_{H^s}\big|\\ &\quad\lesssim\big|\langle\mathcal{B}K_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{1}{2}}\big[U_{0,2}\cdot\nabla_L|\nabla_L|^{-\frac{1}{2}}K_{\neq,2}\big]\rangle_{H^s}\big| \\ &\qquad+ \big|\langle\mathcal{B}K_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{1}{2}}|\nabla_{L,h}|^{-1}\big[\partial_yU_{0,2}\cdot\nabla_L|\nabla_L|^{-\frac{1}{2}}K_{\neq,2}\big]\rangle_{H^s}\big|\\ &\quad\lesssim F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{2}}(\delta\nu^a)F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{2}} + (\delta\nu^b)F^{\frac{1}{2}}_{0,2}(t)\nu^{-\frac{1}{2}}F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{2}}\\ &\quad\lesssim \delta\nu^{a-1}F_{\neq,2}(t) + \delta F_{\neq,2}(t)+\delta\nu^{2b-2}F_{0,2}(t), \end{align}\] and \[\begin{align} &\sqrt{\frac{\beta}{\beta-1}}\big|\langle\mathcal{B}K_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{1}{2}}|\nabla_{L,h}|^{-1}\big[U_{\neq,2}\cdot\nabla\partial_yU_{0,2}^1\big]\rangle_{H^s}\big|\\ &\quad\lesssim\sqrt{\frac{\beta}{\beta-1}}\left(\big|\langle\mathcal{B}K_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{1}{2}}\big[U_{\neq,2}\cdot\nabla U_{0,2}^1\big]\rangle_{H^s}\big| + \big|\langle\mathcal{B}K_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{1}{2}}|\nabla_{L,h}|^{-1}\big[\partial_y^LU_{\neq,2}\cdot\nabla U_{0,2}^1\big]\rangle_{H^s}\big|\right)\\ &\quad\lesssim \sqrt{\frac{\beta}{\beta-1}}\left(F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{2}}(\delta\nu^b)F^{\frac{1}{2}}_{0,2}(t)\nu^{-\frac{1}{2}} + (\delta\nu^b)F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{2}}F^{\frac{1}{2}}_{0,2}(t)\nu^{-\frac{1}{2}}\right)\\&\quad\lesssim \delta F_{\neq,2}(t) + \frac{\beta}{\beta-1}\delta\nu^{2b-2}F_{0,2}(t). \end{align}\]

Finally, for the equation of \(H_{\neq,2}\), recall that \(H_{\neq}=-|\nabla_L|^{\frac{1}{2}}\Theta_{\neq}\) and using the fact that \(mp^{\frac{1}{4}}\lesssim \min\{\nu^{-\frac{1}{6}}p^{\frac{1}{4}},p^{\frac{1}{2}}\}\), we obtain \[\begin{align} &\big|\langle\mathcal{B}H_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{1}{2}}\big[U_{\neq,2}\cdot\nabla_L\Theta_{\neq,2} + U_{0,2}\cdot\nabla_L\Theta_{\neq,2}\big]\rangle_{H^s}\big|\\ &\quad\lesssim F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{2}}(\delta\nu^b+\delta\nu^a)F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{3}}\nu^{-\frac{1}{6}}\\&\quad=\delta(\nu^{b-1}+\nu^{a-1})F_{\neq,2}(t), \end{align}\] and \[\begin{align} &\big|\langle\mathcal{B}H_{\neq,2},\mathcal{B}|\nabla_L|^{\frac{1}{2}}\big[U_{\neq,2}\cdot\nabla\Theta_{0,2}\big]\rangle_{H^s}\big|\\ &\quad\lesssim F^{\frac{1}{2}}_{\neq,2}(t)\nu^{-\frac{1}{2}}(\delta\nu^b)F^{\frac{1}{2}}_{0,2}(t)\nu^{-\frac{1}{2}}\\&\quad\lesssim \delta F_{\neq,2}(t) + \delta\nu^{2b-2}F_{0,2}(t). \end{align}\] Integrating the above inequality respect to \(t\) immediately gives ?? and the proof of Proposition 25 is finished. \(\square\)

Now, we let \(c=5/6+\kappa/(6+6\kappa)\). Combing ?? with ?? obtain that \(a=b=1\), \(c=14/15,\;\kappa/(6+6\kappa)=1/10\) and finish the bootstrap argument.

5 Dynamics of nonzero modes when \(B_{\beta}>0\)↩︎

In fact, when \(B_{\beta}>0\), we can also prove the linear stability of the system of equations (1.6). Let us talk about the differences in the proof process here. At this point, it is sufficient to demonstrate the enhanced dissipation and and the inviscid damping effect at non-zero frequencies. We introduce a new group of good unknowns \[Q^*_{\neq}:=-|\nabla_L|^2|\nabla_{L,h}|^{-1}U^3_{\neq},\;K^*_{\neq}:=-i\sqrt{\frac{\beta}{\beta-1}}|\nabla_L||\nabla_{L,h}|^{-1}W_{\neq}^3,\;H_{\neq}^*:=-|\nabla_L|\Theta_{\neq},\] and then rewrite 25 as \[\begin{align} \label{Non0L1} \begin{cases} \partial_t \widehat{Q_{\neq}^*} + \nu p \widehat{Q_{\neq}^*} - \frac{1}{2} \frac{\partial_t p_{h}}{p_{h}}\widehat{Q_{\neq}^*} - \sqrt{B_{\beta}}\frac{l}{p^{1/2}}\widehat{K_{\neq}^*} + \alpha \frac{p_h^{1/2}}{p^{1/2}}\widehat{H_{\neq}^*} + 2\sqrt{\frac{\beta-1}{\beta}}\frac{k^2}{p_h}\frac{l}{p^{1/2}}\widehat{K_{\neq}^*} = 0, \\ \partial_t\widehat{K_{\neq}^*} + \nu p\widehat{K_{\neq}^*} +\frac{1}{2}\frac{l^{2}}{p}\frac{\partial_t p_h}{p_h}\widehat{K_{\neq}^*}+ \sqrt{B_{\beta}}\frac{l}{p^{1/2}}\widehat{Q_{\neq}^*} = 0,\\ \partial_t\widehat{H_{\neq}^*} + \nu p\widehat{H_{\neq}^*} -\frac{1}{2}\frac{\partial_{t}p}{p} \widehat{H_{\neq}^*}+ \alpha \frac{p_h^{1/2}}{p^{1/2}}\widehat{Q_{\neq}^*} = 0. \end{cases} \end{align}\tag{82}\] We use a suitably constructed Fourier multiplier \(m^*\) with symbol \(m^*=m^*(t,k,\eta,l)\), defined by \[\begin{align} \label{A46464646464} -\frac{\dot{m^*}}{m^*}=\left\{\begin{array}{ll} \frac{| k \left( \eta- kt \right)|}{k^2+\left(\eta- kt\right)^2}, & \text{ if } |t-\frac{\eta}{k}|\leq 1000\nu^{-\frac{1}{3}}, \\ 0, & \text{ if } |t-\frac{\eta}{k}|> 1000\nu^{-\frac{1}{3}}. \end{array}\right. \end{align}\tag{83}\] with the initial data \[m^*(0,k,\eta,l)=\begin{cases} 1,\;\text{ if } \;\frac{\eta}{k}\leq 0,\\ \left(\frac{k^2+\eta^2}{k^2}\right)^{\frac{1}{2}},\;\text{ if } \;0<\frac{\eta}{k}<1000\nu^{-\frac{1}{3}},\\ \left(\frac{k^2+(1000\nu^{-\frac{1}{3}}k)^2}{k^2}\right)^{\frac{1}{2}},\;\text{ if } \;1000\nu^{-\frac{1}{3}}\leq\frac{\eta}{k}. \end{cases}\] The role of \(m^{*}\) is to handle the slowly decaying term \(\frac{1}{2} \frac{\partial_t p_{h}}{p_{h}}\widehat{Q_{\neq}^*}\), \(\frac{1}{2}\frac{l^{2}}{p}\frac{\partial_{t}p_{h}}{p_{h}}\widehat{K_{\neq}^*}\) and \(\frac{1}{2} \frac{\partial_t p}{p}\widehat{H_{\neq}^*}\) in 82 , balancing the growth that solutions experience in the transition between the inviscid damping regime \(t<\frac{\eta}{k}\), and the dissipative regime \(t\gtrsim \frac{\eta}{k}+\nu^{-\frac{1}{3}}\).

Define the multiplier \[M^{*}=M_{1}M_{5},\] with \(M_{1}\) and \(M_{5}\) as in 27 and 31 respectively.

The proof of ?? ?? as in Theorem 1 can be directly obtained from the following Lemma, which gives the linear enhanced dissipation and inviscid damping on \(U_{\neq}\).

Proof.To control the evolution of the nonzero modes, we introduce the Fourier multiplier \[\label{A88} \mathcal{A}^*:=m^*M^{*}e^{\lambda^* \nu^{\frac{1}{3}}t},\qquad{(33)}\] which incorporates the multiplier \(m^*\) already used in the linear analysis, additional ghost weights \({M^*}\) as well as the time weight \(e^{\lambda^*\nu^{\frac{1}{3}}t}\), which \(\lambda^*\) can be confirmed later. Recall the system 82 , energy estimates yield that \[\label{A99} \begin{align} \frac{1}{2}&\frac{d}{dt}|\mathcal{A}(\widehat{Q_{\neq}^*},\widehat{K_{\neq}^*},\widehat{H_{\neq}^*})|^2 + \nu p |\mathcal{A}(\widehat{Q_{\neq}^*},\widehat{K_{\neq}^*},\widehat{H_{\neq}^*})|^2 +\left[(-\frac{\dot{M^{*}}}{M^*})+(-\frac{\dot{m^*}}{m^*})-\lambda^*\nu^{\frac{1}{3}}\right]|\mathcal{A}(\widehat{Q_{\neq}^*},\widehat{K_{\neq}^*},\widehat{H_{\neq}^*})|^2\\ &-\frac{1}{2}\frac{\partial_t p_{h}}{p_{h}}|\widehat{\mathcal{A}Q_{\neq}^*}|^2+\frac{1}{2}\frac{l^{2}}{p}\frac{\partial_t p_h}{p_h}|\widehat{\mathcal{A}K_{\neq}^*}|^2-\frac{1}{2}\frac{\partial_{t}{p}}{p}|\widehat{\mathcal{A}H_{\neq}^*}|+2\sqrt{\frac{\beta-1}{\beta}}\frac{k^2}{p_h}\frac{l}{\sqrt{p}}\widehat{\mathcal{A}K_{\neq}^*}\overline{\widehat{\mathcal{A}Q_{\neq}^*}}=0. \end{align}\qquad{(34)}\] Then we have \[\begin{align} \frac{1}{2}&\frac{d}{dt}|\mathcal{A}(\widehat{Q_{\neq}^*},\widehat{K_{\neq}^*},\widehat{H_{\neq}^*})|^2 +\left[ \nu p+\left(-\frac{\dot{M^*}}{M^*}\right)\right]|\mathcal{A}(\widehat{Q_{\neq}^*},\widehat{K_{\neq}^*},\widehat{H_{\neq}^*})|^2 \\ &=\lambda^*\nu^{\frac{1}{3}}|\mathcal{A}(\widehat{Q_{\neq}^*},\widehat{K_{\neq}^*},\widehat{H_{\neq}^*})|^2-\left[(-\frac{\dot{m^*}}{m^*})-\frac{1}{2} \frac{\partial_t p_{h}}{p_{h}}\right]|\widehat{\mathcal{A}Q_{\neq}^*}|^2-\left[(-\frac{\dot{m^*}}{m^*})+\frac{1}{2}\frac{l^{2}}{p}\frac{\partial_t p_h}{p_h}\right]|\widehat{\mathcal{A}K_{\neq}^*}|^2\\ &\quad-\left[(-\frac{\dot{m^*}}{m^*})-\frac{1}{2} \frac{\partial_t p}{p}\right]|\widehat{\mathcal{A}H_{\neq}^*}|^2 -2\sqrt{\frac{\beta-1}{\beta}}\frac{k^2}{p_h}\frac{l}{\sqrt{p}}\widehat{\mathcal{A}K_{\neq}^*}\overline{\widehat{\mathcal{A}Q_{\neq}^*}}. \end{align}\]

Donate that \(e_{\neq}^*(t):=|\mathcal{A}(\widehat{Q_{\neq}^*},\widehat{K_{\neq}^*},\widehat{H_{\neq}^*})|^2\). Notice that the definition of \(m^*\) as in 83 , we have \(-\frac{\dot{m^*}}{m^*}+\frac{\nu p}{2}-\frac{1}{2} \big|\frac{\partial_t p}{p_{h}}\big|\gtrsim 0\). A direct calculation yields \[\begin{align}\label{A010} &\frac{d}{dt}e_{\neq}^*(t) + \left[\nu p+(-\frac{\dot{M^*}}{M^*})\right]|\mathcal{A}(\widehat{Q_{\neq}^*},\widehat{K_{\neq}^*},\widehat{H_{\neq}^*})|^2\\ &\quad\leq 2\lambda^*\nu^{\frac{1}{3}}|\mathcal{A}(\widehat{Q_{\neq}^*},\widehat{K_{\neq}^*},\widehat{H_{\neq}^*})|^2-\left[(-\frac{\dot{M_5}}{M_5})+2\sqrt{\frac{\beta-1}{\beta}}\frac{k^2}{p_h}\frac{l}{\sqrt{p}}\right]2\widehat{\mathcal{A}K_{\neq}^*}\overline{\widehat{\mathcal{A}Q_{\neq}^*}}, \end{align}\tag{84}\] In the following, we introduce two new notations \[\begin{align} E_{\neq}^*(t):=&\|\mathcal{A}(Q_{\neq}^*,K_{\neq}^*,H_{\neq}^*)\|_{H^r}^2,\\ F_{\neq}^*(t):=&\nu\|\mathcal{A}\nabla_L (Q_{\neq}^*,K_{\neq}^*,H_{\neq}^*)\|_{H^r}^2+\left\|\mathcal{A}\sqrt{-\frac{\dot{M^*}}{M^*}} \left(Q_{\neq}^*,K_{\neq}^*,H_{\neq}^*\right)\right\|_{H^r}^2 , \end{align}\] for any \(r \geq 0\). Thus, recalling 84 and going back to the physical space, we obtain that \[\begin{align} &\frac{\rm d}{{\rm d}t}E_{\neq}^*(t) + F^*_{\neq}(t) \leq 2\lambda^*\nu^{\frac{1}{3}}\|\mathcal{A}(Q_{\neq}^*,K_{\neq}^*,H_{\neq}^*)\|_{H^r}^2. \end{align}\] Taking \(\lambda^*=\frac{1}{16}\) and integrating the above inequality in time yields that \[\begin{align} &\|\mathcal{A}(Q_{\neq}^*,K_{\neq}^*,H_{\neq}^*)\|_{H^r}^2 + \nu\|\mathcal{A}\nabla_L(Q_{\neq}^*,K_{\neq}^*,H_{\neq}^*)\|_{L^2_tH^r}^2 + \left\|\mathcal{A}\sqrt{-\frac{\dot{M^*}}{M^*}}(Q_{\neq}^*,K_{\neq}^*,H_{\neq}^*)\right\|_{L^2_tH^r}^2\\ & \quad\leq \|\mathcal{A}(Q_{\neq}^*,K_{\neq}^*,H_{\neq}^*)(t=0)\|_{H^r}^2 +\frac{\nu^{\frac{1}{3}}}{8}\|\mathcal{A}(Q_{\neq}^*,K_{\neq}^*,H_{\neq}^*)\|_{L^2_tH^r}^2. \end{align}\] Due to \(\mathcal{A}^*=m^*M^*e^{\lambda^* \nu^{\frac{1}{3}} t }\), ?? and ?? , we have \[\label{Ax460} \begin{align} &\|m^* M^*(Q_{\neq}^*,K_{\neq}^*,H_{\neq}^*)\|_{H^r}^2 + \frac{ \nu}{2}\|m^*M^*\nabla_L(Q_{\neq}^*,K_{\neq}^*,H_{\neq}^*)\|_{L^2 H^r}^2 + \frac{1}{2} \left\|\sqrt{-\dot{M^*} M^*}m^*(Q_{\neq}^*,K_{\neq}^*,H_{\neq}^*)\right\|_{L^2 H^r}^2 \\ & \quad\leq e^{-2\lambda^* \nu^{\frac{1}{3}} t} \| (Q_{\mathrm{in}}^*,K_{\mathrm{in}}^*,H_{\mathrm{in}}^*)\|_{H^{r+1}}^2. \end{align}\tag{85}\]

Next, recalling the definition of \(Q_{\neq}^{*}\), \(K_{\neq}^{*}\) and \(H_{\neq}^{*}\) and combining with ?? , we have the following fact \[\begin{align} \label{Ax461} |\widehat{U^3_{\neq}}|= \frac{p_h^{\frac{1}{2}}}{p}|\widehat{Q_{\neq}^*}| \lesssim \frac{p_h}{p|k|}|m^*M^*\widehat{Q_{\neq}^*}|\lesssim \frac{1}{|k|}|m^*M^*\widehat{Q_{\neq}^*}|, \end{align}\tag{86}\] and \[\begin{align} \label{Ax462} |\widehat{\Theta_{\neq}}|=\frac{1}{p^{\frac{1}{2}}}|H_{\neq}^*|\lesssim \frac{1}{|k|}|m^*M\widehat{H_{\neq}^*}|. \end{align}\tag{87}\] Hence, together 85 with 86 87 gives \[\begin{align}\label{Ax463} \left\|{U^3_{\neq}}\right\|_{L^2}^2\lesssim&\left\| m^*M^*{Q_{\neq}^*} \right\|_{L^2}^2 \\ \lesssim & e^{-2\lambda^* \nu^{\frac{1}{3}} t} \| (Q_{\mathrm{in}}^*,K_{\mathrm{in}}^*,H_{\mathrm{in}}^*)\|_{H^{1}}^2 \\ \lesssim& e^{-2\lambda^* \nu^\frac{1}{3} t} \left(\| u^3_{\mathrm{in}}\|_{H^3}^2+{\frac{\beta}{\beta-1}} \left\| \partial_xu^2_{\mathrm{in}}-\partial_yu^1_{\mathrm{in}}\right\|_{H^2}^2+\| \theta_{\mathrm{in}}\|_{H^2}^2 \right), \end{align}\tag{88}\] similarly, we have \[\begin{align}\label{Ax464} \left\|{\Theta_{\neq}}\right\|_{L^2}^2 \lesssim e^{-2\lambda^* \nu^{\frac{1}{3}} t} \left(\| u^3_{\mathrm{in}}\|_{H^3}^2+{\frac{\beta}{\beta-1}} \left\| \partial_xu^2_{\mathrm{in}}-\partial_yu^1_{\mathrm{in}}\right\|_{H^2}^2+\| \theta_{\mathrm{in}}\|_{H^2}^2 \right). \end{align}\tag{89}\]

Using 42 and ?? , we get \(\widehat{U_{\neq}^1}\) and \(\widehat{U_{\neq}^2}\) which satisfy \[\label{Al6} \begin{align} |\widehat{U_{\neq}^1}|\leq &\frac{1}{p_h}\left( \sqrt{\frac{\beta-1}{\beta}} \frac{|\eta-kt| p_h^{\frac{1}{2}}}{p^{\frac{1}{2}}} |\widehat{K_{\neq}^*}|+ \frac{|k||l| p_h^{\frac{1}{2}}}{p} |\widehat{Q_{\neq}^*}|\right)\\ \lesssim & \frac{1}{|k|} \left(\sqrt{\frac{\beta-1}{\beta}}|m^*M^*\widehat{K_{\neq}^*}| + |m^*M^*\widehat{Q_{\neq}^*}|\right), \end{align}\tag{90}\] and \[\label{Al7} \begin{align} |\widehat{U_{\neq}^2}|\leq &\frac{1}{p_h}\left( \sqrt{\frac{\beta-1}{\beta}} \frac{|k| p_h^{\frac{1}{2}}}{p^{\frac{1}{2}}} |\widehat{K_{\neq}^*}|+ \frac{|\eta-kt||l| p_h^{\frac{1}{2}}}{p} |\widehat{Q_{\neq}^*}|\right)\\ \lesssim & \frac{1}{p^{\frac{1}{2}}}\left(\sqrt{\frac{\beta-1}{\beta}} |m^*M^*\widehat{K_{\neq}^*}| +\frac{|l|}{|k|} |m^*M^*\widehat{Q_{\neq}^*}|\right). \end{align}\tag{91}\] Thus, this together 85 leads to \(L^2\)-norm of \(U_{\neq}^1(t)\) and \(U_{\neq}^2(t)\) \[\label{Ax465} \begin{align} \left\|{U^1_{\neq}}\right\|_{L^2}^2\lesssim & \left({\frac{\beta-1}{\beta}}\left\|m^*M^*\widehat{K_{\neq}^*}\right\|_{L^2}^2 + \left\|m^*M^*\widehat{Q_{\neq}^*}\right\|_{L^2}^2\right) \\ \lesssim& \max \left\{ {\frac{\beta-1}{\beta}}, {\frac{\beta}{\beta-1}} \right\} e^{-2\lambda^* \nu^{\frac{1}{3}} t} \left(\| u_{\mathrm{in}}\|_{H^3}^2+\| \theta_{\mathrm{in}}\|_{H^2}^2 \right), \end{align}\tag{92}\] and \[\label{Ax466} \begin{align} \left\|{U^2_{\neq}}\right\|_{L^2}^2\lesssim & {\left\langle {t} \right\rangle}^{-2} \left({\frac{\beta-1}{\beta}}\left\|m^*M^*\widehat{K_{\neq}^*}\right\|_{H^{2}}^2 + \left\|m^*M^*\widehat{Q_{\neq}^*}\right\|_{H^{2}}^2\right) \\ \lesssim& {\left\langle {t} \right\rangle}^{-2} \,\max \left\{ {\frac{\beta-1}{\beta}}, {\frac{\beta}{\beta-1}} \right\} e^{-2\lambda^* \nu^{\frac{1}{3}} t} \left(\| u_{\mathrm{in}}\|_{H^{5}}^2+\| \theta_{\mathrm{in}}\|_{H^{4}}^2 \right), \end{align}\tag{93}\] where we have used the fact \(p^{-1} \lesssim {\left\langle {t} \right\rangle}^{-2} |k, \eta, l|^2\). Hence, together 88 , 89 , 92 with 93 gives ?? ?? and the proof of Lemma 27 is finished. \(\square\)

With Lemma 27 at hand, we directly give the enhanced dissipation of the \(L^2\)-norm of \(U_{\neq}\) and \(\Theta_{\neq}\) and the inviscid damping effect of \(U_{\neq}^2\) when \(s=0\). In fact, when \(s>0\), the enhanced dissipation of \(U_{\neq}\) and \(\Theta_{\neq}\) can also be obtained by using a similar treatment. Acknowledgements. Sun was partially supported by the National Natural Science Foundation of China (grant 12501296). Xu was partially supported by the National Key R&D Program of China (grant 2020YFA0712900) and the National Natural Science Foundation of China (grants 12171040, 11771045 and 12071069).

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