Time-asymptotic stability of composite weak planar waves for a general $n\times n$ multi-D viscous system

Jiayun Meng Department of Mathematics,
The University of Texas at Austin, Austin, TX 78712, USA [email protected]

Abstract.

We prove the time-asymptotic stability of the superposition of a weak planar viscous 1-shock and either a weak planar n-rarefaction or a weak planar viscous n-shock for a general $n\times n$ multi-D viscous system. In 2023, Kang-Vasseur-Wang [11] showed the stability of the superposition of a viscous shock and a rarefaction for 1-D compressible barotropic Navier-Stokes equations and solved a long-standing open problem officially introduced by Matsumura-Nishihara [23] in 1992. Our work is an extension of [11], where a general $n\times n$ multi-D viscous system is studied. Same as in [11], we apply the $a$ -contraction method with shifts, an energy based method invented by Kang and Vasseur in [9], for both viscous shock and rarefaction at the level of the solution. In such a way, we can work with general perturbations and compositions of waves. Finally, a technique to classify and control higher-order terms is developed to work in multi-D.

We would like to thank our advisor Alexis Vasseur for suggesting this problem and for valuable discussions and encouragements during the completion of this project.

Acknowledgment.This work was partially funded by NSF-DMS 2306852, NSF-RTG 1840314, and NSF-EPSRC 2219434.

1. Introduction

We consider a general $n\times n$ multi-D viscous system

(1.1)

\partial_{t}U+\partial_{x_{1}}f(U)+\sum_{j=2}^{d}\partial_{x_{j}}g_{j}(U)=\sum_{j=1}^{d}\partial_{x_{j}}\bigl{(}B_{j}(U)\partial_{x_{j}}\eta^{\prime}(U)\bigl{)}\,,

where $U:\mathbb{R}_{\geq 0}\times\mathbb{R}\times\mathbb{T}^{d-1}\rightarrow\mathcal{V}\subseteq\mathbb{R}^{n}$ with an open convex phase space $\mathcal{V}$ . We assume the periodic boundary condition in the transverse direction $\mathbb{T}^{d-1}$ . The notation for the physical space is $x=(x_{1},y)$ with $y$ denoting the transverse direction $\mathbb{T}^{d-1}$ . The flux functions $f,g_{j}:\mathcal{V}\rightarrow\mathbb{R}^{n}$ are assumed to be smooth. In addition, the flux function $f$ is assumed to be strictly hyperbolic and genuinely nonlinear. The viscosity coefficient matrix $B_{j}:\mathcal{V}\rightarrow\mathbb{R}^{n\times n}$ is assumed to be smooth. For any $V\in\mathcal{V}$ , $B_{j}$ is assumed to be an $n\times n$ positive definite matrix. We assume that the entropy $\eta:\mathcal{V}\rightarrow\mathbb{R}$ is smooth and strictly convex, and for any $1\leq j\leq d$ , there exists an entropy flux $q_{j}:\mathcal{V}\rightarrow\mathbb{R}$ of $\eta$ such that

(1.2)

\partial_{i}q_{j}(U)=\begin{cases}\sum_{l=1}^{n}\partial_{l}\eta(U)\partial_{i}(f)_{l}(U)\,,\textup{ if }j=1\,,\\ \sum_{l=1}^{n}\partial_{l}\eta(U)\partial_{i}(g_{j})_{l}(U)\,,\textup{ if }2\leq j\leq d\,,\end{cases}

for any $1\leq i\leq n$ .

Let us endow the system with an initial value

U(0,x)=U_{0}(x)\,,\quad x\in\mathbb{R}\times\mathbb{T}^{d-1}\,,

with fixed end states $U_{\pm}\in\mathbb{R}^{n}$ , i.e.,

(1.3)

U_{0}(x)\rightarrow U_{\pm}\,,\quad\textup{as }x_{1}\rightarrow\pm\infty\,.

This general framework includes the 3-D barotropic Brenner-Navier-Stokes equations:

(1.4)

\begin{cases}\partial_{t}\rho+div(\rho v)=0\,,\\ \partial_{t}\rho u+div(\rho u\otimes v)+\nabla\rho^{\gamma}=\nu\Delta u\,,\end{cases}

where $\nu>0$ , $\gamma>1$ , $\rho$ is the density, $u$ is the velocity, and $v$ is the corrected velocity defined by

\displaystyle v=u-\frac{1}{\rho}\nabla Q^{\prime}(\rho)\,,\quad\textup{where }Q(\rho)=\frac{\rho^{\gamma}}{\gamma-1}\,.

Such a system was introduced by Brenner as a correction of the Navier-Stokes equations. In [1], Brenner mentioned some extreme situations where the Navier-Stokes equations did not describe the compressible fluid well, and based on those results, Brenner proposed that the specific momentum density of the fluid equals the corrected velocity $v$ (the volume velocity) instead of the velocity $u$ (the mass velocity).

In this paper, we study the long-time asymptotic behavior of solutions of (1.1) with initial value satisfying (1.3). This long-time asymptotic behavior is closely related to the following 1-D Riemann problem:

(1.5)

\partial_{t}U_{1}+\partial_{x_{1}}f(U_{1})=0\,,

with an associated initial value

(1.6)

U_{1}(0,x_{1})=\begin{cases}U_{-}\,,\quad x_{1}<0\,,\\ U_{+}\,,\quad x_{1}>0\,.\end{cases}

Elementary solutions of the 1-D Riemann problem. Since $f$ is genuinely nonlinear, (1.5) has two elementary solutions: shock wave and rarefaction wave. We denote the eigenvalues of $f^{\prime}$ as $\lambda_{1}<...<\lambda_{n}$ and the corresponding right eigenvectors as $r_{1},...,r_{n}$ . Let $U_{-}\in\mathbb{R}^{n}$ .

For any $1\leq i\leq n$ , there exists an integral curve $\mathcal{R}_{i}(U_{-})$ such that for any $U_{+}\in\mathcal{R}_{i}(U_{-})$ that is close enough to $U_{-}$ , there exists a solution $\mathbf{R}_{i}$ to (1.5) with initial value (1.6) that is defined by

\lambda_{i}\bigl{(}\mathbf{R}_{i}(t,x_{1})\bigl{)}=\begin{cases}\lambda_{i}(U_{-})\,,\quad x_{1}<\lambda_{i}(U_{-})t\,,\\ \frac{x_{1}}{t}\,,\quad\lambda_{i}(U_{-})t\leq x_{1}\leq\lambda_{i}(U_{+})t\,,\\ \lambda_{i}(U_{+})\,,\quad x_{1}>\lambda_{i}(U_{+})t\,,\end{cases}

with

z_{i}\bigl{(}\mathbf{R}_{i}(t,x)\bigl{)}=z_{i}(U_{-})=z_{i}(U_{+})\,,

for any $i$ -Riemann invariant $z_{i}$ . The existence of $(n-1)$ $n$ -Riemann invariants can be found in [28]. We call $\mathbf{R}_{i}$ the i-rarefaction wave.

We define the shock set

\mathcal{S}(U_{-})=\{U|-\sigma(U-U_{-})+f(U)-f(U_{-})=0\text{ for a constant }\sigma\}\,.

In a neighborhood of $U_{-}$ , $\mathcal{S}$ consists of n smooth curves $\mathcal{S}_{1}(U_{-}),...,\mathcal{S}_{n}(U_{-})$ that intersects at $U_{-}$ . We call $\mathcal{S}_{i}(U_{-})$ the i-shock curve. For any $U_{+}\in\mathcal{S}_{i}(U_{-})$ that is close enough to $U_{-}$ , there exists a solution $\tilde{\mathbf{S}}_{i}$ to (1.5) with initial value (1.6) that is defined by

\tilde{\mathbf{S}}_{i}(t,x_{1})=\begin{cases}U_{-}\,,\quad x_{1}<\sigma t\,,\\ U_{+}\,,\quad x_{1}>\sigma t\,.\\ \end{cases}

We call $\tilde{\mathbf{S}}_{i}(U_{-})$ the i-shock wave and $\sigma$ the speed of the i-shock wave. In particular, $\sigma$ is close to $\lambda_{i}(U_{-})$ . We denote $\sigma$ as $\sigma_{i}$ .

Stability of elementary solutions for 1-D viscous systems. If the Riemann solution to an inviscid system is a rarefaction wave (respectively a shock wave), then the asymptotic state of the solution to the corresponding viscous system with a perturbed initial value is the rarefaction wave (respectively the viscous shock wave).

While the rarefaction wave spreads much faster than the diffusion process, the jump of the shock is smoothed by the viscosity and becomes a thin transition layer. The viscous i-shock wave $\mathbf{S}_{i}(x-\sigma_{i}t)$ refers to the smoothened i-shock wave, and $\mathbf{S}_{i}$ satisfies

(1.7)

-\sigma_{i}\partial_{x_{1}}\mathbf{S}_{i}+\partial_{x_{1}}f(\mathbf{S}_{i})=\partial_{x_{1}}\bigl{(}B(\mathbf{S}_{i})\partial_{x_{1}}\eta^{\prime}(\mathbf{S}_{i})\bigl{)}\,.

Ilin-Oleinik [7] proved the stability of elementary solutions for the scalar case in 1960, but the maximum principle they used did not work for systems (see [20]). Later, the energy method has become the main tool. When studying the stability of rarefaction waves, the energy method can be applied directly at the level of the solutions. We call such a method the direct energy method. The stability of rarefaction waves was first proved by Matsumura-Nishihara [22, 23] for the 1-D compressible Navier-Stokes equations. Later, Liu-Xin [16] and Nishihara-Yang-Zhao [25] pushed the stability result of rarefaction waves to the Navier-Stokes-Fourier system.

However, the direct energy method fails when working with viscous shock waves. The first results were based on the energy method applied at the level of the antiderivative. Matsumura-Nishihara [21] in 1985, and Goodman [4] in 1986 independently proved the stability of viscous shock waves. Matsumura and Nishihara proved the stability for the 1-D compressible barotropic Navier-Stokes system, while Goodman proved the stability for a general 1-D system with a positive definite viscosity. As both papers worked with the antiderivative, they need to assume the zero mass condition, i.e., that the mass of the initial perturbation is zero.

Two very fruitful approaches removed the stringent zero mass condition. The Green’s function method was started by Liu [15] in 1985. It involves a constant shift on the viscous shock and the introduction of a diffusion wave and a coupled diffusion wave in the transverse characteristic fields. Szepessy-Xin [29] showed the stability of viscous shock waves for a 1-D general system with a nondegenerate artificial viscosity, and Liu-Zeng [17] applied the Green’s function method to 1-D systems with degenerate viscosity. Another approach is the Evans function method. In 2004, Mascia-Zumbrun [19] showed the spectral stability of viscous shock waves for the 1-D compressible Navier-Stokes system. The study of spectral stability is very advanced now. In 2017, Humpherys-Lyng-Zumbrun [6] proved the spectral stability of large-amplitude planar viscous shock waves for the compressible Navier-Stokes equations in multi-D by the numerical Evans function method. Although this technique gives stability only for single elementary waves, the stability result works in the whole space $\mathbb{R}^{d}$ .

Both Green’s function method and Evans function method can deal with more general perturbations and provide pointwise estimates but fail to give global-in-time stability results for composite waves. Assuming the strengths of the two viscous shock waves are suitably small with the same order, Huang-Matsumura [5] showed the stability of two viscous shock waves for the 1-D Navier-Stokes-Fourier equations under a more relaxed condition than the zero mass condition.

Matsumura Conjecture. Even in 1-D, the stability of the composition of a viscous shock wave and a rarefaction wave is very difficult to study. Matsumura-Nishihara [22] mentioned the problem in 1986 and officially introduced it as an open problem in 1992 in [23]. In 2018, Matsumura [20] classified the problem as a very hard open problem. There are two difficulties. First, the direct energy method used to study the stability of rarefaction waves does not match very well with the methods developed for viscous shock waves. At the same time, the rarefaction wave is not an exact solution to the viscous system and any spatial shift of the rarefaction wave has the same asymptotic state. Hence, it is hard to analyze the interaction between the rarefaction wave and the viscous shock wave.

The breakthrough happened in 2023. Using the $a$ -contraction method with shifts, Kang-Vasseur-Wang proved the stability of the composition of a viscous shock wave and a rarefaction wave for the 1-D compressible barotropic Navier-Stokes equations in [11] and the stability of the generic Riemann solutions for the 1-D compressible Navier-Stokes-Fourier equations in [12]. The $a$ -contraction method with shifts is an energy method that can be applied at the level of the solution for contact waves, rarefaction waves, and viscous shock waves, so it unifies the methods for elementary waves. This paper is an extension of their work [11], in which the stability of a planar viscous 1-shock wave and either a planar n-rarefaction wave or a planar viscous n-shock wave is proved for a general $n\times n$ multi-D viscous system. Note that we consider here only extremal waves.

Multi-D results. The stability of planar rarefaction waves for the 3-D compressible Navier-Stokes-Fourier system was shown by Li-Wang-Wang [14] in 2018. In 2023, the stability of planar viscous shock waves for the 3-D compressible Navier-Stokes equations was proved by Wang-Wang [30]. Later in 2024, Kang-Lee [8] generalized Wang-Wang’s stability result to the compositions of planar viscous shocks for the same system. Note that all these multi-D results are based on a-contraction method with shifts and work only with periodic transversal variables and weak elementary waves.

Result of the paper. Let $m$ be the smallest integer that is strictly bigger than $\frac{d}{2}$ , i.e.,

(1.8)

m=\begin{cases}\frac{d+1}{2}\,,\textup{ if }d\textup{ is odd,}\\ \frac{d+2}{2}\,,\textup{ if }d\textup{ is even.}\end{cases}

The main result of the paper is the following theorem.

Theorem 1.1.

For any $U_{-}\in\mathbb{R}^{n}$ , there exist constants $\delta_{0},\epsilon_{0}>0$ such that the following is true.

Let $U_{m}\in\mathcal{S}_{1}(U_{-})$ and $U_{+}\in\mathcal{S}_{n}(U_{m})$ or $\mathcal{R}_{n}(U_{m})$ be such that

|U_{m}-U_{-}|+|U_{+}-U_{m}|<\delta_{0}\,.

Let $\mathbf{S}_{1}$ be the viscous 1-shock wave solution to (1.7) with end states $U_{-}$ and $U_{m}$ , and $\mathbf{W}_{n}$ be the viscous n-shock wave solution $\mathbf{S}_{n}$ to (1.7) or the n-rarefaction wave solution $\mathbf{R}_{n}$ to (1.5) with end states $U_{m}$ and $U_{+}$ . Let $\mathcal{I}_{\mathbf{W}_{n}}$ be $\{1,n\}$ if $\mathbf{W}_{n}=\mathbf{S}_{n}$ and be $\{1\}$ if $\mathbf{W}_{n}=\mathbf{R}_{n}$ .

Let $U_{0}$ be an initial value such that

(1.9)

\sum_{\pm}\|U_{0}-U_{\pm}\|_{L^{2}(\mathbb{R}_{\pm}\times\mathbb{T}^{d-1})}+\|U_{0}\|_{\dot{H}^{m}(\mathbb{R}\times\mathbb{T}^{d-1})}<\epsilon_{0}\,,

where $\mathbb{R}_{+}=-\mathbb{R}_{-}=(0,\infty)$ .

Then the viscous system (1.1) has a unique global-in-time solution $U$ . Moreover, there exist absolutely continuous shifts $\mathbf{X}_{i}(t)$ for $i\in\mathcal{I}_{\mathbf{W}_{n}}$ such that

(1.10)		$\displaystyle U(t,x)-\mathbf{\tilde{U}}(t,x)\in C\bigl{(}0,\infty;H^{m}(\mathbb{R}\times\mathbb{T}^{d-1})\bigl{)}\,,$
(1.11)		$\displaystyle\underset{x\in\mathbb{R}\times\mathbb{T}^{d-1}}{\textup{sup}}\,\|U(t,x)-\mathbf{\tilde{U}}(t,x)\|\rightarrow 0\;\text{ as }\;t\rightarrow 0\,,$

where

\mathbf{\tilde{U}}(t,x)=\begin{cases}\mathbf{S}_{1}\bigl{(}x_{1}-{\sigma}_{1}t+\mathbf{X}_{1}(t)\bigl{)}+\mathbf{S}_{n}\bigl{(}x_{1}-\sigma_{n}t+\mathbf{X}_{n}(t)\bigl{)}-U_{m}\,,\textup{ if }\mathbf{W}_{n}=\mathbf{S}_{n}\,,\\ \mathbf{S}_{1}\bigl{(}x_{1}-{\sigma}_{1}t+\mathbf{X}_{1}(t)\bigl{)}+\mathbf{R}_{n}\bigl{(}\frac{x_{1}}{t}\bigl{)}-U_{m}\,,\textup{ if }\mathbf{W}_{n}=\mathbf{R}_{n}\,.\end{cases}

In addition,

(1.12)

\displaystyle U-\mathbf{\tilde{U}}\in L^{2}\bigl{(}0,\infty;\dot{H}^{m+1}(\mathbb{R}\times\mathbb{T}^{d-1})\bigl{)}\,,

and for any $i\in\mathcal{I}_{\mathbf{W}_{n}}$ ,

(1.13)

\underset{t\rightarrow\infty}{\textup{lim}}\dot{\mathbf{X}}_{i}(t)=0\,.

Remark. Theorem 1.1 shows that if $U_{-}$ and $U_{+}$ are connected by a composition of a planar viscous 1-shock wave and either a planar viscous n-shock wave or a planar n-rarefaction wave, then the asymptotic state of the solution to the viscous system (1.1) with a perturbed initial value is the composition with viscous shocks shifted.

Proposition 1.2.

The 3-D barotropic Brenner-Navier-Stokes equations (1.4) can be transformed into the form of the viscous system (1.1).

Remark. The detailed transformation is in the appendix. Proposition 1.2 implies that Theorem 1.1 can be applied to the 3-D barotropic Brenner-Navier-Stokes equations.

Structure of the paper. Section 2 discusses the properties of the viscous shock wave and the approximate rarefaction wave and introduces the weight functions, shift functions, and the superposition wave.

In section 3, we show how Theorem 1.1 is proved by local-in-time estimates and a priori estimates. Proposition 3.1 provides local-in-time estimates which could be shown in the same way as previous work, while a priori estimates are given in Proposition 3.3 which is to be shown in sections 4 and 5.

We get the $L^{2}$ estimates by the $a$ -contraction method with shifts in section 4 and go from the $L^{2}$ estimates to the $H^{m}$ estimates (a priori estimates) by induction in section 5.

The $a$ -contraction method with shifts. The method of $a$ -contraction relies on the ad-hoc construction of the shifts $\mathbf{X}_{i}$ for $i\in\mathcal{I}_{\mathbf{W}_{n}}$ solving special ODEs. For the scalar case, the method can be applied directly on the $L^{2}$ norm (see [9]). However, it was shown that the result is not true for systems in [27]. To work on systems, we need to introduce weight functions $a_{\mathbf{S}_{1}},a_{\mathbf{W}_{n}}$ . In this paper, we follow the method of [11] written for the special case of the 1-D compressible barotropic Navier-Stokes equations. Our extension allows to clarify and explain why the method works at a deeper level.

To get the $L^{2}$ estimates, we study the evolution of a pseudo-distance given by the physical structure of the problem

\displaystyle\frac{d}{dt}\int(a_{\mathbf{S}_{1}}+a_{\mathbf{W}_{n}})\,\eta(U|\mathbf{\tilde{U}})\,dx\,,

where the relative entropy $\eta(\cdot|\cdot)$ is defined by

(1.14)

\eta(U|V)=\eta(U)-\eta(V)-\eta^{\prime}(V)(U-V)\,.

We define two bases at the beginning of subsection 4.3. The basis (4.3) (respectively (4.8)) is designed for the wave $\mathbf{S}_{1}$ (respectively $\mathbf{W}_{n}$ ). It contains the special direction $r_{1}(U_{-})$ (respectively $r_{n}(U_{-})$ ) of the wave $\mathbf{S}_{1}$ (respectively $\mathbf{W}_{n}$ ) and is orthogonal with respect to the viscous matrix $B_{1}$ in order to be consistent with the viscosity.

In Lemma 4.3, we apply the relative entropy method introduced by Dafermos [2] and DiPerna [3] and project the perturbation $U-\mathbf{\tilde{U}}$ onto the bases. We get

\displaystyle\frac{d}{dt}\int(a_{\mathbf{S}_{1}}+a_{\mathbf{W}_{n}})\,\eta(U|\tilde{U})\,dx\leq\mathcal{Z}(U)-\mathcal{D}(U)+\mathcal{H}(U)+\mathcal{E}(U)\,.

The shift term $\mathcal{Z}$ represents the new terms induced by the shifts $\mathbf{X}_{i}$ for $i\in\mathcal{I}_{\mathbf{W}_{n}}$ . The viscosity operation (the right-hand side of (1.1)) gives the viscous term $\mathcal{D}$ . The hyperbolic term $\mathcal{H}$ comes from the flux functions $f,g_{j}$ . The interaction between waves $\mathbf{S}_{1}$ and $\mathbf{W}_{n}$ creates the interaction term $\mathcal{E}$ .

Lemma 4.3 discusses the hyperbolic “scalarization”. The weight function $a_{\mathbf{S}_{1}}$ (respectively $a_{\mathbf{W}_{n}}$ ) activates the spectral gap, creates new negative hyperbolic terms, and initiates cancellation for hyperbolic terms corresponding to the perturbation $U-\mathbf{\tilde{U}}$ in all directions except the special direction $r_{1}(U_{-})$ (respectively $r_{n}(U_{-})$ ) of the wave $\mathbf{S}_{1}$ (respectively $\mathbf{W}_{n}$ ). We get

\displaystyle\mathcal{H}(U)\leq-C_{a}\mathcal{H}_{C}(U)+C\mathcal{H}_{\mathbf{S}_{1}}(U)+C\mathcal{H}_{\mathbf{W}_{n}}(U)\,,

where $C_{a}>0$ is a constant that depends on the strengths of $a_{\mathbf{S}_{1}},a_{\mathbf{W}_{n}}$ , and $C>0$ represents the constants that depend only on $B_{j},f,\eta,U_{-}$ . $\mathcal{H}_{\mathbf{S}_{1}}$ (respectively $\mathcal{H}_{\mathbf{W}_{n}}$ ) is the hyperbolic term corresponding to the projection of the perturbation in the special direction $r_{1}(U_{-})$ (respectively $r_{n}(U_{-})$ ) of the wave $\mathbf{S}_{1}$ (respectively $\mathbf{W}_{n}$ ), and $\mathcal{H}_{C}$ denotes the sum of the absolute value of the hyperbolic terms corresponding to the other orthogonal directions of the perturbation. We see the hyperbolic “scalarization” scalarizes the problem to the special direction $r_{1}(U_{-})$ (respectively $r_{n}(U_{-})$ ) of the wave $\mathbf{S}_{1}$ (respectively $\mathbf{W}_{n}$ ).

The hyperbolic remainder due to the perturbation in the special direction $r_{n}(U_{-})$ of the rarefaction wave $\mathbf{R}_{n}$ is negative, so the rarefaction wave $\mathbf{R}_{n}$ is contractive at the hyperbolic level. However, the hyperbolic remainder due to the perturbation in the special direction of a viscous shock wave is positive and needs to be depleted using the viscous term $\mathcal{D}$ . This is done by introducing a Poincaré type inequality in Lemma 4.4. For $i\in\mathcal{I}_{\mathbf{W}_{n}}$ , we show

\displaystyle\mathcal{H}_{\mathbf{S}_{i}}(U)\lesssim\mathcal{D}(U)-\mathcal{Z}(U)+\mathcal{H}_{C}(U)\,.

The viscous term $\mathcal{D}$ controls the $L^{2}$ norm of the derivative, while the strengths of the shifts $\mathbf{X}_{i}$ for $i\in\mathcal{I}_{\mathbf{W}_{n}}$ are chosen to be big enough that the shift term $\mathcal{Z}$ handles the average with the help of $\mathcal{H}_{C}$ .

In Lemma 4.5, we choose weight functions $a_{\mathbf{S}_{1}},a_{\mathbf{W}_{n}}$ and shift functions $\mathbf{X}_{i}$ for $i\in\mathcal{I}_{\mathbf{W}_{n}}$ that work for both the hyperbolic “scalarization” and the Poincaré type inequality. We get

\displaystyle\mathcal{Z}(U)-\mathcal{D}(U)+\mathcal{H}(U)\leq 0\,.

By Lemma 2.2 and Lemma 4.1, we can be bound the interaction term $\mathcal{E}(U)$ by a small and time integrable function depending on the strengths of the waves $\mathbf{S}_{1},\mathbf{W}_{n}$ . In all, we obtain the $L^{2}$ estimates at the end of section 4.

Remark. In Lemma 4.4, we reduce the Poincaré type inequality to the following lemma proved in [10].

Lemma 1.3.

For any $f:[0,1]\rightarrow\mathbb{R}$ satisfying $\int_{0}^{1}y(1-y)|f^{\prime}|^{2}\,dy<\infty$ ,

(1.15)

\int_{0}^{1}|f-\int_{0}^{1}f\,dy|^{2}\,dy\leq\frac{1}{2}\int_{0}^{1}y(1-y)|f^{\prime}|^{2}\,dy\,.

Notations and a remark. Before we go to the proofs, let us fix some notations. Let the eigenvalues of $f^{\prime}(U_{-})$ be $\lambda_{1}<...<\lambda_{n}$ . For any $1\leq i\leq n$ , let $\mathbf{r_{i}}$ and $\mathbf{l_{i}}$ be the right and left eigenvectors corresponding to $\lambda_{i}$ such that $\mathbf{r_{i}}$ is tangent to $\mathcal{S}_{i}(U_{-})$ , $\mathbf{l_{i}}=\eta^{\prime\prime}(U_{-})\mathbf{r_{i}}$ , and $\mathbf{r_{i}}\cdot\mathbf{l_{i}}=1$ . We define

(1.16)

\displaystyle c_{f}^{(i)}:=(f^{\prime\prime}(U_{-}):\mathbf{r_{i}}\otimes\mathbf{r_{i}})\cdot\mathbf{l_{i}}

\displaystyle=\lambda_{i}^{\prime}(U_{-})\cdot\mathbf{r_{i}}<0\,.

Let $C$ denote the positive constants that depend only on $B_{j},f,\eta,U_{-}$ . For $\alpha=(a_{1},...,a_{d})\in\mathbb{N}^{d}$ , we define

(1.17)

\displaystyle(\alpha)_{j}=a_{j}\,\textup{ for any }\,1\leq j\leq d\;\textup{ and }\;\partial_{x}^{\alpha}=\partial_{x_{1}}^{a_{1}}...\partial_{x_{d}}^{a_{d}}\,.

As $\eta^{\prime\prime}f^{\prime}$ is symmetry, we know

(1.18)

\eta^{\prime\prime}(U_{-})\mathbf{r_{i}}\cdot\mathbf{r_{j}}=0\text{ if }i\neq j\,.

2. Preliminaries

2.1. Viscous shock wave

Let $i\in\{1,n\}$ . We examine the viscous i-shock wave $\mathbf{S}_{i}$ satisfying

(2.1)

\begin{cases}-{\sigma}_{i}\partial_{x_{1}}\mathbf{S}_{i}+\partial_{x_{1}}f(\mathbf{S}_{i})=\partial_{x_{1}}\bigl{(}B_{1}(\mathbf{S}_{i})\partial_{x_{1}}\eta^{\prime}(\mathbf{S}_{i})\bigl{)}\,,\\ \mathbf{S}_{i}(-\infty)=U_{L}\,,\;\mathbf{S}_{i}(+\infty)=U_{R}\,.\end{cases}

Recall the Rankine-Hugoniot condition and the Lax inequality

\lambda_{1}(U_{R})<{\sigma}_{i}<\lambda_{1}(U_{L})\,.

Let the wave strength of $\mathbf{S}_{i}$ be

\displaystyle\delta_{S_{i}}=(U_{R}-U_{L})\cdot\mathbf{l_{i}}\,.

The proof of the existence of the viscous i-shock wave can be found in [18]. The following results are proved in [13].

Lemma 2.1.

For any $U_{L}\in\mathbb{R}^{n}$ , there exist $\epsilon,C>0$ and $C_{j}>0$ for any $j\geq 2$ such that the following is true.

For any $U_{R}\in\mathcal{S}_{i}(U_{L})$ such that $|U_{L}-U_{R}|<\epsilon$ , there exists a unique solution $\mathbf{S}_{i}$ to (2.1) such that

	$\displaystyle\|\mathbf{S}_{i}({x_{1}})-U_{L}\|$	$\displaystyle\leq C\delta_{S_{i}}e^{-C\delta_{S_{i}}\|{x_{1}}\|},\quad\forall{x_{1}}<0\,,$
	$\displaystyle\|\mathbf{S}_{i}({x_{1}})-U_{R}\|$	$\displaystyle\leq C\delta_{S_{i}}e^{-C\delta_{S_{i}}\|{x_{1}}\|},\quad\forall{x_{1}}>0\,,$
	$\displaystyle\|\partial_{x_{1}}\mathbf{S}_{i}\|$	$\displaystyle\leq C\delta_{S_{i}}^{2}e^{-C\delta_{S_{i}}\|{x_{1}}\|},\quad\forall{x_{1}}\in\mathbb{R}\,,$
	$\displaystyle\|\partial_{x_{1}}^{j}\mathbf{S}_{i}\|$	$\displaystyle\leq C_{j}\delta_{S_{i}}\|\partial_{x_{1}}\mathbf{S}_{i}\|,\quad\forall{x_{1}}\in\mathbb{R}\,,\;\forall j\geq 2\,.$

We define

\displaystyle S_{i}(t,x)=\mathbf{S}_{i}(x_{1}-\sigma_{i}t)\,,

and the projection of $S_{i}$ onto $\mathbf{l_{i}}$

(2.2)

\displaystyle k_{S_{i}}(t,x)=\frac{\bigl{(}S_{i}(t,x)-U_{L}\bigl{)}\cdot\mathbf{l_{i}}}{\delta_{S_{i}}}\,.

From now on, we call $S_{i}$ the planar viscous i-shock wave. We know $S_{i}$ satisfies

(2.3)

\displaystyle\partial_{t}S_{i}+\partial_{x_{1}}f(S_{i})

\displaystyle=\partial_{x_{1}}\bigl{(}B_{1}(S_{i})\partial_{x_{1}}\eta^{\prime}(S_{i})\bigl{)}\,.

In [13], we show

(2.4)

\displaystyle\big{|}\partial_{x_{1}}k_{S_{i}}+\frac{c_{f}^{(1)}}{2B(U_{-})\mathbf{l_{i}}\cdot\mathbf{l_{i}}}\delta_{S_{i}}k_{S_{i}}(1-k_{S_{i}})\big{|}\leq C\delta_{S_{i}}^{2}k_{S_{i}}(1-k_{S_{i}})\,,

and

(2.5)

\displaystyle|\partial_{x_{1}}S_{i}-\delta_{S_{i}}\partial_{x_{1}}k_{S_{i}}\mathbf{r_{i}}|\leq C\delta_{S_{i}}^{2}\partial_{x_{1}}k_{S_{i}}\,.

In particular, $k_{S_{i}}$ is strictly increasing.

2.2. Construction of approximate rarefaction wave

As in [11], we will consider a smooth approximation of the planar n-rarefaction wave with the help of the smooth solution to the Burgers’ equation

(2.6)

\begin{cases}w_{t}+ww_{x_{1}}=0\,,\\ w(0,x_{1})=w_{0}(x_{1})=\frac{w_{+}+w_{m}}{2}+\frac{w_{+}-w_{m}}{2}\text{tanh}x\,.\end{cases}

The smooth approximate planar n-rarefaction wave $R_{n}$ is defined by

(2.7)			$\displaystyle\lambda_{n}(U_{m})=w_{m}\,,\;\lambda_{n}(U_{+})=w_{+}\,,$
			$\displaystyle\lambda_{n}\bigl{(}R_{n}(t,x)\bigl{)}=w(1+t,x_{1})\,,$
			$\displaystyle z_{n}\bigl{(}R_{n}(t,x)\bigl{)}=z_{n}(U_{m})=z_{n}(U_{+})\,,$

where $w(t,x_{1})$ is the smooth solution to the Burgers’ equation (2.6) and $z_{n}$ is any $n$ -Riemann invariant to (1.5).

It is easy to check that $R_{n}$ is the solution to the inviscid system, i.e.,

(2.8)

\partial_{t}R_{n}+\partial_{x_{1}}f(R_{n})=0\,.

We define the wave strength of the rarefaction

\delta_{R_{n}}=-(U_{+}-U_{m})\cdot\mathbf{l_{n}}\,,

and

(2.9)

\displaystyle k_{R_{n}}=\frac{-(R_{n}-U_{m})\cdot\mathbf{l_{n}}}{\delta_{R_{n}}}\,.

The following properties of the approximate planar n-rarefaction wave $R_{n}$ follow from the properties of the smooth solution to the Burgers’ equations proved in [22].

Lemma 2.2.

The smooth approximate n-rarefaction wave $R_{n}$ defined in (2.7) satisfies the following properties.

1)

$\partial_{x_{1}}R_{n}\cdot\mathbf{l_{n}}>0$ and $|\partial_{x_{1}}R_{n}\cdot\mathbf{l_{i}}|\leq C\delta_{R_{n}}\partial_{x_{1}}R_{n}\cdot\mathbf{l_{n}}$ for any $1\leq i\leq n-1$ .

For any $t\geq 0$ and $j\in\mathbb{N}^{*}$ ,

	$\displaystyle\\|\partial_{x_{1}}^{j}R_{n}\\|_{L^{p}(\mathbb{R}\times\mathbb{T}^{d-1})}$	$\displaystyle\leq C_{p,j}\,\textup{min}\bigl{\{}\delta_{R_{n}},\delta_{R_{n}}^{1/p}(1+t)^{-1+1/p}\bigl{\}}\,,\;\forall p\in[1,\infty]\,,$
	$\displaystyle\\|\partial_{x_{1}}^{2j}R_{n}\\|_{L^{p}(\mathbb{R}\times\mathbb{T}^{d-1})}$	$\displaystyle\leq C_{p,j}\,\textup{min}\bigl{\{}\delta_{R_{n}},(1+t)^{-1}\bigl{\}}\,,\;\forall p\in[1,\infty)\,,$
	$\displaystyle\|\partial_{x_{1}}^{j}R_{n}\|$	$\displaystyle\leq C_{j}\|\partial_{x_{1}}R_{n}\|\,,$

where $C_{j}>0$ depends on $j$ and $C_{p,j}>0$ depends on $p,j$ .

For any $t\geq 0$ ,

	$\displaystyle\|R_{n}(t,x)-U_{m}\|$	$\displaystyle\leq C\delta_{R_{n}}e^{-2\|x_{1}-\lambda_{n}(U_{m})t\|}\,,\quad\forall x_{1}\leq\lambda_{n}(U_{m})t\,,$
	$\displaystyle\|\partial_{x_{1}}R_{n}(t,x)\|$	$\displaystyle\leq C\delta_{R_{n}}e^{-2\|x_{1}-\lambda_{n}(U_{m})t\|}\,,\quad\forall x_{1}\leq\lambda_{n}(U_{m})t\,,$
	$\displaystyle\|R_{n}(t,x)-U_{+}\|$	$\displaystyle\leq C\delta_{R_{n}}e^{-2\|x_{1}-\lambda_{n}(U_{+})t\|}\,,\quad\forall x_{1}\geq\lambda_{n}(U_{+})t\,,$
	$\displaystyle\|\partial_{x_{1}}R_{n}(t,x)\|$	$\displaystyle\leq C\delta_{R_{n}}e^{-2\|x_{1}-\lambda_{n}(U_{+})t\|}\,,\quad\forall x_{1}\geq\lambda_{n}(U_{+})t\,.$

4)

$\underset{t\rightarrow\infty}{\textup{lim}}\,\underset{x\in\mathbb{R}\times\mathbb{T}^{d-1}}{\textup{sup}}\,|R_{n}(t,x)-\mathbf{R}_{n}(\frac{x_{1}}{t})|=0\,.$

In particular, $k_{R_{n}}$ is strictly increasing,

(2.10)

\displaystyle|\partial_{x_{1}}R_{n}+\delta_{R_{n}}\partial_{x_{1}}k_{R_{n}}\mathbf{r_{n}}|

\displaystyle\leq C\delta_{R_{n}}^{2}\partial_{x_{1}}k_{R_{n}}\,,

and

(2.11)		$\displaystyle\int_{0}^{\infty}\\|\partial_{x_{1}}R_{n}\\|_{L^{4}}^{4}\,dt\leq C\delta_{R_{n}}^{3}\,,\;\int_{0}^{\infty}\\|\partial_{x_{1}}R_{n}\\|_{L^{4}}^{2}\,dt\leq C\delta_{R_{n}}\,,$
(2.11)		$\displaystyle\int_{0}^{\infty}\\|\partial_{x_{1}}^{2j}R_{n}\\|_{L^{2}}^{2}\,dt\leq C\delta_{R_{n}}\,,\;\int_{0}^{\infty}\\|\partial_{x_{1}}^{2}R_{n}\\|_{L^{1}}^{4/3}\,dt\leq C\delta_{R_{n}}^{1/3}\,.$

For convenience, we call $R_{n}$ the planar n-rarefaction wave from now on.

2.3. Construction of weight functions, shift functions and the superposition wave

We are ready to introduce the weight functions, the shift functions and the superposition wave.

Let $W_{n}$ be either the planar viscous n-shock wave $S_{n}$ or the planar n-rarefaction wave $R_{n}$ . Recall (2.2) and (2.9). We define the weight functions $a_{S_{1}},a_{W_{n}}$ by

(2.12)		$\displaystyle a_{S_{1}}(t,x)$	$\displaystyle=1-\Lambda_{S_{1}}\delta_{S_{1}}{k_{S_{1}}}(t,x)\,,$
(2.12)		$\displaystyle a_{W_{n}}(t,x)$	$\displaystyle=1+\Lambda_{W_{n}}\delta_{W_{n}}k_{W_{n}}(t,x)\,,$

for large enough constants $\Lambda_{S_{1}},\Lambda_{W_{n}}>0$ that depend only on $B_{j},f,\eta,U_{-}$ . As $k_{S_{1}},k_{W_{n}}$ are increasing, the weight function $a_{S_{1}}$ is decreasing and the weight function $a_{W_{n}}$ is increasing. Also, we have $\|a_{S_{1}}\|_{C^{1}},\|a_{W_{n}}\|_{C^{1}}\leq 2$ by taking $\delta_{S_{1}},\delta_{W_{n}}$ small enough.

For any function $h:\mathbb{R}_{\geq 0}\times\mathbb{R}\times\mathbb{T}^{d-1}\rightarrow\mathbb{R}^{n}$ and $\mathbf{X}:\mathbb{R}_{\geq 0}\rightarrow\mathbb{R}$ , we define

\displaystyle h^{\mathbf{X}}\bigl{(}t,(x_{1},y)\bigl{)}=h\bigl{(}t,(x_{1}+\mathbf{X}(t),y)\bigl{)}\,.

We define the superposition wave

(2.13)

\tilde{U}(t,x)={S}_{1}^{{\mathbf{X}_{1}}}(t,x)+W_{n}^{\mathbf{X}_{n}}(t,x)-U_{m}\,,

and the weight function

(2.14)

\displaystyle a(t,x)=a_{S_{1}}^{\mathbf{X}_{1}}(t,x)+a_{W_{n}}^{\mathbf{X}_{n}}(t,x)\,,

where $a_{S_{1}},a_{W_{n}}$ are defined in (2.12), and the shift $\mathbf{X}_{i}$ is defined as the solution to the ODE

(2.15)

\begin{cases}\dot{\mathbf{X}}_{i}(t)=\frac{\tilde{C}_{i}}{\delta_{S_{i}}}\int a\,\eta^{\prime\prime}(\tilde{U})(U-\tilde{U})\partial_{x_{1}}S_{i}^{\mathbf{X}_{i}}\,dx\,,\\ \mathbf{X}_{i}(0)=0\,,\end{cases}

for a large enough constant $\tilde{C_{i}}>0$ that depends only on $B_{j},f,\eta,U_{-}$ for $i\in\{1,n\}$ if $W_{n}=S_{n}$ and for $i=1$ if $W_{n}=R_{n}$ . As the planar n-rarefaction wave $R_{n}$ is not shifted, we define $\mathbf{X}_{n}=0$ if $W_{n}=R_{n}$ for consistency. The existence and uniqueness of shifts $\mathbf{X}_{i}$ are proved in Proposition 3.1.

By (2.3) and (2.8), we have

(2.16)

\partial_{t}\tilde{U}+\partial_{x_{1}}f(\tilde{U})=\partial_{x_{1}}\bigl{(}B_{1}(\tilde{U})\partial_{x_{1}}\eta^{\prime}(\tilde{U})\bigl{)}+Z+E_{1}+E_{2}\,,

where

{E_{1}}=\partial_{x_{1}}f(\tilde{U})-\partial_{x_{1}}f(S_{1}^{\mathbf{X}_{1}})-\partial_{x_{1}}f(W_{n}^{\mathbf{X}_{n}})\,,

if ${W}_{n}=S_{n}$ , then

	$\displaystyle{Z}$	$\displaystyle=\dot{\mathbf{X}}_{1}\partial_{x_{1}}S_{1}^{\mathbf{X}_{1}}+\dot{\mathbf{X}}_{n}\partial_{x_{1}}S_{n}^{\mathbf{X}_{n}}\,,$
	$\displaystyle{E_{2}}$	$\displaystyle=\partial_{x_{1}}\bigl{(}B_{1}(S_{1}^{\mathbf{X}_{1}})\partial_{x_{1}}\eta^{\prime}(S_{1}^{\mathbf{X}_{1}})\bigl{)}+\partial_{x_{1}}\bigl{(}B_{1}(S_{n}^{\mathbf{X}_{n}})\partial_{x_{1}}\eta^{\prime}(S_{n}^{\mathbf{X}_{n}})\bigl{)}$
		$\displaystyle\quad-\partial_{x_{1}}\bigl{(}B_{1}(\tilde{U})\partial_{x_{1}}\eta^{\prime}(\tilde{U})\bigl{)}\,,$

and if ${W}_{n}={R}_{n}$ , then

	$\displaystyle{Z}$	$\displaystyle=\dot{\mathbf{X}}_{1}\partial_{x_{1}}S_{1}^{\mathbf{X}_{1}}\,,$
	$\displaystyle{E_{2}}$	$\displaystyle=\partial_{x_{1}}\bigl{(}B_{1}(S_{1}^{\mathbf{X}_{1}})\partial_{x_{1}}\eta^{\prime}(S_{1}^{\mathbf{X}_{1}})\bigl{)}-\partial_{x_{1}}\bigl{(}B_{1}(\tilde{U})\partial_{x_{1}}\eta^{\prime}(\tilde{U})\bigl{)}\,.$

The terms $E_{1},E_{2}$ are error terms caused by the fact that $\tilde{U}$ is not an exact solution of the system (1.1). The term $Z$ comes from the shifts. We see $Z,E_{2}$ depend on the choice of $W_{n}$ , because while the planar viscous shock wave is shifted and is a solution to the viscous system (2.3), the planar rarefaction wave is not shifted and is a solution to the inviscid model (2.8).

As $\tilde{U}$ does not depend on the transverse direction $y$ , (1.1) and (2.16) give

(2.17)		$\displaystyle\partial_{t}(U-\tilde{U})+\partial_{x_{1}}\bigl{(}f(U)-f(\tilde{U})\bigl{)}+\sum_{j=2}^{d}\partial_{x_{j}}\bigl{(}g_{j}(U)-g_{j}(\tilde{U})\bigl{)}$
(2.17)		$\displaystyle=\sum_{j=1}^{d}\partial_{x_{j}}\bigl{(}B_{j}(U)\partial_{x_{j}}\eta^{\prime}(U)-B_{j}(\tilde{U})\partial_{x_{j}}\eta^{\prime}(\tilde{U})\bigl{)}-{Z}-{E_{1}}-{E_{2}}\,.$

3. Proof of Theorem 1.1

First, we introduce local-in-time estimates in Proposition 3.1 and a priori estimates in Proposition 3.3. Then we discuss how the two propositions prove the global existence and the asymptotic behavior results stated in Theorem 1.1.

3.1. Local-in-time estimates and a priori estimates

Proposition 3.1.

For any $0<\epsilon_{1}<\epsilon_{2}$ and any $t_{0}\geq 0$ , there exists $T_{0}>0$ that depends on $\epsilon_{1},\epsilon_{2}$ such that the following is true.

If $\|U(t_{0},\cdot)-\tilde{U}(t_{0},\cdot)\|_{H^{m}(\mathbb{R}\times{\mathbb{T}^{d-1}})}\leq\epsilon_{1}$ , then

1)

(1.1) has a unique solution $U$ on $[t_{0},t_{0}+T_{0}]$ ,
2)

(2.15) has a unique absolutely continuous solution $\mathbf{X}_{i}$ on $[t_{0},t_{0}+T_{0}]$ ,
3)

$U-\tilde{U}\in C\bigl{(}[t_{0},t_{0}+T_{0}];H^{m}(\mathbb{R}\times{\mathbb{T}^{d-1}})\bigl{)}$ ,
4)

$\|U(t,\cdot)-\tilde{U}(t,\cdot)\|_{H^{m}(\mathbb{R}\times{\mathbb{T}^{d-1}})}\leq\epsilon_{2}$ for any $t_{0}\leq t\leq t_{0}+T_{0}$ .

The local-in-time existence and uniqueness of the solution $U$ can be done similarly as in Serre’s paper [26]. Since the viscosity matrices of our system (1.1) are positive definite, the proof will be simpler. The existence and uniqueness of shifts $\mathbf{X}_{i}$ can be shown in the same way as in subsection 3.3 of [11].

Before introducing a priori estimates, we first establish the assumption of a priori estimates. Since sections 4 and 5 give the proof of a priori estimates, the following assumption will be the assumption of all lemmas in both sections.

Assumption 3.2.

Let $U$ be solution to (1.1) on $[0,T]$ for some $T>0$ . Let $\tilde{U}$ be the superposition wave defined in (2.13) with absolutely continuous shifts $\mathbf{X}_{1},\mathbf{X}_{n}$ defined in (2.15) and weight function $a$ defined in (2.14). Assume $\delta_{S_{1}},\delta_{W_{n}}<\delta_{0}$ ,

U-\tilde{U}\in C\bigl{(}[0,T];H^{m}(\mathbb{R}\times{\mathbb{T}^{d-1}})\bigl{)}\,,

and

(3.1)

\|U-\tilde{U}\|_{L^{\infty}(0,T;\,H^{m}(\mathbb{R}\times{\mathbb{T}^{d-1}}))}\leq\epsilon_{2}\,.

By Sobolev embedding, (3.1) in Assumption 3.2 implies

(3.2)

\|U-\tilde{U}\|_{L^{\infty}((0,T)\times(\mathbb{R}\times{\mathbb{T}^{d-1}}))}\leq C\epsilon_{2}\,.

Proposition 3.3.

For any $U_{-}\in\mathbb{R}^{n}$ , there exist $\delta_{0},\epsilon_{2},C_{0},\Lambda_{S_{1}},\Lambda_{W_{n}},\tilde{C}_{1},\tilde{C}_{n}>0$ such that the following is true.

Assume Assumption 3.2. Then

(3.3)

\displaystyle\begin{split}\underset{t\in[0,T]}{\textup{sup}}\|U(t,\cdot)-\tilde{U}(t,\cdot)\|_{H^{m}(\mathbb{R}\times{\mathbb{T}^{d-1}})}\\ +\sqrt{\int_{0}^{T}\sum_{k=0}^{m}D_{k}(U)+G_{S_{1}}(U)+G_{W_{n}}(U)+Ydt}\\ \leq C_{0}\|U_{0}-\tilde{U}_{0}\|_{H^{m}(\mathbb{R}\times{\mathbb{T}^{d-1}})}+C_{0}E\,,\end{split}

and for any $\beta\in\mathbb{N}^{d}$ such that $1\leq|\beta|\leq m$ ,

(3.4)			$\displaystyle\int_{0}^{T}\Big{\|}\frac{d}{dt}\int\big{\|}\partial_{x}^{\beta}\bigl{(}U(t,\cdot)-\tilde{U}(t,\cdot)\bigl{)}\big{\|}^{2}\,dx\Big{\|}\,dt$
(3.4)			$\displaystyle\leq C_{0}\int_{0}^{T}\Bigl{(}\sum_{k=0}^{m}D_{k}(U)+G_{S_{1}}(U)+G_{W_{n}}(U)+Y\Bigl{)}\,dt+C_{0}E^{2}\,,$

where

(3.5)	$\displaystyle D_{k}(U)$	$\displaystyle=\sum_{\alpha\in\mathbb{N}^{d},\|\alpha\|=k+1}\int\big{\|}\partial_{x}^{\alpha}(U-\tilde{U})\big{\|}^{2}\,dx\,,$
	$\displaystyle G_{S_{1}}(U)$	$\displaystyle=\int\|U-\tilde{U}\|^{2}\|\partial_{x_{1}}S_{1}^{\mathbf{X}_{1}}\|\,dx\,,$
	$\displaystyle G_{W_{n}}(U)$	$\displaystyle=\int\|U-\tilde{U}\|^{2}\|\partial_{x_{1}}W_{n}^{\mathbf{X}_{n}}\|\,dx\,,$
	$\displaystyle Y$	$\displaystyle=\begin{cases}\delta_{S_{1}}\|\dot{\mathbf{X}}_{1}\|^{2}+\delta_{W_{n}}\|\dot{\mathbf{X}}_{n}\|^{2}\,,\textup{ if }W_{n}=S_{n}\,,\\ \delta_{S_{1}}\|\dot{\mathbf{X}}_{1}\|^{2}\,,\textup{ if }W_{n}=R_{n}\,,\end{cases}$
	$\displaystyle E$	$\displaystyle=\begin{cases}\delta_{S_{n}}+\delta_{S_{1}}\,,\textup{ if }W_{n}=S_{n}\,,\\ \delta_{R_{n}}^{1/6}\,,\textup{ if }W_{n}=R_{n}\,.\end{cases}$

In addition,

(3.6)

|\dot{\mathbf{X}}_{1}(t)|+|\dot{\mathbf{X}}_{n}(t)|\leq C_{0}\|U(t,\cdot)-\tilde{U}(t,\cdot)\|_{L^{\infty}(\mathbb{R}\times{\mathbb{T}^{d-1}})}\,,\quad\forall\,t\in[0,T]\,.

3.2. Global existence and estimates

We define

\mathbf{E}(t)=\|U(t,\cdot)-\tilde{U}(t,\cdot)\|_{H^{m}(\mathbb{R}\times{\mathbb{T}^{d-1}})}\,.

We fix $\epsilon_{2},C_{0}$ as in Proposition 3.3. We take the strength of the initial perturbation $\epsilon_{0}$ and the wave strength $\delta_{0}$ in Theorem 1.1 and $\epsilon_{1}$ in Proposition 3.1 to be small enough such that

(3.7)

0<C_{0}\mathbf{E}(0)+C_{0}E<\epsilon_{1}<\frac{\epsilon_{2}}{2}\,.

We define

\displaystyle T=\textup{sup}\Bigl{\{}t\geq 0:U-\tilde{U}\in C\bigl{(}[0,T];H^{m}(\mathbb{R}\times{\mathbb{T}^{d-1}})\bigl{)}\,,\;\mathbf{E}(t)<\epsilon_{2}\Bigl{\}}\,.

Assume $T<\infty$ . Then Proposition 3.3 gives

\displaystyle\mathbf{E}(T)\leq C_{0}\mathbf{E}(0)+C_{0}E<\epsilon_{1}\,.

By Proposition 3.1, there exists $T_{0}>0$ such that

\displaystyle U-\tilde{U}\in C\bigl{(}[T,T+T_{0}];H^{m}(\mathbb{R}\times{\mathbb{T}^{d-1}})\bigl{)}\,\textup{ and }\,\mathbf{E}(t)\leq\frac{\epsilon_{2}}{2}\textup{ for any }t\in[T,T+T_{0}]\,.

Contradiction! Hence, we get $T=\infty$ and (1.10) in Theorem 1.1. Now we can apply Proposition 3.3 on $[0,\infty)$ and get

(3.8)

\displaystyle\begin{split}\underset{t\in[0,\infty)}{\textup{sup}}\|U-\tilde{U}\|_{H^{m}(\mathbb{R}\times{\mathbb{T}^{d-1}})}\\ +\sqrt{\int_{0}^{\infty}\sum_{k=0}^{m}D_{k}(U)+G_{S_{1}}(U)+G_{W_{n}}(U)+Ydt}\\ \leq C_{0}\|U_{0}-\tilde{U}_{0}\|_{H^{m}(\mathbb{R}\times{\mathbb{T}^{d-1}})}+C_{0}E<\infty\,,\end{split}

and for any $\beta=\mathbb{N}^{d}$ such that $1\leq|\beta|\leq m$ ,

(3.9)			$\displaystyle\int_{0}^{\infty}\Big{\|}\frac{d}{dt}\int\big{\|}\partial_{x}^{\beta}\bigl{(}U(t,\cdot)-\tilde{U}(t,\cdot)\bigl{)}\big{\|}^{2}\,dx\Big{\|}\,dt$
(3.9)			$\displaystyle\leq C_{0}\int_{0}^{\infty}\Bigl{(}\sum_{k=0}^{m}D_{k}(U)+G_{S_{1}}(U)+G_{W_{n}}(U)+Y\Bigl{)}\,dt+C_{0}E^{2}<\infty\,.$

In addition,

(3.10)

|\dot{\mathbf{X}}_{1}(t)|+|\dot{\mathbf{X}}_{n}(t)|\leq C_{0}\|U(t,\cdot)-\tilde{U}(t,\cdot)\|_{L^{\infty}(\mathbb{R}\times{\mathbb{T}^{d-1}})}\,,\quad\forall\,t\geq 0\,.

By (3.8), we get (1.12) in Theorem 1.1.

3.3. Time-asymptotic behavior

Let $\beta\in\mathbb{N}^{d}$ be such that $1\leq|\beta|\leq m$ . We define

g(t)=\big{\|}\partial_{x}^{\beta}\bigl{(}U(t,\cdot)-\tilde{U}(t,\cdot)\bigl{)}\big{\|}_{L^{2}(\mathbb{R}\times{\mathbb{T}^{d-1}})}^{2}\,.

We show the classical estimate

(3.11)

\int_{0}^{\infty}|g(t)|+|{g}^{\prime}(t)|\,dt<\infty\,.

By (3.8), we get

\displaystyle\int_{0}^{\infty}|g(t)|\,dt\leq\int_{0}^{\infty}\sum_{k=0}^{m-1}D_{k}(U)\,dt<\infty\,.

By (3.9), we have

\displaystyle\int_{0}^{\infty}|g^{\prime}(t)|\,dt=\int_{0}^{\infty}\Big{|}\frac{d}{dt}\int\big{|}\partial_{x}^{\beta}\bigl{(}U(t,\cdot)-\tilde{U}(t,\cdot)\bigl{)}\big{|}^{2}\,dx\Big{|}\,dt<\infty.

The classical estimate (3.11) gives

\underset{t\rightarrow\infty}{\textup{lim}}\big{\|}\partial_{x}^{\beta}\bigl{(}U(t,\cdot)-\tilde{U}(t,\cdot)\bigl{)}\big{\|}_{L^{2}(\mathbb{R}\times{\mathbb{T}^{d-1}})}=0\,.

The Gagliardo–Nirenberg inequality proved in [24], the periodicity in the transverse direction, and (3.8) give

(3.12)

\underset{t\rightarrow\infty}{\textup{lim}}\|U(t,\cdot)-\tilde{U}(t,\cdot)\|_{L^{\infty}(\mathbb{R}\times{\mathbb{T}^{d-1}})}=0\,.

By (3.12) and Lemma 2.2, we get (1.11) in Theorem 1.1. By (3.10) and (3.12), we get (1.13) in Theorem 1.1.

4. Energy estimate

We show the $L^{2}$ estimates by the $a$ -contraction method with shifts in this section. Later in section 5, induction will help us get the $H^{m}$ estimates and finish the proof of Proposition 3.3.

First, we develop tools to handle interaction terms and higher-order terms in subsections 4.1 and 4.2. Then we apply the $a$ -contraction method with shifts. The hyperbolic “scalarization” is discussed in subsection 4.3. The positive hyperbolic remainder corresponding to the special direction of the planar viscous shock wave motivates the Poincaré type inequality introduced in subsection 4.4. Finally in subsection 4.5, we choose the constants $\Lambda_{S_{1}},\Lambda_{W_{n}},\tilde{C}_{1},\tilde{C}_{n}$ defining weight functions and shift functions in a way that makes both hyperbolic “scalarization” and Poincaré type inequality work.

4.1. Wave interaction estimates

To control the interaction between waves, the idea is to take the shifts small enough that the main layer regions do not overlap.

Lemma 4.1.

Let $\mathbf{X}_{1},\mathbf{X}_{n}$ be the shifts defined in (2.15). Assume Assumption 3.2. Then for any $0\leq t\leq T$ ,

	$\displaystyle\big{\\|}\|\partial_{x_{1}}S_{1}^{\mathbf{X}_{1}}\|\|R_{n}-U_{m}\|\big{\\|}_{L^{2}}+\big{\\|}\|\partial_{x_{1}}R_{n}\|\|S_{1}^{\mathbf{X}_{1}}-U_{m}\|\big{\\|}_{L^{2}}+\big{\\|}\|\partial_{x_{1}}R_{n}\|\|\partial_{x_{1}}S_{1}^{\mathbf{X}_{1}}\|\big{\\|}_{L^{2}}$
	$\displaystyle\leq C\delta_{R_{n}}\delta_{S_{1}}e^{-C\delta_{S_{1}}t}\,,\quad\textup{ if }W_{n}=R_{n}\,,$
	$\displaystyle\big{\\|}\|\partial_{x_{1}}S_{1}^{\mathbf{X}_{1}}\|\|S_{n}^{\mathbf{X}_{n}}-U_{m}\|^{2}\big{\\|}_{L^{1}}+\big{\\|}\|\partial_{x_{1}}S_{n}^{\mathbf{X}_{n}}\|\|S_{1}^{\mathbf{X}_{1}}-U_{m}\|^{2}\big{\\|}_{L^{1}}+\big{\\|}\|\partial_{x_{1}}S_{n}^{\mathbf{X}_{n}}\|\|\partial_{x_{1}}S_{1}^{\mathbf{X}_{1}}\|\big{\\|}_{L^{1}}$
	$\displaystyle\leq C(\delta_{S_{n}}^{2}\delta_{S_{1}}e^{-C\delta_{S_{1}}t}+\delta_{S_{n}}\delta_{S_{1}}^{2}e^{-C\delta_{S_{n}}t})\,,\quad\textup{ if }W_{n}=S_{n}\,.$

Proof.

First, we define the spectral gap

\displaystyle\Delta=\lambda_{n}(U_{-})-\lambda_{1}(U_{-})\,.

By (2.15) and (3.2), we take $\epsilon_{2}$ small enough such that for any $i\in\{1,n\}$ ,

|\mathbf{X}_{i}(t)|\leq C\tilde{C}_{i}\epsilon_{2}t<\frac{\Delta t}{8}\,.

For any $x_{1}>\frac{(\lambda_{1}(U_{-})+\lambda_{n}(U_{-}))t}{2}$ , we have

\displaystyle x_{1}-{\sigma}_{1}t+\mathbf{X}_{1}(t)>\frac{\Delta t}{4}\,.

Lemma 2.1 implies that for any $x_{1}>\frac{(\lambda_{1}(U_{-})+\lambda_{n}(U_{-}))t}{2}$ ,

	$\displaystyle\|S_{1}^{\mathbf{X}_{1}}-U_{m}\|$	$\displaystyle\leq C\delta_{S_{1}}e^{-C\delta_{S_{1}}\|x_{1}-{\sigma}_{1}t+\mathbf{X}_{1}(t)\|}$
		$\displaystyle\leq C\delta_{S_{1}}\textup{exp}\bigl{(}\frac{-C\delta_{S_{1}}\|x_{1}-{\sigma}_{1}t+\mathbf{X}_{1}(t)\|}{2}\bigl{)}\textup{exp}\bigl{(}\frac{-C\delta_{S_{1}}\Delta t}{8}\bigl{)}\,,$
	$\displaystyle\|\partial_{x_{1}}S_{1}^{\mathbf{X}_{1}}\|$	$\displaystyle\leq C\delta_{S_{1}}^{2}e^{-C\delta_{S_{1}}\|x_{1}-{\sigma}_{1}t+\mathbf{X}_{1}(t)\|}$
		$\displaystyle\leq C\delta_{S_{1}}^{2}\textup{exp}\bigl{(}\frac{-C\delta_{S_{1}}\|x_{1}-{\sigma}_{1}t+\mathbf{X}_{1}(t)\|}{2}\bigl{)}\textup{exp}\bigl{(}\frac{-C\delta_{S_{1}}\Delta t}{8}\bigl{)}\,.$

For any $x_{1}\leq\frac{(\lambda_{1}(U_{-})+\lambda_{n}(U_{-}))t}{2}$ , we have

	$\displaystyle x_{1}-\lambda_{n}(U_{m})t$	$\displaystyle\leq-\frac{\Delta t}{4}\,,$
	$\displaystyle x_{1}-\sigma_{n}t+\mathbf{X}_{n}(t)$	$\displaystyle\leq-\frac{\Delta t}{4}\,.$

Lemma 2.2 implies that for any $x_{1}\leq\frac{(\lambda_{1}(U_{-})+\lambda_{n}(U_{-}))t}{2}$ ,

	$\displaystyle\|R_{n}-U_{m}\|\,,\;\|\partial_{x_{1}}R_{n}\|$	$\displaystyle\leq C\delta_{R_{n}}e^{-2\|x_{1}-\lambda_{n}(U_{m})t\|}$
		$\displaystyle\leq C\delta_{R_{n}}\textup{exp}\bigl{(}-\|x_{1}-\lambda_{n}(U_{m})t\|\bigl{)}\textup{exp}\bigl{(}-\frac{\Delta t}{4}\bigl{)}\,,$

and Lemma 2.1 implies that for any $x_{1}\leq\frac{(\lambda_{1}(U_{-})+\lambda_{n}(U_{-}))t}{2}$ ,

	$\displaystyle\|S_{n}^{\mathbf{X}_{n}}-U_{m}\|$	$\displaystyle\leq C\delta_{S_{n}}e^{-C\delta_{S_{n}}\|x_{1}-\sigma_{n}t+\mathbf{X}_{n}(t)\|}$
		$\displaystyle\leq C\delta_{S_{n}}\textup{exp}\bigl{(}\frac{-C\delta_{S_{n}}\|x_{1}-\sigma_{n}t+\mathbf{X}_{n}(t)\|}{2}\bigl{)}\textup{exp}\bigl{(}\frac{-C\delta_{S_{n}}\Delta t}{8}\bigl{)}\,,$
	$\displaystyle\|\partial_{x_{1}}S_{n}^{\mathbf{X}_{n}}\|$	$\displaystyle\leq C\delta_{S_{n}}^{2}e^{-C\delta_{S_{n}}\|x_{1}-\sigma_{n}t+\mathbf{X}_{n}(t)\|}$
		$\displaystyle\leq C\delta_{S_{n}}^{2}\textup{exp}\bigl{(}\frac{-C\delta_{S_{n}}\|x_{1}-\sigma_{n}t+\mathbf{X}_{n}(t)\|}{2}\bigl{)}\textup{exp}\bigl{(}\frac{-C\delta_{S_{n}}\Delta t}{8}\bigl{)}\,.$

We have

	$\displaystyle\|\partial_{x_{1}}S_{1}^{\mathbf{X}_{1}}\|\bigl{(}\|R_{n}-U_{m}\|+\|\partial_{x_{1}}R_{n}\|\bigl{)}$
	$\displaystyle\leq\begin{cases}C\delta_{R_{n}}\delta_{S_{1}}^{2}e^{-C\delta_{S_{1}}\|x_{1}-{\sigma}_{1}t+\mathbf{X}_{1}(t)\|}e^{-C\delta_{S_{1}}t}\,,\;\textup{ if }x_{1}>\frac{(\lambda_{1}(U_{-})+\lambda_{n}(U_{-}))t}{2}\,,\\ C\delta_{R_{n}}\delta_{S_{1}}^{2}e^{-\|x_{1}-\lambda_{n}(U_{m})t\|}e^{-Ct}\,,\;\textup{ if }x_{1}\leq\frac{(\lambda_{1}(U_{-})+\lambda_{n}(U_{-}))t}{2}\,,\end{cases}$
	$\displaystyle\|\partial_{x_{1}}R_{n}\|\|S_{1}^{\mathbf{X}_{1}}-U_{m}\|$
	$\displaystyle\leq\begin{cases}C\|\partial_{x_{1}}R_{n}\|\delta_{S_{1}}e^{-C\delta_{S_{1}}\|{x_{1}}-{\sigma}_{1}t+\mathbf{X}_{1}(t)\|}e^{-C\delta_{S_{1}}t}\,,\;\textup{ if }{x_{1}}>\frac{(\lambda_{1}(U_{-})+\lambda_{n}(U_{-}))t}{2}\,,\\ C\delta_{R_{n}}\delta_{S_{1}}e^{-\|{x_{1}}-\lambda_{n}(U_{m})t\|}e^{-Ct}\,,\;\textup{ if }{x_{1}}\leq\frac{(\lambda_{1}(U_{-})+\lambda_{n}(U_{-}))t}{2}\,.\end{cases}$

Hence, we get

	$\displaystyle\big{\\|}\|\partial_{x_{1}}S_{1}^{\mathbf{X}_{1}}\|\|R_{n}-U_{m}\|\big{\\|}_{L_{x_{1}}^{2}}^{2}+\big{\\|}\|\partial_{x_{1}}R_{n}\|\|\partial_{x_{1}}S_{1}^{\mathbf{X}_{1}}\|\big{\\|}_{L_{x_{1}}^{2}}^{2}$
	$\displaystyle\leq C\delta_{R_{n}}^{2}\delta_{S_{1}}^{3}e^{-C\delta_{S_{1}}t}\int\delta_{S_{1}}(e^{-C\delta_{S_{1}}\|{x_{1}}\|}+e^{-\|{x_{1}}\|})\,d{x_{1}}\leq C\delta_{R_{n}}^{2}\delta_{S_{1}}^{3}e^{-C\delta_{S_{1}}t}\,,$
	$\displaystyle\big{\\|}\|\partial_{x_{1}}R_{n}\|\|S_{1}^{\mathbf{X}_{1}}-U_{m}\|\big{\\|}_{L_{x_{1}}^{2}}^{2}$
	$\displaystyle\leq C\delta_{R_{n}}\delta_{S_{1}}^{2}e^{-C\delta_{S_{1}}t}\int\|\partial_{x_{1}}R_{n}\|\,d{x_{1}}+C\delta_{R_{n}}^{2}\delta_{S_{1}}^{2}e^{-Ct}\int e^{-\|{x_{1}}\|}\,d{x_{1}}$
	$\displaystyle\leq C\delta_{R_{n}}^{2}\delta_{S_{1}}^{2}e^{-C\delta_{S_{1}}t}\,.$

We have

	$\displaystyle\|\partial_{x_{1}}S_{1}^{\mathbf{X}_{1}}\|\|S_{n}^{\mathbf{X}_{n}}-U_{m}\|^{2}+\|\partial_{x_{1}}S_{n}^{\mathbf{X}_{n}}\|\|S_{1}^{\mathbf{X}_{1}}-U_{m}\|^{2}+\|\partial_{x_{1}}S_{n}^{\mathbf{X}_{n}}\|\|\partial_{x_{1}}S_{1}^{\mathbf{X}_{1}}\|$
	$\displaystyle\leq\begin{cases}C\delta_{S_{n}}^{2}\delta_{S_{1}}^{2}e^{-C\delta_{S_{1}}\|x_{1}-{\sigma}_{1}t+\mathbf{X}_{1}(t)\|}e^{-C\delta_{S_{1}}t}\,,\;\textup{ if }x_{1}>\frac{(\lambda_{1}(U_{-})+\lambda_{n}(U_{-}))t}{2}\,,\\ C\delta_{S_{n}}^{2}\delta_{S_{1}}^{2}e^{-C\delta_{S_{n}}\|x_{1}-\sigma_{n}t+\mathbf{X}_{n}(t)\|}e^{-C\delta_{S_{n}}t}\,,\;\textup{ if }x_{1}\leq\frac{(\lambda_{1}(U_{-})+\lambda_{n}(U_{-}))t}{2}\,.\end{cases}$

Hence, we get

	$\displaystyle\big{\\|}\|\partial_{x_{1}}S_{1}^{\mathbf{X}_{1}}\|\|S_{n}^{\mathbf{X}_{n}}-U_{m}\|^{2}\big{\\|}_{L_{x_{1}}^{1}}+\big{\\|}\|\partial_{x_{1}}S_{n}^{\mathbf{X}_{n}}\|\|S_{1}^{\mathbf{X}_{1}}-U_{m}\|^{2}\big{\\|}_{L_{x_{1}}^{1}}+\big{\\|}\|\partial_{x_{1}}S_{n}^{\mathbf{X}_{n}}\|\|\partial_{x_{1}}S_{1}^{\mathbf{X}_{1}}\|\big{\\|}_{L_{x_{1}}^{1}}$
	$\displaystyle\leq C\delta_{S_{n}}^{2}\delta_{S_{1}}e^{-C\delta_{S_{1}}t}\int\delta_{S_{1}}e^{-C\delta_{S_{1}}\|{x_{1}}\|}\,d{x_{1}}+C\delta_{S_{n}}\delta_{S_{1}}^{2}e^{-C\delta_{S_{n}}t}\int\delta_{S_{n}}e^{-C\delta_{S_{n}}\|{x_{1}}\|}\,d{x_{1}}$
	$\displaystyle\leq C(\delta_{S_{n}}^{2}\delta_{S_{1}}e^{-C\delta_{S_{1}}t}+\delta_{S_{n}}\delta_{S_{1}}^{2}e^{-C\delta_{S_{n}}t})\,.$

∎

4.2. Higher derivatives estimates

Let

(4.1)

\psi=U-\tilde{U}\,.

For any $k\in\mathbb{N}^{*}$ , we define

(4.2)

\displaystyle\mathcal{L}^{k}=\Bigl{\{}\Pi_{j=1}^{l}|\partial_{x}^{\beta_{j}}\psi|:l\in\mathbb{N}^{*}\,,\;\beta_{j}\in\mathbb{N}^{d}\,,\;1\leq|\beta_{j}|\leq k\,,\;\sum_{j=1}^{l}|\beta_{j}|\leq k+1\Bigl{\}}\,.

Lemma 4.2 plays an important role in section 5 when working with higher derivatives. It singles out terms that need to be handled differently and unifies the rest in the same form. The last two inequalities in the lemma will be used in the proof of Lemma 4.3 to evaluate the interaction terms induced by $E_{1},E_{2}$ .

Lemma 4.2.

Assume Assumption 3.2. Let $1\leq k\leq m$ where $m$ is defined in (1.8). Let $M:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n\times n}$ and $F:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ be smooth functions. Then there exists a constant $C>0$ such that for any $\alpha_{k},\alpha_{1}\in\mathbb{N}^{d}$ such that $|\alpha_{k}|=k\,,\;|\alpha_{1}|=1$ ,

	$\displaystyle\Big{\|}\partial_{x}^{\alpha_{k}}\bigl{(}M(U)\partial_{x}^{\alpha_{1}}U-M(\tilde{U})\partial_{x}^{\alpha_{1}}\tilde{U}\bigl{)}-\bigl{(}M(U)\partial_{x}^{\alpha_{k}+\alpha_{1}}U-M(\tilde{U})\partial_{x}^{\alpha_{k}+\alpha_{1}}\tilde{U}\bigl{)}\Big{\|}$
	$\displaystyle\leq C\bigl{(}\sum_{L\in\mathcal{L}^{k}}L+\|\psi\|\|\partial_{x_{1}}\tilde{U}\|\bigl{)}\,,$
	$\displaystyle\Big{\|}\partial_{x}^{\alpha_{k}}\bigl{(}F(U)-F(\tilde{U})\bigl{)}\Big{\|}\leq C\bigl{(}\sum_{L\in\mathcal{L}^{k}}L+\|\psi\|\|\partial_{x_{1}}\tilde{U}\|\bigl{)}\,,$
	$\displaystyle\Big{\|}\partial_{x}^{\alpha_{k}}\bigl{(}M(\tilde{U})\partial_{x_{1}}\tilde{U}-M(S_{1}^{\mathbf{X}_{1}})\partial_{x_{1}}S_{1}^{\mathbf{X}_{1}}\bigl{)}\Big{\|}$
	$\displaystyle\leq C\bigl{(}\|\partial_{x_{1}}^{k+1}W_{n}^{\mathbf{X}_{n}}\|+\|\partial_{x_{1}}W_{n}^{\mathbf{X}_{n}}\|^{2}+\|\partial_{x_{1}}W_{n}^{\mathbf{X}_{n}}\|\|\partial_{x_{1}}S_{1}^{\mathbf{X}_{1}}\|+\|W_{n}^{\mathbf{X}_{n}}-U_{m}\|\|\partial_{x_{1}}S_{1}^{\mathbf{X}_{1}}\|\bigl{)}\,,$
	$\displaystyle\Big{\|}\partial_{x}^{\alpha_{k}}\bigl{(}F(\tilde{U})-F(S_{1}^{\mathbf{X}_{1}})-F(W_{n}^{\mathbf{X}_{n}})\bigl{)}\Big{\|}$
	$\displaystyle+\Big{\|}\partial_{x}^{\alpha_{k}}\bigl{(}M(\tilde{U})\partial_{x_{1}}\tilde{U}-M(S_{1}^{\mathbf{X}_{1}})\partial_{x_{1}}S_{1}^{\mathbf{X}_{1}}-M(W_{n}^{\mathbf{X}_{n}})\partial_{x_{1}}W_{n}^{\mathbf{X}_{n}}\bigl{)}\Big{\|}$
	$\displaystyle\leq C\bigl{(}\|\partial_{x_{1}}W_{n}^{\mathbf{X}_{n}}\|\|\partial_{x_{1}}S_{1}^{\mathbf{X}_{1}}\|+\|W_{n}^{\mathbf{X}_{n}}-U_{m}\|\|\partial_{x_{1}}S_{1}^{\mathbf{X}_{1}}\|+\|S_{1}^{\mathbf{X}_{1}}-U_{m}\|\|\partial_{x_{1}}W_{n}^{\mathbf{X}_{n}}\|\bigl{)}\,.$

Proof.

For any $k_{1},k_{2}\in\mathbb{N}^{*}$ and any function $V,\tilde{V}:\mathbb{R}_{\geq 0}\times\mathbb{R}\times\mathbb{T}^{d-1}\rightarrow\mathbb{R}^{n}$ , we define

	$\displaystyle\mathcal{L}^{k_{1},k_{2}}(V,\tilde{V})$	$\displaystyle=\Bigl{\{}\Pi_{j=1}^{l}\|\partial_{x}^{\beta_{j}}V\|\,\Pi_{j=1}^{\tilde{l}}\|\partial_{x}^{\tilde{\beta_{j}}}\tilde{V}\|:l\in\mathbb{N}^{*},\,\tilde{l}\in\mathbb{N},\,\beta_{j},\tilde{\beta_{j}}\in\mathbb{N}^{d},\,$
		$\displaystyle\quad\quad 1\leq\|\beta_{j}\|,\|\tilde{\beta_{j}}\|\leq k_{1},\,\sum_{j=1}^{l}\|\beta_{j}\|+\sum_{j=1}^{\tilde{l}}\|\tilde{\beta_{j}}\|=k_{2}\,,\;\sum_{j=1}^{l}\|\beta_{j}\|\geq 1\Bigl{\}}\,.$

We show the first inequality. If we apply the chain rule to

\partial_{x}^{\alpha_{k}}\bigl{(}M(\cdot)\partial_{x}^{\alpha_{1}}(\cdot)\bigl{)}\,,

then either all $\partial_{x}^{\alpha_{k}}$ fall on $\partial_{x}^{\alpha_{1}}(\cdot)$ or some fall on $M(\cdot)$ . Therefore, we know that

\displaystyle\Big{|}\partial_{x}^{\alpha_{k}}\bigl{(}M(U)\partial_{x}^{\alpha_{1}}U-M(\tilde{U})\partial_{x}^{\alpha_{1}}\tilde{U}\bigl{)}-\bigl{(}M(U)\partial_{x}^{\alpha_{k}+\alpha_{1}}\partial_{x}U-M(\tilde{U})\partial_{x}^{\alpha_{k}+\alpha_{1}}\tilde{U}\bigl{)}\Big{|}

is bounded by the sum of the absolute value of the terms in the following form

\displaystyle\mathbf{M}:=\bigl{(}M^{\beta_{j}}(U)\otimes_{j=1}^{l-1}\partial_{x}^{\beta_{j}}U\bigl{)}\partial_{x}^{\beta_{l}}U-\bigl{(}M^{\beta_{j}}(\tilde{U})\otimes_{j=1}^{l-1}\partial_{x}^{\beta_{j}}\tilde{U}\bigl{)}\partial_{x}^{\beta_{l}}\tilde{U}\,,

where $l\geq 2$ , $\beta_{j}\in\mathbb{N}^{d}$ , $1\leq|\beta_{j}|\leq k$ , $\sum_{j=1}^{l}|\beta_{j}|=k+1$ , and $M^{\beta_{j}}$ is some derivative of $M$ . By (3.2), Lemma 2.1, and Lemma 2.2, we have

	$\displaystyle\|\mathbf{M}\|$	$\displaystyle=\Big{\|}\bigl{(}M^{\beta_{j}}(U)\otimes_{j=1}^{l-1}\partial_{x}^{\beta_{j}}(\psi+\tilde{U})\bigl{)}\partial_{x}^{\beta_{l}}(\psi+\tilde{U})-\bigl{(}M^{\beta_{j}}(\tilde{U})\otimes_{j=1}^{l-1}\partial_{x}^{\beta_{j}}\tilde{U}\bigl{)}\partial_{x}^{\beta_{l}}\tilde{U}\Big{\|}$
		$\displaystyle\leq C\sum_{\mathcal{L}^{k,k+1}(\psi,\tilde{U})}L+\Big{\|}\Bigl{(}\bigl{(}M^{\beta_{j}}(U)-M^{\beta_{j}}(\tilde{U})\bigl{)}\otimes_{j=1}^{l-1}\partial_{x}^{\beta_{j}}\tilde{U}\Bigl{)}\partial_{x}^{\beta_{l}}\tilde{U}\Big{\|}$
		$\displaystyle\leq C\sum_{\mathcal{L}^{k}}L+C\|\psi\|\|\partial_{x_{1}}\tilde{U}\|\,.$

Note the constants $C$ in this proof depend on $B_{j},f,\eta,U_{-},k$ , but the dependency on $k$ does not matter as $m$ is fixed and finite.

Similarly, we can show

\displaystyle\Big{|}\partial_{x}^{\alpha_{k}}\bigl{(}F(U)-F(\tilde{U})\bigl{)}\Big{|}\leq C\bigl{(}\sum_{\mathcal{L}^{k,k}(\psi,\tilde{U})}L+|\psi||\partial_{x_{1}}\tilde{U}|\bigl{)}\leq C\bigl{(}\sum_{\mathcal{L}^{k}}L+|\psi||\partial_{x_{1}}\tilde{U}|\bigl{)}\,,

and

	$\displaystyle\Big{\|}\partial_{x}^{\alpha_{k}}\bigl{(}M(\tilde{U})\partial_{x}\tilde{U}-M(S_{1}^{\mathbf{X}_{1}})\partial_{x}S_{1}^{\mathbf{X}_{1}}\bigl{)}\Big{\|}$
	$\displaystyle\leq C\bigl{(}\sum_{\mathcal{L}^{k+1,k+1}(W_{n}^{\mathbf{X}_{n}},\,S_{1}^{\mathbf{X}_{1}})}L+\|W_{n}^{\mathbf{X}_{n}}-U_{m}\|\|\partial_{x_{1}}S_{1}^{\mathbf{X}_{1}}\|\bigl{)}$
	$\displaystyle\leq C\bigl{(}\|\partial_{x_{1}}^{k+1}W_{n}^{\mathbf{X}_{n}}\|+\|\partial_{x_{1}}W_{n}^{\mathbf{X}_{n}}\|^{2}+\|\partial_{x_{1}}W_{n}^{\mathbf{X}_{n}}\|\|\partial_{x_{1}}S_{1}^{\mathbf{X}_{1}}\|+\|W_{n}^{\mathbf{X}_{n}}-U_{m}\|\|\partial_{x_{1}}S_{1}^{\mathbf{X}_{1}}\|\bigl{)}\,.$

Finally, we show the last inequality. We know

\displaystyle\Big{|}\partial_{x}^{\alpha_{k}}\bigl{(}F(\tilde{U})-F(S_{1}^{\mathbf{X}_{1}})-F(W_{n}^{\mathbf{X}_{n}})\bigl{)}\Big{|}

is bounded by the sum of the absolute value of the terms in the following form

\displaystyle\mathbf{F}:=

\displaystyle F^{\beta_{j}}(\tilde{U})\otimes_{j=1}^{l}\partial_{x}^{\beta_{j}}\tilde{U}-F^{\beta_{j}}(S_{1}^{\mathbf{X}_{1}})\otimes_{j=1}^{l}\partial_{x}^{\beta_{j}}S_{1}^{\mathbf{X}_{1}}-F^{\beta_{j}}(W_{n}^{\mathbf{X}_{n}})\otimes_{j=1}^{l}\partial_{x}^{\beta_{j}}W_{n}^{\mathbf{X}_{n}}\,,

where $l\geq 1$ , $\beta_{j}\in\mathbb{N}^{d}$ , $1\leq|\beta_{j}|\leq k$ , $\sum_{j=1}^{l}|\beta_{j}|=k$ , and $F^{\beta_{j}}$ is some derivative of $F$ . By Lemma 2.1 and Lemma 2.2, we have

	$\displaystyle\|\mathbf{F}\|$	$\displaystyle=\Big{\|}F^{\beta_{j}}(\tilde{U})\otimes_{j=1}^{l}\partial_{x}^{\beta_{j}}(S_{1}^{\mathbf{X}_{1}}+W_{n}^{\mathbf{X}_{n}})-F^{\beta_{j}}(S_{1}^{\mathbf{X}_{1}})\otimes_{j=1}^{l}\partial_{x}^{\beta_{j}}S_{1}^{\mathbf{X}_{1}}$
		$\displaystyle\quad-F^{\beta_{j}}(W_{n}^{\mathbf{X}_{n}})\otimes_{j=1}^{l}\partial_{x}^{\beta_{j}}W_{n}^{\mathbf{X}_{n}}\Big{\|}$
		$\displaystyle\leq C\bigl{(}\|\partial_{x_{1}}W_{n}^{\mathbf{X}_{n}}\|\|\partial_{x_{1}}S_{1}^{\mathbf{X}_{1}}\|+\|W_{n}^{\mathbf{X}_{n}}-U_{m}\|\|\partial_{x_{1}}S_{1}^{\mathbf{X}_{1}}\|+\|S_{1}^{\mathbf{X}_{1}}-U_{m}\|\|\partial_{x_{1}}W_{n}^{\mathbf{X}_{n}}\|\bigl{)}\,.$

Similarly, we can bound

\displaystyle\Big{|}\partial_{x}^{\alpha_{k}}\bigl{(}M(\tilde{U})\partial_{x_{1}}\tilde{U}-M(S_{1}^{\mathbf{X}_{1}})\partial_{x_{1}}S_{1}^{\mathbf{X}_{1}}-M(W_{n}^{\mathbf{X}_{n}})\partial_{x_{1}}W_{n}^{\mathbf{X}_{n}}\bigl{)}\Big{|}\,.

∎

4.3. Relative entropy method

For each layer corresponding to $S_{1}^{\mathbf{X}_{1}}$ and $W_{n}^{\mathbf{X}_{n}}$ , we will construct a basis of the phase space that is well-adapted to both the special direction of the wave $\mathbf{r_{1}}$ (respectively $\mathbf{r_{n}}$ ) and the dissipation matrix $B_{1}(U_{-})$ . As the dissipation takes place in the so-called entropic variables $\eta^{\prime}(U)-\eta^{\prime}(\tilde{U})$ , we project such quantity onto the bases. Since $U-\tilde{U}$ is small, we know

\displaystyle\eta^{\prime}(U)-\eta^{\prime}(\tilde{U})\approx\eta^{\prime\prime}(U_{-})(U-\tilde{U})\,.

Hence, the corresponding natural special direction of the wave for the entropic variables is $\eta^{\prime\prime}(U_{-})\mathbf{r_{1}}=\mathbf{l_{1}}$ (respectively $\eta^{\prime\prime}(U_{-})\mathbf{r_{n}}=\mathbf{l_{n}}$ ). The point is to work with a basis that is orthogonal with respect to the dissipation matrix and contains $\mathbf{l_{1}}$ (respectively $\mathbf{l_{n}}$ ).

When working with the layer of $S_{1}^{\mathbf{X}_{1}}$ , we complete $\mathbf{l_{1}}$ into an orthogonal basis of $\mathbb{R}^{n}$ with respect to the dissipation. In particular, we choose a basis $(\mathbf{v_{1}^{(1)}}=\mathbf{l_{1}},\mathbf{v_{2}^{(1)}},...,\mathbf{v_{n}^{(1)}})$ such that for any $2\leq i\leq n$ and any $1\leq j\leq n$ ,

\displaystyle\tilde{B}_{1}(U_{-})\mathbf{v_{i}^{(1)}}\cdot\mathbf{v_{j}^{(1)}}=0\,,\;B_{1}(U_{-})\mathbf{v_{i}^{(1)}}\cdot{\mathbf{v_{i}^{(1)}}}=1\,,

where $\tilde{B}_{1}=B_{1}+B_{1}^{T}$ . This is always possible thanks to the Gram-Schmidt process. We project the perturbation in entropic variables onto this basis:

(4.3)

\eta^{\prime}(U)(t,x)-\eta^{\prime}(\tilde{U})(t,x)=\mu_{1}(t,x)\mathbf{l_{1}}+\sum_{i=2}^{n}\mu_{i}(t,x)\mathbf{v_{i}^{(1)}}\,.

When working with the hyperbolic terms, we will work with the conserved quantity $U-\tilde{U}$ . By Taylor expansion and (3.2), we have

(4.4)

\displaystyle\big{|}(U-\tilde{U})-\bigl{(}\mu_{1}\mathbf{r_{1}}+\sum_{i=2}^{n}\mu_{i}\,\eta^{\prime\prime}(U_{-})^{-1}\mathbf{v_{i}^{(1)}}\bigl{)}\big{|}\leq C(\epsilon_{2}+\delta_{0})|U-\tilde{U}|\,.

As $(\mathbf{r_{1}},\eta^{\prime\prime}(U_{-})^{-1}\mathbf{v_{2}^{(1)}},...,\eta^{\prime\prime}(U_{-})^{-1}\mathbf{v_{n}^{(1)}})$ is a basis, we have

(4.5)

\displaystyle c\sum_{i=2}^{n}|\mu_{i}|^{2}\leq\bigl{|}\sum_{i=2}^{n}\mu_{i}\,\eta^{\prime\prime}(U_{-})^{-1}\mathbf{v_{i}^{(1)}}\bigl{|}^{2}\leq C\sum_{i=2}^{n}|\mu_{i}|^{2}\,,

for some constants $c,C>0$ that depend only on $B_{j},f,\eta,U_{-}$ .

Let $P$ be the projection onto $\textup{span}\{\mathbf{r_{2}},...,\mathbf{r_{n}}\}$ , i.e.,

(4.6)

\displaystyle P(v)=v-(v\cdot\mathbf{r_{1}})\mathbf{r_{1}}\textup{ for any }v\in\mathbb{R}^{n}\,.

For any $2\leq i\leq n$ ,

\displaystyle\eta^{\prime\prime}(U_{-})^{-1}\mathbf{v_{i}^{(1)}}\in\bigl{\{}v:\tilde{B}_{1}(U_{-})\eta^{\prime\prime}(U_{-})\mathbf{r_{1}}\cdot\eta^{\prime\prime}(U_{-})v=0\bigl{\}}=:V\,.

As $\mathbf{r_{1}}\notin V$ , there exists $C>0$ such that

(4.7)

\displaystyle|P(v)|^{2}\geq C|v|^{2}\,\textup{ for any }v\in V\,.

When working with the layer of $W_{n}^{\mathbf{X}_{n}}$ , we complete $\mathbf{l_{n}}$ into an orthogonal basis of $\mathbb{R}^{n}$ with respect to the dissipation. In particular, we choose a basis $(\mathbf{v_{1}^{(n)}},...,\mathbf{v_{n-1}^{(n)}},\mathbf{v_{n}^{(n)}}=\mathbf{l_{n}})$ such that for any $1\leq i\leq n-1$ and any $1\leq j\leq n$ ,

\displaystyle\tilde{B}_{1}(U_{-})\mathbf{v_{i}^{(n)}}\cdot\mathbf{v_{j}^{(n)}}=0\,,\;B_{1}(U_{-})\mathbf{v_{i}^{(n)}}\cdot{\mathbf{v_{i}^{(n)}}}=1\,.

We project the perturbation in entropic variables onto this basis:

(4.8)

\eta^{\prime}(U)(t,x)-\eta^{\prime}(\tilde{U})(t,x)=\nu_{n}(t,x){\mathbf{l_{n}}}+\sum_{i=1}^{n-1}\nu_{i}(t,x)\mathbf{v_{i}^{(n)}}\,.

Such projection has similar properties as (4.3).

We choose such bases to make the best use of dissipation in the special directions of the planar shock waves. In the case that $W_{n}$ is a planar rarefaction wave, we could work with the natural hyperbolic basis $(\mathbf{r_{1}},...,\mathbf{r_{n}})$ since we do not need the Poincaré inequality. However, for the sake of consistency, we will use the basis $(\mathbf{v_{1}^{(n)}},...,\mathbf{v_{n-1}^{(n)}},\mathbf{l_{n}})$ in the case $W_{n}=R_{n}$ too.

Relative functions. The relative flux $(f,g_{2},...,g_{n})$ is defined by

(4.9)			$\displaystyle f(U\|V)=f(U)-f(V)-f^{\prime}(V)(U-V)\,,$
(4.9)			$\displaystyle g_{j}(U\|V)=g_{j}(U)-g_{j}(V)-g_{j}^{\prime}(V)(U-V)\,,\textup{ for }2\leq j\leq d\,.$

The flux of the relative entropy $q=(q_{1},...,q_{d})$ is defined by

(4.10)

q_{j}(U;V)=\begin{cases}q_{1}(U)-q_{1}(V)-\eta^{\prime}(V)\bigl{(}f(U)-f(V)\bigl{)}\,,\textup{ if }j=1\,,\\ q_{j}(U)-q_{j}(V)-\eta^{\prime}(V)\bigl{(}g_{j}(U)-g_{j}(V)\bigl{)}\,,\textup{ if }2\leq j\leq d\,,\end{cases}

where $q_{j}(\cdot)$ , the entropy flux of $\eta$ , is defined in (1.2).

We want to study the evolution of the weighted relative entropy

\int a(t,x)\,\eta\bigl{(}U(t,x)|\tilde{U}(t,x)\bigl{)}\,dx\,.

It involves controlling layer quantities of the form:

\displaystyle\int|\partial_{x_{1}}S_{1}^{\mathbf{X}_{1}}|F_{1}\,dx\,,\;\int|\partial_{x_{1}}W_{n}^{\mathbf{X}_{n}}|F_{n}\,dx\,.

In each of these layers, a “scalarization” effect takes place. Such effect damps the perturbation $U-\tilde{U}$ in the transverse direction well-adapted to both the special direction of the wave $\mathbf{r_{1}}$ (respectively $\mathbf{r_{n}}$ ) and the diffusion eigen-direction.

The main lemma of this section is the following. We postpone our choice of $\Lambda_{S_{1}},\Lambda_{W_{n}},\tilde{C}_{1},\tilde{C}_{n}$ to subsection 4.5, where we unify the choice of constants for both hyperbolic “scalarization” and Poincaré inequality.

Lemma 4.3.

For any $U_{-}\in\mathbb{R}^{n}$ , any $0<\delta<1$ , and any $\Lambda_{S_{1}},\Lambda_{W_{n}},\tilde{C}_{1},\tilde{C}_{n}>0$ , there exist $\delta_{0},\epsilon_{2},C>0$ such that the following is true.

Assume Assumption 3.2. Then for any $t\in[0,T]$ ,

\frac{d}{dt}\int a(t,x)\,\eta\bigl{(}U(t,x)|\tilde{U}(t,x)\bigl{)}\,dx\leq\mathcal{Z}(U)-\mathcal{D}(U)+\mathcal{H}(U)+\mathcal{E}(U)\,,

where

	$\displaystyle\mathcal{Z}(U)$	$\displaystyle=\begin{cases}\dot{\mathbf{X}}_{1}\mathcal{Y}_{1}(U)+\dot{\mathbf{X}}_{n}\mathcal{Y}_{n}(U)\,,\textup{ if }W_{n}=S_{n}\,,\\ \dot{\mathbf{X}}_{1}\mathcal{Y}_{1}(U)\,,\textup{ if }W_{n}=R_{n}\,,\end{cases}$
	$\displaystyle\mathcal{D}(U)$	$\displaystyle=(1+2\gamma)\mathcal{D}_{x_{1}}^{p}(U)+C\mathcal{D}_{x_{1}}^{r}(U)+C\mathcal{D}_{y}(U)\,,$
	$\displaystyle\mathcal{H}(U)$	$\displaystyle=\bigl{(}-\mathbf{C}_{1}\textup{min}\{\Lambda_{S_{1}},\Lambda_{W_{n}}\}+\frac{\mathbf{C}_{2}}{\delta}\bigl{)}\mathcal{H}_{C}$
		$\displaystyle\quad+(1+\delta)\mathcal{H}_{S_{1}}\begin{cases}+(1+\delta)\mathcal{H}_{S_{n}}\,,\textup{ if }W_{n}=S_{n}\,,\\ -(1-\delta)\mathcal{H}_{R_{n}}\,,\textup{ if }W_{n}=R_{n}\,,\end{cases}$
	$\displaystyle\mathcal{E}(U)$	$\displaystyle=\begin{cases}\frac{C}{\delta}(\delta_{S_{n}}^{2}\delta_{S_{1}}e^{-C\delta_{S_{1}}t}+\delta_{S_{n}}\delta_{S_{1}}^{2}e^{-C\delta_{S_{n}}t})\,,\textup{ if }W_{n}=S_{n}\,,\\ C\epsilon_{2}\delta_{R_{n}}\delta_{S_{1}}e^{-C\delta_{S_{1}}t}+C\epsilon_{2}\\|\partial_{x_{1}}R_{n}\\|_{L^{4}}^{2}+\frac{C}{\gamma^{1/3}}\epsilon_{2}^{2/3}\\|\partial_{x_{1}}^{2}R_{n}\\|_{L^{1}}^{4/3}\,,\textup{ if }W_{n}=R_{n}\,,\end{cases}$

where $\gamma,\mathbf{C}_{1},\mathbf{C}_{2}>0$ are constants that depend only on $B_{j},f,\eta,U_{-}$ , and

	$\displaystyle\mathcal{Y}_{i}(U)$	$\displaystyle=\int\partial_{x_{1}}a_{S_{i}}^{\mathbf{X}_{i}}\,\eta({U\|\tilde{U}})\,dx-\int a\eta^{\prime\prime}(\tilde{U})(U-\tilde{U})\partial_{x_{1}}S_{i}^{\mathbf{X}_{i}}\,dx\,,$
	$\displaystyle\mathcal{D}_{x_{1}}^{p}(U)$	$\displaystyle={B_{1}(U_{-})\mathbf{l_{1}}\cdot\mathbf{l_{1}}}\int\|\partial_{x_{1}}{\mu_{1}}\|^{2}\,dx+{B_{1}(U_{-}){\mathbf{l_{n}}}\cdot{\mathbf{l_{n}}}}\int\|\partial_{x_{1}}\nu_{n}\|^{2}\,dx\,,$
	$\displaystyle\mathcal{D}_{x_{1}}^{r}(U)$	$\displaystyle=\sum_{i=2}^{n}\int\|\partial_{x_{1}}\mu_{i}\|^{2}\,dx+\sum_{i=1}^{n-1}\int\|\partial_{x_{1}}\nu_{i}\|^{2}\,dx\,,$
	$\displaystyle\mathcal{D}_{y}(U)$	$\displaystyle=\sum_{j=2}^{d}\int\big{\|}\partial_{x_{j}}(U-\tilde{U})\big{\|}^{2}\,dx\,,$
	$\displaystyle\mathcal{H}_{C}(U)$	$\displaystyle=\sum_{i=2}^{n}\delta_{S_{1}}\int\mu_{i}^{2}\;\partial_{x_{1}}{k_{S_{1}}^{\mathbf{X}_{1}}}dx+\sum_{i=1}^{n-1}\delta_{W_{n}}\int\nu_{i}^{2}\;\partial_{x_{1}}{k_{W_{n}}^{\mathbf{X}_{n}}}\,dx\,,$
	$\displaystyle\mathcal{H}_{S_{1}}(U)$	$\displaystyle=-c_{f}^{(1)}\delta_{S_{1}}\int\mu_{1}^{2}\;\partial_{x_{1}}{k_{S_{1}}^{\mathbf{X}_{1}}}\,dx\,,$
	$\displaystyle\mathcal{H}_{W_{n}}(U)$	$\displaystyle=-c_{f}^{(n)}\delta_{W_{n}}\int\nu_{n}^{2}\;\partial_{x_{1}}{k_{W_{n}}^{\mathbf{X}_{n}}}\,dx\,.$

Remark. Recall (1.16). We have

\displaystyle\mathcal{D}_{x_{1}}^{p}\,,\;\mathcal{D}_{x_{1}}^{r}\,,\;\mathcal{D}_{y}\,,\;\mathcal{H}_{C}\,,\;\mathcal{H}_{S_{1}}\,,\;\mathcal{H}_{W_{n}}\geq 0\,.

We have four families of terms: the shift family $\mathcal{Z}$ , the viscous family $\mathcal{D}$ , the hyperbolic family $\mathcal{H}$ , and the interaction family $\mathcal{E}$ .

The shift family $\mathcal{Z}$ corresponds to the new terms induced by the shifts. The viscous family $\mathcal{D}$ comes from the dissipation. The viscous term $\mathcal{D}_{y}$ corresponds to the transverse direction, and the viscous terms $\mathcal{D}_{x_{1}}^{p},\mathcal{D}_{x_{1}}^{r}$ correspond to the $x_{1}$ direction. If we examine $\mathcal{D}_{x_{1}}^{p},\mathcal{D}_{x_{1}}^{r}$ in the phase space, $\mathcal{D}_{x_{1}}^{p}$ corresponds to the special direction $\mathbf{r_{1}}$ (respectively $\mathbf{r_{n}}$ ) of the wave $S_{1}$ (respectively $W_{n}$ ), and $D_{x_{1}}^{r}$ corresponds to the other orthogonal directions. In Lemma 4.4 in the next section, we see how $D_{x_{1}}^{p}$ helps us get the Poincaré type inequality.

The flux functions give the hyperbolic family $\mathcal{H}$ . The hyperbolic term $\mathcal{H}_{S_{1}}$ (respectively $\mathcal{H}_{W_{n}}$ ) corresponds to the special direction $\mathbf{r_{1}}$ (respectively $\mathbf{r_{n}}$ ) of the wave $S_{1}$ (respectively $W_{n}$ ). The hyperbolic term $\mathcal{H}_{C}$ corresponds to the other orthogonal directions. The hyperbolic “scalarization” happens in all directions except the special direction $\mathbf{r_{1}}$ (respectively $\mathbf{r_{n}}$ ) of the wave $S_{1}$ (respectively $W_{n}$ ). More precisely, the coefficient of $\mathcal{H}_{C}$ becomes negative if we choose sufficiently large $\Lambda_{S_{1}},\Lambda_{W_{n}}$ , while the coefficients of $\mathcal{H}_{S_{1}},\mathcal{H}_{W_{n}}$ are independent of the choice of $\Lambda_{S_{1}},\Lambda_{W_{n}}$ . In short, the hyperbolic “scalarization” reduces the problem to the scalar case with hyperbolic remainders in the form of $\mathcal{H}_{S_{1}},\mathcal{H}_{W_{n}}$ .

Such hyperbolic remainder is negative if it corresponds to a planar rarefaction wave, but it is positive if it corresponds to a planar viscous shock wave. In Lemma 4.4 in the next section, we show in detail how the dissipation $D_{x_{1}}^{p}$ , the shift family $\mathcal{Z}$ , and the hyperbolic terms corresponding to the other orthogonal directions $\mathcal{H}_{C}$ control the positive hyperbolic remainder by the Poincaré type inequality. While the dissipation $D_{x_{1}}^{p}$ dominates the $L^{2}$ norm of the derivative, the shifts will be chosen in a way that the shift family $\mathcal{Z}$ controls the average with the help of the hyperbolic terms corresponding to the other orthogonal directions $\mathcal{H}_{C}$ .

Finally, we can bound interaction terms corresponding to $E_{1},E_{2}$ in (2.16) in the form of $\mathcal{E}$ by Lemma 4.1.

Proof.

By the definition of relative entropy function (1.14), (1.1), and (2.16), we have

	$\displaystyle\frac{d}{dt}\int a(t,x)\,\eta\bigl{(}U(t,x)\|\tilde{U}(t,x)\bigl{)}\,dx$
	$\displaystyle=\int\partial_{t}a\,\eta({U\|\tilde{U}})\,dx+\int a\,\Bigr{[}\bigl{(}\eta^{\prime}(U)-\eta^{\prime}(\tilde{U})\bigl{)}\partial_{t}U-\eta^{\prime\prime}(\tilde{U})(U-\tilde{U})\partial_{t}\tilde{U}\Bigr{]}dx$
	$\displaystyle=\int\partial_{t}a\,\eta({U\|\tilde{U}})\,dx+\int a\,\biggr{[}\bigl{(}\eta^{\prime}(U)-\eta^{\prime}(\tilde{U})\bigl{)}\Bigl{(}\sum_{j=1}^{d}\partial_{x_{j}}\bigl{(}B_{j}(U)\partial_{x_{j}}\eta^{\prime}(U)\bigl{)}$
	$\displaystyle\quad-\partial_{x}f(U)-\sum_{j=2}^{d}g_{j}(U)\Bigl{)}-\eta^{\prime\prime}(\tilde{U})(U-\tilde{U})\Bigl{(}\partial_{x_{1}}\bigl{(}B_{1}(\tilde{U})\partial_{x_{1}}\eta^{\prime}(\tilde{U})\bigl{)}$
	$\displaystyle\quad-\partial_{x}f(\tilde{U})+Z+E_{1}+E_{2}\Bigl{)}\biggr{]}\,dx\,.$

According to definitions of (4.9) and (4.10), we have

\frac{d}{dt}\int a(t,x)\,\eta\bigl{(}U(t,x)|\tilde{U}(t,x)\bigl{)}\,dx=\mathcal{Z}_{1}+\sum_{i=1}^{4}\mathcal{D}_{i}+\sum_{i=1}^{2}\mathcal{E}_{i}+\sum_{i=1}^{4}\mathcal{H}_{i}\,,

where the shift term is

\displaystyle\mathcal{Z}_{1}

\displaystyle=\int\partial_{t}a\,\eta({U|\tilde{U}})\,dx-\begin{cases}\sum_{i=1,n}\dot{\mathbf{X}}_{i}\int a\,\eta^{\prime\prime}(\tilde{U})(U-\tilde{U})\partial_{x_{1}}S_{i}^{\mathbf{X}_{i}}\,dx\,,\textup{ if }W_{n}=S_{n}\,,\\ \dot{\mathbf{X}}_{1}\int a\,\eta^{\prime\prime}(\tilde{U})(U-\tilde{U})\partial_{x_{1}}S_{1}^{\mathbf{X}_{1}}\,dx\,,\textup{ if }W_{n}=R_{n}\,,\end{cases}

the dissipation gives

	$\displaystyle\mathcal{D}_{1}$	$\displaystyle=\int a\,\bigl{(}\eta^{\prime}(U)-\eta^{\prime}(\tilde{U})\bigl{)}\partial_{x_{1}}\Bigl{(}B_{1}(U)\partial_{x_{1}}\bigl{(}\eta^{\prime}(U)-\eta^{\prime}(\tilde{U})\bigl{)}\Bigl{)}\,dx\,,$
	$\displaystyle\mathcal{D}_{2}$	$\displaystyle=\int a\,\bigl{(}\eta^{\prime}(U)-\eta^{\prime}(\tilde{U})\bigl{)}\partial_{x_{1}}\Bigl{(}\bigl{(}B_{1}(U)-B_{1}(\tilde{U})\bigl{)}\partial_{x_{1}}\eta^{\prime}(\tilde{U})\Bigl{)}\,dx\,,$
	$\displaystyle\mathcal{D}_{3}$	$\displaystyle=\int a\,\eta^{\prime}(U\|\tilde{U})\partial_{x_{1}}\bigl{(}B_{1}(\tilde{U})\partial_{x_{1}}\eta^{\prime}(\tilde{U})\bigl{)}\,dx\,,$
	$\displaystyle\mathcal{D}_{4}$	$\displaystyle=\sum_{j=2}^{d}\int a\,\bigl{(}\eta^{\prime}(U)-\eta^{\prime}(\tilde{U})\bigl{)}\partial_{x_{j}}\bigl{(}B_{j}(U)\partial_{x_{j}}\eta^{\prime}(U)\bigl{)}\,dx\,,$

the wave interaction causes

	$\displaystyle\mathcal{E}_{1}$	$\displaystyle=-\int a\,\eta^{\prime\prime}(\tilde{U})(U-\tilde{U})E_{1}\,dx\,,$
	$\displaystyle\mathcal{E}_{2}$	$\displaystyle=-\int a\,\eta^{\prime\prime}(\tilde{U})(U-\tilde{U})E_{2}\,dx\,,$

and the flux induces

	$\displaystyle\mathcal{H}_{1}$	$\displaystyle=\int\partial_{x_{1}}a\,q_{1}(U;\tilde{U})\,dx\,,$
	$\displaystyle\mathcal{H}_{2}$	$\displaystyle=-\int a\,\partial_{x_{1}}\eta^{\prime}(\tilde{U})f(U\|\tilde{U})\,dx\,,$
	$\displaystyle\mathcal{H}_{3}$	$\displaystyle=\sum_{j=2}^{d}\int\partial_{x_{j}}a\,q_{j}(U;\tilde{U})\,dx\,,$
	$\displaystyle\mathcal{H}_{4}$	$\displaystyle=-\sum_{j=2}^{d}\int a\,\partial_{x_{j}}\eta^{\prime}(\tilde{U})g_{j}(U\|\tilde{U})\,dx\,.$

We are going to analyze those terms and get $\mathcal{Z},\mathcal{D},\mathcal{H},\mathcal{E}$ defined in the lemma.

Hyperbolic parts and dissipation in $y$ . As $S_{1}^{\mathbf{X}_{1}}$ and $W_{n}^{\mathbf{X}_{n}}$ are planar waves, $a$ and $\tilde{U}$ do not depend on $y$ . Also, we know $B_{j}$ are positive definite and $\eta^{\prime\prime}$ is strictly convex. Therefore, we have

	$\displaystyle\mathcal{H}_{3}$	$\displaystyle=\mathcal{H}_{4}=0\,,$
	$\displaystyle\mathcal{D}_{4}$	$\displaystyle=-\sum_{j=2}^{d}\int a\,\partial_{x_{j}}\bigl{(}\eta^{\prime}(U)-\eta^{\prime}(\tilde{U})\bigl{)}B_{j}(U)\partial_{x_{j}}\bigl{(}\eta^{\prime}({U})-\eta^{\prime}(\tilde{U})\bigl{)}\,dx$
		$\displaystyle\leq-CD_{y}\,.$

Dissipation in $x_{1}$ . By integration by parts, we have

	$\displaystyle\mathcal{D}_{1}=$	$\displaystyle-\int a\,\partial_{x_{1}}\bigl{(}\eta^{\prime}(U)-\eta^{\prime}(\tilde{U})\bigl{)}B_{1}(U)\partial_{x_{1}}\bigl{(}\eta^{\prime}(U)-\eta^{\prime}(\tilde{U})\bigl{)}\,dx$
		$\displaystyle-\int\partial_{x_{1}}a\,\bigl{(}\eta^{\prime}(U)-\eta^{\prime}(\tilde{U})\bigl{)}B_{1}(U)\partial_{x_{1}}\bigl{(}\eta^{\prime}(U)-\eta^{\prime}(\tilde{U})\bigl{)}\,dx$
	$\displaystyle=$	$\displaystyle:\,\mathcal{D}_{11}+\mathcal{D}_{12}\,,$
	$\displaystyle\mathcal{D}_{2}=$	$\displaystyle-\int a\,\partial_{x_{1}}\bigl{(}\eta^{\prime}(U)-\eta^{\prime}(\tilde{U})\bigl{)}\bigl{(}B_{1}(U)-B_{1}(\tilde{U})\bigl{)}\partial_{x_{1}}\eta^{\prime}(\tilde{U})\,dx$
		$\displaystyle-\int\partial_{x_{1}}a\,\bigl{(}\eta^{\prime}(U)-\eta^{\prime}(\tilde{U})\bigl{)}\bigl{(}B_{1}(U)-B_{1}(\tilde{U})\bigl{)}\partial_{x_{1}}\eta^{\prime}(\tilde{U})\,dx$
	$\displaystyle=$	$\displaystyle:\,\mathcal{D}_{21}+\mathcal{D}_{22}\,.$

We apply projections (4.3) and (4.8) to $\mathcal{D}_{11}$ . By (3.2) and the fact that $B_{1}$ is positive definite, we take $\epsilon_{2},\delta_{0}$ small enough and get

	$\displaystyle\mathcal{D}_{11}$	$\displaystyle\leq-\bigl{(}1-C(\epsilon_{2}+\delta_{0})-C(\Lambda_{S_{1}}+\Lambda_{W_{n}})\delta_{0}\bigl{)}$
		$\displaystyle\quad\Bigl{(}\int\bigl{(}\mathbf{l_{1}}\partial_{x_{1}}\mu_{1}+\sum_{i=2}^{n}\mathbf{v_{i}^{(1)}}\partial_{x_{1}}\mu_{i}\bigl{)}B_{1}(U_{-})\bigl{(}\mathbf{l_{1}}\partial_{x_{1}}\mu_{1}+\sum_{i=2}^{n}\mathbf{v_{i}^{(1)}}\partial_{x_{1}}\mu_{i}\bigl{)}\,dx$
		$\displaystyle\quad+\int\bigl{(}\mathbf{l_{n}}\partial_{x_{1}}\nu_{n}+\sum_{i=1}^{n-1}\mathbf{v_{i}^{(n)}}\partial_{x_{1}}\nu_{i}\bigl{)}B_{1}(U_{-})\bigl{(}\mathbf{l_{n}}\partial_{x_{1}}\nu_{n}+\sum_{i=1}^{n-1}\mathbf{v_{i}^{(n)}}\partial_{x_{1}}\nu_{i}\bigl{)}\,dx\Bigl{)}$
		$\displaystyle\leq-\bigl{(}1-C(\epsilon_{2}+\delta_{0}^{1/2})\bigl{)}(\mathcal{D}_{x_{1}}^{p}+\mathcal{D}_{x_{1}}^{r})\,.$

Compared with $\mathcal{D}_{11}$ , $\partial_{x_{1}}a$ and $\partial_{x_{1}}\eta^{\prime}(\tilde{U})$ give extra smallness to $\mathcal{D}_{12},\mathcal{D}_{21},\mathcal{D}_{22}$ .By Young’s inequality, (2.5), and (2.10), we take $\epsilon_{2},\delta_{0}$ small enough and get

	$\displaystyle\mathcal{D}_{1}+\mathcal{D}_{2}$	$\displaystyle\leq-\bigl{(}1-C(\epsilon_{2}+\delta_{0}^{1/4})\bigl{)}(\mathcal{D}_{x_{1}}^{p}+\mathcal{D}_{x_{1}}^{r})$
		$\displaystyle\quad+C\delta_{0}^{1/2}\int\bigl{(}\|\partial_{x_{1}}S_{1}^{\mathbf{X}_{1}}\|+\|\partial_{x_{1}}W_{n}^{\mathbf{X}_{n}}\|\bigl{)}\|U-\tilde{U}\|^{2}\,dx\,.$

We have

\displaystyle\eta^{\prime}(U)-\eta^{\prime}(\tilde{U})=\mu_{1}\mathbf{l_{1}}+\sum_{i=2}^{n}\mu_{i}\mathbf{v_{i}^{(1)}}=\nu_{n}\mathbf{l_{n}}+\sum_{i=1}^{n-1}\nu_{i}\mathbf{v_{i}^{(n)}}\,,

which implies

	$\displaystyle\mu_{1}\mathbf{l_{1}}\cdot\mathbf{r_{1}}+\sum_{i=2}^{n}\mu_{i}\mathbf{v_{i}^{(1)}}\cdot\mathbf{r_{1}}=\sum_{i=1}^{n-1}\nu_{i}\mathbf{v_{i}^{(n)}}\cdot\mathbf{r_{1}}\,,$
	$\displaystyle\sum_{i=2}^{n}\mu_{i}\mathbf{v_{i}^{(1)}}\cdot\mathbf{r_{n}}=\nu_{n}\mathbf{l_{n}}\cdot\mathbf{r_{n}}+\sum_{i=1}^{n-1}\nu_{i}\mathbf{v_{i}^{(n)}}\cdot\mathbf{r_{n}}\,.$

As $\mathbf{l_{1}}\cdot\mathbf{r_{1}},\mathbf{l_{n}}\cdot\mathbf{r_{n}}\neq 0$ , we get

\displaystyle\sum_{i=2}^{n}|\partial_{x_{1}}\mu_{i}|^{2}+\sum_{i=1}^{n-1}|\partial_{x_{1}}\nu_{i}|^{2}\geq C\bigl{(}|\partial_{x_{1}}\mu_{1}|^{2}+|\partial_{x_{1}}\nu_{n}|^{2}\bigl{)}\,,

for some $C>0$ . Thus there exists some constant $\gamma>0$ that depends only on $B_{j},f,\eta,U_{-}$ such that

\displaystyle\frac{1}{2}\mathcal{D}_{x_{1}}^{r}\geq 4\gamma\mathcal{D}_{x_{1}}^{p}\,.

Taking $\epsilon_{2},\delta_{0}$ small enough, we get

	$\displaystyle\mathcal{D}_{1}+\mathcal{D}_{2}$	$\displaystyle\leq-(1+3\gamma)\mathcal{D}_{x_{1}}^{p}-\frac{1}{4}\mathcal{D}_{x_{1}}^{r}$
		$\displaystyle\quad+C\delta_{0}^{1/2}\int\bigl{(}\|\partial_{x_{1}}S_{1}^{\mathbf{X}_{1}}\|+\|\partial_{x_{1}}W_{n}^{\mathbf{X}_{n}}\|\bigl{)}\|U-\tilde{U}\|^{2}\,dx\,.$

By the chain rule, Lemma 2.1, Lemma 2.2, and (3.2), we have

\displaystyle|\mathcal{D}_{3}|\leq C\delta_{0}\int\bigl{(}|\partial_{x_{1}}S_{1}^{\mathbf{X}_{1}}|+|\partial_{x_{1}}W_{n}^{\mathbf{X}_{n}}|\bigl{)}|U-\tilde{U}|^{2}\,dx+C\epsilon_{2}\int|U-\tilde{U}||\partial_{x_{1}}^{2}R_{n}|\,dx\,.

Hyperbolic parts in $x_{1}$ . By the definition of the flux of the relative entropy (4.10), (3.2), (2.5), and (2.10), we take $\epsilon_{2},\delta_{0}$ small enough and get

	$\displaystyle\mathcal{H}_{1}$	$\displaystyle=\int\partial_{x_{1}}a\,q_{1}(U;\tilde{U})\,dx$
		$\displaystyle=\frac{1}{2}(I_{1}+I_{2})+C(\epsilon_{2}^{1/2}+\delta_{0}^{1/2})\int\bigl{(}\|\partial_{x_{1}}S_{1}^{\mathbf{X}_{1}}\|+\|\partial_{x_{1}}W_{n}^{\mathbf{X}_{n}}\|\bigl{)}\|U-\tilde{U}\|^{2}\,dx\,,$

where

	$\displaystyle I_{1}=\int\partial_{x_{1}}a_{S_{1}}^{\mathbf{X}_{1}}(U-\tilde{U})\eta^{\prime\prime}(U_{-})f^{\prime}(U_{-})(U-\tilde{U})\,dx\,,$
	$\displaystyle I_{2}=\int\partial_{x_{1}}a_{W_{n}}^{\mathbf{X}_{n}}(U-\tilde{U})\eta^{\prime\prime}(U_{-})f^{\prime}(U_{-})(U-\tilde{U})\,dx\,.$

We apply (4.4) to $I_{1}$ . Recall (4.6). Let

\displaystyle v=\sum_{i=2}^{n}\mu_{i}\eta^{\prime\prime}(U_{-})^{-1}\mathbf{v_{i}^{(1)}}\in V\,.

By (1.18), (4.4), (4.5), and (4.7), we take $\epsilon_{2},\delta_{0}$ small enough and get

	$\displaystyle I_{1}$	$\displaystyle=\int\partial_{x_{1}}a_{S_{1}}^{\mathbf{X}_{1}}(U-\tilde{U})\eta^{\prime\prime}(U_{-})\bigl{(}f^{\prime}(U_{-})-\lambda_{1}I\bigl{)}(U-\tilde{U})\,dx+2\mathcal{Z}_{2}$
		$\displaystyle\leq-{\Lambda_{S_{1}}}(\lambda_{2}-\lambda_{1})\delta_{S_{1}}\int\eta^{\prime\prime}(U_{-})P(v)\cdot P(v)\;\partial_{x_{1}}{k_{S_{1}}^{\mathbf{X}_{1}}}\,dx$
		$\displaystyle\quad+C(\epsilon_{2}^{1/2}+\delta_{0}^{1/2})\int\|\partial_{x_{1}}S_{1}^{\mathbf{X}_{1}}\|\|U-\tilde{U}\|^{2}\,dx+2\mathcal{Z}_{2}$
		$\displaystyle\leq-C{\Lambda_{S_{1}}}(\lambda_{2}-\lambda_{1})\sum_{i=2}^{n}\delta_{S_{1}}\int\mu_{i}^{2}\;\partial_{x_{1}}{k_{S_{1}}^{\mathbf{X}_{1}}}\,dx$
		$\displaystyle\quad+C(\epsilon_{2}^{1/2}+\delta_{0}^{1/2})\int\|\partial_{x_{1}}S_{1}^{\mathbf{X}_{1}}\|\|U-\tilde{U}\|^{2}\,dx+2\mathcal{Z}_{2}\,,$

where

\displaystyle\mathcal{Z}_{2}=\frac{\lambda_{1}}{2}\int\partial_{x_{1}}a_{S_{1}}^{\mathbf{X}_{1}}(U-\tilde{U})\eta^{\prime\prime}(U_{-})(U-\tilde{U})\,dx\,.

Similarly, we get

	$\displaystyle I_{2}$	$\displaystyle=\int\partial_{x_{1}}a_{W_{n}}^{\mathbf{X}_{n}}(U-\tilde{U})\eta^{\prime\prime}(U_{-})\bigl{(}f^{\prime}(U_{-})-\lambda_{n}I\bigl{)}(U-\tilde{U})\,dx+2\mathcal{Z}_{3}$
		$\displaystyle\leq C\Lambda_{W_{n}}(\lambda_{n-1}-\lambda_{n})\sum_{i=1}^{n-1}\delta_{W_{n}}\int\nu_{i}^{2}\;\partial_{x_{1}}{k_{W_{n}}^{\mathbf{X}_{n}}}\,dx$
		$\displaystyle\quad+C(\epsilon_{2}^{1/2}+\delta_{0}^{1/2})\int\|\partial_{x_{1}}W_{n}^{\mathbf{X}_{n}}\|\|U-\tilde{U}\|^{2}\,dx+2\mathcal{Z}_{3}\,,$

where

\displaystyle\mathcal{Z}_{3}=\frac{\lambda_{n}}{2}\int\partial_{x_{1}}a_{W_{n}}^{\mathbf{X}_{n}}(U-\tilde{U})\eta^{\prime\prime}(U_{-})(U-\tilde{U})\,dx\,.

As $\lambda_{2}-\lambda_{1},\lambda_{n}-\lambda_{n-1}>0$ , we get

	$\displaystyle\mathcal{H}_{1}$	$\displaystyle\leq-C\textup{min}\{\Lambda_{S_{1}},\Lambda_{W_{n}}\}\mathcal{H}_{C}+\mathcal{Z}_{2}+\mathcal{Z}_{3}$
		$\displaystyle\quad+C(\epsilon_{2}^{1/2}+\delta_{0}^{1/2})\int\bigl{(}\|\partial_{x_{1}}S_{1}^{\mathbf{X}_{1}}\|+\|\partial_{x_{1}}W_{n}^{\mathbf{X}_{n}}\|\bigl{)}\|U-\tilde{U}\|^{2}\,dx\,.$

By the definition of the relative flux (4.9) and (3.2), we take $\epsilon_{2},\delta_{0}$ small enough and get

	$\displaystyle\mathcal{H}_{2}$	$\displaystyle=-\int a\,\partial_{x_{1}}\eta^{\prime}(\tilde{U})f(U\|\tilde{U})\,dx$
		$\displaystyle\leq I_{3}+I_{4}+C(\epsilon_{2}+\delta_{0}^{1/2})\int\bigl{(}\|\partial_{x_{1}}S_{1}^{\mathbf{X}_{1}}\|+\|\partial_{x_{1}}W_{n}^{\mathbf{X}_{n}}\|\bigl{)}\|U-\tilde{U}\|^{2}\,dx\,,$

where

	$\displaystyle I_{3}$	$\displaystyle=-\delta_{S_{1}}\int\;\partial_{x_{1}}{k_{S_{1}}^{\mathbf{X}_{1}}}\,\mathbf{l_{1}}\cdot f^{\prime\prime}(U_{-}):(U-\tilde{U})\otimes(U-\tilde{U})\,dx\,,$
	$\displaystyle I_{4}$	$\displaystyle=\begin{cases}-\delta_{W_{n}}\int\;\partial_{x_{1}}{k_{S_{n}}^{\mathbf{X}_{n}}}\,\mathbf{l_{n}}\cdot f^{\prime\prime}(U_{-}):(U-\tilde{U})\otimes(U-\tilde{U})\,dx\,,\textup{ if }W_{n}=S_{n}\,,\\ \delta_{W_{n}}\int\;\partial_{x_{1}}{k_{R_{n}}}\,\mathbf{l_{n}}\cdot f^{\prime\prime}(U_{-}):(U-\tilde{U})\otimes(U-\tilde{U})\,dx\,,\textup{ if }W_{n}=R_{n}\,.\end{cases}$

We apply (4.4) to $I_{3}$ . As the hyperbolic “scalarization” will happen in all the directions except the special direction $\mathbf{r_{1}}$ of the wave $S_{1}^{\mathbf{X}_{1}}$ , we use Young’s inequality to make the hyperbolic term in the $\mathbf{r_{1}}$ direction as small as possible

		$\displaystyle I_{3}-C(\epsilon_{2}+\delta_{0})\int\bigl{(}\|\partial_{x_{1}}S_{1}^{\mathbf{X}_{1}}\|+\|\partial_{x_{1}}W_{n}^{\mathbf{X}_{n}}\|\bigl{)}\|U-\tilde{U}\|^{2}\,dx$
		$\displaystyle\leq(1+\frac{\delta}{2})\mathcal{H}_{S_{1}}+\frac{C}{\delta}\sum_{i=2}^{n}\delta_{S_{1}}\int\mu_{i}^{2}\;\partial_{x_{1}}{k_{S_{1}}^{\mathbf{X}_{1}}}\,dx\,.$

Similarly, we get

	$\displaystyle I_{4}-C(\epsilon_{2}+\delta_{0})\int\bigl{(}\|\partial_{x_{1}}S_{1}^{\mathbf{X}_{1}}\|+\|\partial_{x_{1}}W_{n}^{\mathbf{X}_{n}}\|\bigl{)}\|U-\tilde{U}\|^{2}\,dx$
	$\displaystyle\leq\frac{C}{\delta}\sum_{i=1}^{n-1}\delta_{W_{n}}\int\nu_{i}^{2}\;\partial_{x_{1}}{k_{W_{n}}^{\mathbf{X}_{n}}}\,dx\begin{cases}+(1+\frac{\delta}{2})\mathcal{H}_{S_{n}}\,,\textup{ if }W_{n}=S_{n}\,,\\ -(1-\frac{\delta}{2})\mathcal{H}_{R_{n}}\,,\textup{ if }W_{n}=R_{n}\,.\end{cases}$

Then we have

		$\displaystyle\mathcal{H}_{2}-C(\epsilon_{2}+\delta_{0}^{1/2})\int\bigl{(}\|\partial_{x_{1}}S_{1}^{\mathbf{X}_{1}}\|+\|\partial_{x_{1}}W_{n}^{\mathbf{X}_{n}}\|\bigl{)}\|U-\tilde{U}\|^{2}\,dx$
		$\displaystyle\leq\frac{C}{\delta}\mathcal{H}_{C}+(1+\frac{\delta}{2})\mathcal{H}_{S_{1}}\begin{cases}+(1+\frac{\delta}{2})\mathcal{H}_{S_{n}}\,,\textup{ if }W_{n}=S_{n}\,,\\ -(1-\frac{\delta}{2})\mathcal{H}_{R_{n}}\,,\textup{ if }W_{n}=R_{n}\,.\end{cases}$

In all, we get

		$\displaystyle\mathcal{H}_{1}+\mathcal{H}_{2}-C(\epsilon_{2}^{1/2}+\delta_{0}^{1/2})\int\bigl{(}\|\partial_{x_{1}}S_{1}^{\mathbf{X}_{1}}\|+\|\partial_{x_{1}}W_{n}^{\mathbf{X}_{n}}\|\bigl{)}\|U-\tilde{U}\|^{2}\,dx-\mathcal{Z}_{2}-\mathcal{Z}_{3}$
		$\displaystyle\leq\bigl{(}-C\textup{min}\{\Lambda_{S_{1}},\Lambda_{W_{n}}\}+\frac{C}{\delta}\bigl{)}\mathcal{H}_{C}+(1+\frac{\delta}{2})\mathcal{H}_{S_{1}}$
		$\displaystyle\quad\begin{cases}+(1+\frac{\delta}{2})\mathcal{H}_{S_{n}}\,,\textup{ if }W_{n}=S_{n}\,,\\ -(1-\frac{\delta}{2})\mathcal{H}_{R_{n}}\,,\textup{ if }W_{n}=R_{n}\,.\end{cases}$

Shift terms. We have

\displaystyle\partial_{t}a=(\dot{\mathbf{X}}_{1}-{\sigma}_{1})\partial_{x_{1}}a_{S_{1}}^{\mathbf{X}_{1}}+\begin{cases}(\dot{\mathbf{X}}_{n}-\sigma_{n})\partial_{x_{1}}a_{S_{n}}^{\mathbf{X}_{n}}\,,\textup{ if }W_{n}=S_{n}\,,\\ \Lambda_{R_{n}}\partial_{x_{1}}f(R_{n})\cdot\mathbf{l_{n}}\,,\textup{ if }W_{n}=R_{n}\,.\end{cases}

By (2.5) and (2.10), we take $\epsilon_{2},\delta_{0}$ small enough and get

\displaystyle\mathcal{Z}_{1}+\mathcal{Z}_{2}+\mathcal{Z}_{3}

\displaystyle=\mathcal{Z}+\mathcal{Z}_{4}+C(\epsilon_{2}^{1/2}+\delta_{0}^{1/2})\int\bigl{(}|\partial_{x_{1}}S_{1}^{\mathbf{X}_{1}}|+|\partial_{x_{1}}W_{n}^{\mathbf{X}_{n}}|\bigl{)}|U-\tilde{U}|^{2}\,dx\,,

where

\displaystyle\mathcal{Z}_{4}=\begin{cases}0\,,\textup{ if }W_{n}=S_{n}\,,\\ \Lambda_{R_{n}}\int\bigl{(}\partial_{x_{1}}f(R_{n})-\lambda_{n}\partial_{x_{1}}R_{n}\bigl{)}\cdot\mathbf{l_{n}}\,\eta(U|\tilde{U})\,dx\,,\textup{ if }W_{n}=R_{n}\,.\end{cases}\,.

By Taylor expansion and Lemma 2.2, we take $\epsilon_{2},\delta_{0}$ small enough and get

	$\displaystyle\|\mathcal{Z}_{4}\|$	$\displaystyle\leq C\Lambda_{R_{n}}\int\big{\|}\bigl{(}f^{\prime}(U_{-})-{\lambda_{n}}I\bigl{)}\partial_{x_{1}}R_{n}\cdot\mathbf{l_{n}}\big{\|}\|U-\tilde{U}\|^{2}\,dx$
		$\displaystyle\quad+C\Lambda_{R_{n}}\delta_{0}\int\|\partial_{x_{1}}R_{n}\|\|U-\tilde{U}\|^{2}\,dx$
		$\displaystyle\leq C\delta_{0}^{1/2}\int\|\partial_{x_{1}}R_{n}\|\|U-\tilde{U}\|^{2}\,dx\,.$

In all, we get

\displaystyle\mathcal{Z}_{1}+\mathcal{Z}_{2}+\mathcal{Z}_{3}\leq\mathcal{Z}+C(\epsilon_{2}^{1/2}+\delta_{0}^{1/2})\int\bigl{(}|\partial_{x_{1}}S_{1}^{\mathbf{X}_{1}}|+|\partial_{x_{1}}W_{n}^{\mathbf{X}_{n}}|\bigl{)}|U-\tilde{U}|^{2}\,dx\,.

Interaction terms. Note $\mathcal{H}_{C},\mathcal{H}_{S_{1}},\mathcal{H}_{W_{n}}\geq 0$ . We have

(4.11)

\displaystyle\int\bigl{(}|\partial_{x_{1}}S_{1}^{\mathbf{X}_{1}}|+|\partial_{x_{1}}W_{n}^{\mathbf{X}_{n}}|\bigl{)}|U-\tilde{U}|^{2}\leq C(\mathcal{H}_{C}+\mathcal{H}_{S_{1}}+\mathcal{H}_{W_{n}})\,.

If $W_{n}=S_{n}$ , then Lemma 4.2 and Lemma 4.1 give

	$\displaystyle\|\mathcal{E}_{1}\|+\|\mathcal{E}_{2}\|$
	$\displaystyle\leq C\int\|U-\tilde{U}\|\Bigl{(}\|\partial_{x_{1}}S_{1}^{\mathbf{X}_{1}}\|\|S_{n}^{\mathbf{X}_{n}}-U_{m}\|+\|\partial_{x_{1}}S_{n}^{\mathbf{X}_{n}}\|\|S_{1}^{\mathbf{X}_{1}}-U_{m}\|$
	$\displaystyle\quad+\|\partial_{x_{1}}S_{n}^{\mathbf{X}_{n}}\|\|\partial_{x_{1}}S_{1}^{\mathbf{X}_{1}}\|\Bigl{)}\,dx$
	$\displaystyle\leq C\Bigl{(}\int\bigl{(}\|\partial_{x_{1}}S_{1}^{\mathbf{X}_{1}}\|+\|\partial_{x_{1}}S_{n}^{\mathbf{X}_{n}}\|\bigl{)}\|U-\tilde{U}\|^{2}\,dx\Bigl{)}^{1/2}\Bigl{(}\big{\\|}\|\partial_{x_{1}}S_{1}^{\mathbf{X}_{1}}\|\|S_{n}^{\mathbf{X}_{n}}-U_{m}\|^{2}\big{\\|}_{L^{1}}^{1/2}$
	$\displaystyle\quad+\big{\\|}\|\partial_{x_{1}}S_{n}^{\mathbf{X}_{n}}\|\|S_{1}^{\mathbf{X}_{1}}-U_{m}\|^{2}\big{\\|}_{L^{1}}^{1/2}+\big{\\|}\|\partial_{x_{1}}S_{n}^{\mathbf{X}_{n}}\|\|\partial_{x_{1}}S_{1}^{\mathbf{X}_{1}}\|\big{\\|}_{L^{1}}^{1/2}\Bigl{)}$
	$\displaystyle\leq\frac{\delta}{4}(\mathcal{H}_{C}+\mathcal{H}_{S_{1}}+\mathcal{H}_{W_{n}})+\frac{C}{\delta}(\delta_{S_{n}}^{2}\delta_{S_{1}}e^{-C\delta_{S_{1}}t}+\delta_{S_{n}}\delta_{S_{1}}^{2}e^{-C\delta_{S_{n}}t})\,.$

If $W_{n}=R_{n}$ , then (3.1), Lemma 4.2, and Lemma 4.1 give

	$\displaystyle\|\mathcal{E}_{1}\|+\|\mathcal{E}_{2}\|$
	$\displaystyle\leq C\int\|U-\tilde{U}\|\Bigl{(}\|\partial_{x_{1}}^{2}R_{n}\|+\|\partial_{x_{1}}R_{n}\|^{2}+\|\partial_{x_{1}}R_{n}\|\|\partial_{x_{1}}S_{1}^{\mathbf{X}_{1}}\|$
	$\displaystyle\quad+\|\partial_{x_{1}}R_{n}\|\|S_{1}^{\mathbf{X}_{1}}-U_{m}\|+\|\partial_{x_{1}}S_{1}^{\mathbf{X}_{1}}\|\|R_{n}-U_{m}\|\Bigl{)}\,dx$
	$\displaystyle\leq C\int\|U-\tilde{U}\|\|\partial_{x_{1}}^{2}R_{n}\|+C\epsilon_{2}\Bigl{(}\\|\partial_{x_{1}}R_{n}\\|_{L^{4}}^{2}+\big{\\|}\|\partial_{x_{1}}R_{n}\|\|\partial_{x_{1}}S_{1}^{\mathbf{X}_{1}}\|\big{\\|}_{L^{2}}$
	$\displaystyle\quad+\big{\\|}\|R_{n}-U_{m}\|\|\partial_{x_{1}}S_{1}^{\mathbf{X}_{1}}\|\big{\\|}_{L^{2}}+\big{\\|}\|S_{1}^{\mathbf{X}_{1}}-U_{m}\|\|\partial_{x_{1}}R_{n}\|\big{\\|}_{L^{2}}\Bigl{)}$
	$\displaystyle\leq C_{E}\int\|U-\tilde{U}\|\|\partial_{x_{1}}^{2}R_{n}\|+C\epsilon_{2}\\|\partial_{x_{1}}R_{n}\\|_{L^{4}}^{2}+C\epsilon_{2}\delta_{R_{n}}\delta_{S_{1}}e^{-C\delta_{S_{1}}t}\,,$

where $C_{E}>0$ is a constant that depends only on $B_{j},f,\eta,U_{-}$ .

Some estimates. By Hölder’s inequality and the Gagliardo-Nirenberg inequality proved in [24], we have

\int|U-\tilde{U}||\partial_{x_{1}}^{2}R_{n}|\,dx\leq C\big{\|}\partial_{x_{1}}(U-\tilde{U})\big{\|}_{L^{2}}^{1/2}\|U-\tilde{U}\|_{L^{2}}^{1/2}\|\partial_{x_{1}}^{2}R_{n}\|_{L^{1}}\,.

The Young’s inequality and (3.1) give

2C_{E}\int|U-\tilde{U}||\partial_{x_{1}}^{2}R_{n}|\,dx\leq\frac{C}{\gamma^{1/3}}\epsilon_{2}^{2/3}\|\partial_{x_{1}}^{2}R_{n}\|_{L^{1}}^{4/3}+{\gamma}D_{x_{1}}^{p}+\frac{1}{8}D_{x_{1}}^{r}\,.

Finally, we can take $\epsilon_{2},\delta_{0}$ small enough and prove the lemma because of (4.11). ∎

4.4. Poincaré type inequality

Lemma 4.4.

For any $U_{-}\in\mathbb{R}^{n}$ , any $0<\gamma<1$ , and any $\Lambda_{S_{1}},\Lambda_{W_{n}},\tilde{C}_{1},\tilde{C}_{n}>0$ , there exist $\delta_{0},\epsilon_{2},C>0$ such that the following is true.

Assume Assumption 3.2. Let $\tilde{\mu}_{i}$ be $\mu_{1}$ if $i=1$ and be $\nu_{n}$ if $i=n$ . Then for any $i\in\{1,n\}$ and any $t\in[0,T]$ ,

	$\displaystyle\mathcal{H}_{S_{i}}(U)-(1+\gamma)D_{x_{1}}^{p,i}(U)+\dot{\mathbf{X}}_{i}(t)\mathcal{Y}_{i}(U)$
	$\displaystyle\leq-C\gamma\int\|\partial_{x_{1}}{\tilde{\mu}_{i}}\|^{2}\,dx-C\gamma\delta_{S_{i}}\int\tilde{\mu}_{i}^{2}\;\partial_{x_{1}}{k_{S_{i}}^{\mathbf{X}_{i}}}\,dx-\frac{\delta_{S_{i}}}{2\tilde{C}_{i}}\|\dot{\mathbf{X}}_{i}(t)\|^{2}$
	$\displaystyle\quad+\bigl{(}-\frac{\mathbf{C}_{3}}{\delta_{S_{i}}}\tilde{C}_{i}+\frac{\mathbf{C}_{4}}{\delta_{S_{i}}}\bigl{)}\bigl{(}\delta_{S_{i}}\int\tilde{\mu}_{i}\;\partial_{x_{1}}{k_{S_{i}}^{\mathbf{X}_{i}}}\,dx\bigl{)}^{2}+\mathbf{C}_{5}\tilde{C}_{i}\mathcal{H}_{C}(U)+C\delta_{S_{i}}D_{y}(U)\,,$

where $\mathbf{C}_{3},\mathbf{C}_{4},\mathbf{C}_{5}>0$ are constants that depend only on $B_{j},f,\eta,U_{-}$ ,

\displaystyle D_{x_{1}}^{p,i}(U)

\displaystyle={B_{1}(U_{-})\mathbf{l_{i}}\cdot\mathbf{l_{i}}}\int|\partial_{x_{1}}{\tilde{\mu}_{i}}|^{2}\,dx\,,

and $\mathcal{H}_{S_{i}},\mathcal{H}_{C},\mathcal{D}_{y},\mathcal{Y}_{i}$ are defined in Lemma 4.3.

Remark. By taking $\tilde{C}_{i}$ large enough, the positive remainders are $\mathbf{C}_{5}\tilde{C}_{i}\mathcal{H}_{C}$ and $C\delta_{S_{i}}D_{y}(U)$ . We will take $\Lambda_{S_{1}},\Lambda_{W_{n}}$ large enough to control $\mathbf{C}_{5}\tilde{C}_{i}\mathcal{H}_{C}$ and $\delta_{0}$ small enough to control $C\delta_{S_{i}}D_{y}(U)$ in Lemma 4.5 in subsection 4.5.

Proof.

First, we fix the time $t$ and the transverse direction $y$ and compactify the problem by changing variables. Let

	$\displaystyle(\mathcal{H}_{S_{i}})^{t,y}$	$\displaystyle=-c_{f}^{(i)}\delta_{S_{i}}\int\tilde{\mu}_{i}\bigl{(}t,(x_{1},y)\bigl{)}^{2}\;\partial_{x_{1}}{k_{S_{i}}^{\mathbf{X}_{i}}}\,d{x_{1}}\,,$
	$\displaystyle(D_{x_{1}}^{p,i})^{t,y}$	$\displaystyle={B_{1}(U_{-})\mathbf{l_{i}}\cdot\mathbf{l_{i}}}\int\big{\|}\partial_{x_{1}}{\tilde{\mu}_{i}}\bigl{(}t,(x_{1},y)\bigl{)}\big{\|}^{2}\,dx_{1}\,.$

As ${k_{S_{i}}^{\mathbf{X}_{i}}}$ is strictly increasing, we define the change of variable

x_{1}\mapsto z:={k_{S_{i}}^{\mathbf{X}_{i}}}\bigl{(}t,(x_{1},y)\bigl{)}\text{ and }h(z)=\tilde{\mu}_{i}\bigl{(}t,(x_{1},y)\bigl{)}\,.

By (2.4), we can take $\delta_{0}$ small enough and get

	$\displaystyle(\mathcal{H}_{S_{i}})^{t,y}$	$\displaystyle=-c_{f}^{(i)}\delta_{S_{i}}\int_{0}^{1}h(z)^{2}\,dz\,,$
	$\displaystyle-(1+\gamma)(D_{x_{1}}^{p,i})^{t,y}$	$\displaystyle\leq(\frac{1}{2}+\frac{\gamma}{4})\,c_{f}^{(i)}\delta_{S_{i}}\int_{0}^{1}h^{\prime}(z)^{2}z(1-z)\,dz\,.$

Lemma 1.3 gives

	$\displaystyle\int_{0}^{1}h(z)^{2}\,dz$	$\displaystyle=\int_{0}^{1}\bigl{(}h(z)-\int_{0}^{1}h(z)\,dz\bigl{)}^{2}\,dz+\bigl{(}\int_{0}^{1}h(z)\,dz\bigl{)}^{2}$
		$\displaystyle\leq\frac{1}{2}\int_{0}^{1}h^{\prime}(z)^{2}z(1-z)\,dz+\bigl{(}\int_{0}^{1}h(z)\,dz\bigl{)}^{2}\,,$

which implies

	$\displaystyle(\mathcal{H}_{S_{i}})^{t,y}-(1+\gamma)(D_{x_{1}}^{p,i})^{t,y}$
	$\displaystyle\leq-C\gamma\delta_{S_{i}}\int\tilde{\mu}_{i}^{2}\;\partial_{x_{1}}{k_{S_{i}}^{\mathbf{X}_{i}}}\,dx_{1}-C\gamma\int\|\partial_{x_{1}}{\tilde{\mu}_{i}}\|^{2}\,dx_{1}+\frac{C}{\delta_{S_{i}}}\bigl{(}\delta_{S_{i}}\int\tilde{\mu}_{i}\;\partial_{x_{1}}{k_{S_{i}}^{\mathbf{X}_{i}}}\,dx_{1}\bigl{)}^{2}\,,$

where $C>0$ is independent of $t,y$ . Therefore, we have

	$\displaystyle\mathcal{H}_{S_{i}}(U)-(1+\gamma)D_{x_{1}}^{p,i}(U)$
	$\displaystyle\leq-C\gamma\delta_{S_{i}}\int\tilde{\mu}_{i}^{2}\;\partial_{x_{1}}{k_{S_{i}}^{\mathbf{X}_{i}}}\,dx-C\gamma\int\|{\partial_{x_{1}}\tilde{\mu}_{i}}\|^{2}\,dx+\frac{C}{\delta_{S_{i}}}\int\bigl{(}\delta_{S_{i}}\int\tilde{\mu}_{i}\;\partial_{x_{1}}{k_{S_{i}}^{\mathbf{X}_{i}}}\,d{x_{1}}\bigl{)}^{2}dy\,.$

Using the classical Poincaré inequality, we find

	$\displaystyle\int\bigl{(}\delta_{S_{i}}\int\tilde{\mu}_{i}\;\partial_{x_{1}}{k_{S_{i}}^{\mathbf{X}_{i}}}\,dx_{1}\bigl{)}^{2}\,dy$
	$\displaystyle\leq C\bigl{(}\delta_{S_{i}}\int\tilde{\mu}_{i}\;\partial_{x_{1}}{k_{S_{i}}^{\mathbf{X}_{i}}}\,dx\bigl{)}^{2}+C\int\big{\|}\nabla_{y}\bigl{(}\delta_{S_{i}}\int\tilde{\mu}_{i}\;\partial_{x_{1}}{k_{S_{i}}^{\mathbf{X}_{i}}}\,dx_{1}\bigl{)}\big{\|}^{2}\,dy$
	$\displaystyle\leq C\bigl{(}\delta_{S_{i}}\int\tilde{\mu}_{i}\;\partial_{x_{1}}{k_{S_{i}}^{\mathbf{X}_{i}}}\,dx\bigl{)}^{2}+C{\delta_{S_{i}}^{2}}\mathcal{D}_{y}\,.$

By Young’s inequality, Hölder’s inequality, (2.5), and (3.2), we take $\epsilon_{2},\delta_{0}$ small enough and get

	$\displaystyle(\frac{\delta_{S_{i}}}{\tilde{C}_{i}})^{2}\|\dot{\mathbf{X}}_{i}(t)\|^{2}$	$\displaystyle=\bigl{(}\int a\,\eta^{\prime\prime}(\tilde{U})(U-\tilde{U})\partial_{x_{1}}S_{i}^{\mathbf{X}_{i}}\,dx\bigl{)}^{2}$
		$\displaystyle\geq\bigl{(}\int\bigl{(}\eta^{\prime}(U)-\eta^{\prime}(\tilde{U})\bigl{)}\partial_{x_{1}}S_{i}^{\mathbf{X}_{i}}\,dx\bigl{)}^{2}$
		$\displaystyle\quad-C(\epsilon_{2}^{1/2}+\delta_{0}^{1/2})\bigl{(}\int\|U-\tilde{U}\|\|\partial_{x_{1}}S_{i}^{\mathbf{X}_{i}}\|\,dx\bigl{)}^{2}$
		$\displaystyle\geq C\bigl{(}\delta_{S_{i}}\int\tilde{\mu}_{i}\;\partial_{x_{1}}{k_{S_{i}}^{\mathbf{X}_{i}}}\,dx\bigl{)}^{2}-C\sum_{j\neq i}\bigl{(}\delta_{S_{i}}\int\tilde{\mu}_{j}\;\partial_{x_{1}}{k_{S_{i}}^{\mathbf{X}_{i}}}\,dx\bigl{)}^{2}$
		$\displaystyle\quad-C(\epsilon_{2}^{1/2}+\delta_{0}^{1/2})\,\delta_{S_{i}}^{2}\int\|U-\tilde{U}\|^{2}\;\partial_{x_{1}}{k_{S_{i}}^{\mathbf{X}_{i}}}\,dx$
		$\displaystyle\geq C\bigl{(}\delta_{S_{i}}\int\tilde{\mu}_{i}\;\partial_{x_{1}}{k_{S_{i}}^{\mathbf{X}_{i}}}\,dx\bigl{)}^{2}-C\sum_{j\neq i}\delta_{S_{i}}^{2}\int\tilde{\mu}_{j}^{2}\;\partial_{x_{1}}{k_{S_{i}}^{\mathbf{X}_{i}}}\,dx$
		$\displaystyle\quad-C(\epsilon_{2}^{1/2}+\delta_{0}^{1/2})\,\delta_{S_{i}}^{2}\int\tilde{\mu}_{i}^{2}\;\partial_{x_{1}}{k_{S_{i}}^{\mathbf{X}_{i}}}\,dx\,,$

where $\tilde{\mu}_{j}=\mu_{j}$ if $i=1$ and $\tilde{\mu}_{j}=\nu_{j}$ if $i=n$ for any $j\neq i$ .

By (2.5), (3.2), and Hölder’s inequality, we take $\epsilon_{2},\delta_{0}$ small enough and get

		$\displaystyle\dot{\mathbf{X}}_{i}\mathcal{Y}_{i}+\frac{\delta_{S_{i}}}{2\tilde{C}_{i}}\|\dot{\mathbf{X}}_{i}(t)\|^{2}$
		$\displaystyle=-\frac{\delta_{S_{i}}}{2\tilde{C}_{i}}\|\dot{\mathbf{X}}_{i}(t)\|^{2}+\dot{\mathbf{X}}_{i}(t)\int\partial_{x_{1}}a_{S_{i}}^{\mathbf{X}_{i}}\,\eta({U\|\tilde{U}})dx$
		$\displaystyle\leq-\frac{\delta_{S_{i}}}{2\tilde{C}_{i}}\|\dot{\mathbf{X}}_{i}(t)\|^{2}+C\epsilon_{2}\Lambda_{S_{i}}\frac{\tilde{C}_{i}}{\delta_{S_{i}}}\bigl{(}\int\|U-\tilde{U}\|\|\partial_{x_{1}}S_{i}^{\mathbf{X}_{i}}\|\,dx\bigl{)}^{2}$
		$\displaystyle\leq-\frac{C}{\delta_{S_{i}}}\tilde{C}_{i}\bigl{(}\delta_{S_{i}}\int\tilde{\mu}_{i}\;\partial_{x_{1}}{k_{S_{i}}^{\mathbf{X}_{i}}}\,dx\bigl{)}^{2}+C\tilde{C}_{i}\sum_{j\neq i}\delta_{S_{i}}\int\tilde{\mu}_{j}^{2}\;\partial_{x_{1}}{k_{S_{i}}^{\mathbf{X}_{i}}}\,dx$
		$\displaystyle\quad+C(\epsilon_{2}^{1/4}+\delta_{0}^{1/4})\,\delta_{S_{i}}\int\tilde{\mu}_{i}^{2}\;\partial_{x_{1}}{k_{S_{i}}^{\mathbf{X}_{i}}}\,dx\,.$

We take $\epsilon_{2},\delta_{0}$ small enough and get the estimate. ∎

4.5. $L^{2}$ estimates

Now it is time to choose $\Lambda_{S_{1}},\Lambda_{W_{n}},\tilde{C}_{1},\tilde{C}_{n}$ that work for both Lemma 4.3 and Lemma 4.4.

Lemma 4.5.

For any $U_{-}\in\mathbb{R}^{n}$ , there exist $\delta_{0},\epsilon_{2},C,\Lambda_{S_{1}},\Lambda_{W_{n}},\tilde{C}_{1},\tilde{C}_{n}>0$ such that the following is true.

Assume Assumption 3.2. Then for any $t\in[0,T]$ ,

\displaystyle\begin{split}\|U(t,\cdot)-\tilde{U}(t,\cdot)\|_{L^{2}(\mathbb{R}\times{\mathbb{T}^{d-1}})}^{2}+\int_{0}^{t}D_{0}(U)+G_{S_{1}}(U)+G_{W_{n}}(U)+Y\,ds\\ \leq C\|U_{0}-\tilde{U}_{0}\|_{L^{2}(\mathbb{R}\times{\mathbb{T}^{d-1}})}^{2}+CE^{2}\,,\end{split}

where $D_{0},G_{S_{1}},G_{W_{n}},Y,E$ are defined in (3.5).

Proof.

We fix $\gamma$ as in Lemma 4.3. We choose $\delta>0$ small enough such that

\displaystyle\delta<1\,\text{ and }\,(1+\delta)(1+\gamma)\leq 1+\frac{3}{2}\gamma\,.

We take

\displaystyle\tilde{C}_{i}=\frac{2\mathbf{C}_{4}}{\mathbf{C}_{3}}\,.

We take $\Lambda_{S_{1}},\Lambda_{W_{n}}=\Lambda$ where

\displaystyle-\mathbf{C}_{1}\Lambda+\frac{\mathbf{C}_{2}}{\delta}+(1+\delta)\mathbf{C}_{5}\tilde{C}_{i}=-1\,.

Finally, we take $\epsilon_{2},\delta_{0}$ small enough such that Lemma 4.3, Lemma 4.4, and (2.11) give the estimate. ∎

5. Proof of Prop 3.3

5.1. Some estimates

Before proving the $H^{m}$ estimates, we bound the higher-order terms evaluated in Lemma 4.2.

Recall $m$ defined in (1.8). Let $\phi:\mathbb{R}\times{\mathbb{T}^{d-1}}\rightarrow\mathbb{R}^{n}$ be a function. The Gagliardo-Nirenberg inequality for functions defined on $\mathbb{R}^{d}$ is proved in [24]. Together with the periodicity in the transverse direction, we get the following two inequalities. If $1\leq q\leq m-1$ , then for any $2\leq p<\frac{2d}{d-2q}$ , there exists $C>0$ such that

(5.1)

\displaystyle\|\phi\|_{L^{p}(\mathbb{R}\times{\mathbb{T}^{d-1}})}\leq C\|\phi\|_{H^{q}(\mathbb{R}\times{\mathbb{T}^{d-1}})}\,.

If $q=m$ , then for any $2\leq p<\infty$ , there exists $C>0$ such that

(5.2)

\displaystyle\|\phi\|_{L^{p}(\mathbb{R}\times{\mathbb{T}^{d-1}})}\leq C\|\phi\|_{H^{q}(\mathbb{R}\times{\mathbb{T}^{d-1}})}\,.

Lemma 5.1.

Assume Assumption 3.2. Let $1\leq k\leq m$ where $m$ is defined in (1.8). Then there exists a constant $C>0$ such that for any $t\in[0,T]$ and any $L\in\mathcal{L}^{k}$ where $\mathcal{L}^{k}$ is defined in (4.2),

\displaystyle\|L\|_{L^{2}(\mathbb{R}\times{\mathbb{T}^{d-1}})}^{2}\leq C\epsilon_{2}D_{k}+C\sum_{j=0}^{k-1}D_{j}\,,

where $D_{j}$ are defined in (3.5).

Proof.

If $l=1$ , then

\displaystyle\|L\|_{L^{2}}^{2}\leq C\sum_{j=0}^{k-1}D_{j}\,.

Assume $l\geq 2$ . Without loss of generality, assume $|\beta_{1}|\leq...\leq|\beta_{l}|$ . For any $1\leq j\leq l-1$ , (5.1) and (3.1) give

(5.3)

\displaystyle\|\partial_{x}^{\beta_{j}}\psi\|_{L^{p}}\leq C\|\psi\|_{H^{m}}\leq C\epsilon_{2}<1\,,

for any $2\leq p<\frac{2d}{d-2m+2|\beta_{j}|}:=p_{j}$ . We will consider 3 cases.

1) case $k+1-|\beta_{l}|\leq m-1$ . By (5.1), we have

\displaystyle\|\partial_{x}^{\beta_{l}}\psi\|_{L^{p}}\leq C\|\psi\|_{H^{k+1}}\,,

for any $2\leq p<\frac{2d}{d-2k-2+2|\beta_{l}|}:=p_{l}$ .

Since

\displaystyle\sum_{j=1}^{l}\frac{1}{p_{j}}\leq\frac{1}{2}+\frac{(d-2m)(l-1)}{2d}<\frac{1}{2}\;\text{ and }\;\frac{l}{2}\geq\frac{1}{2}\,,

there exist $p_{1}^{*},...,p_{l}^{*}$ such that

\displaystyle 2\leq p_{j}^{*}<p_{j}\;\text{ and }\;\sum_{j=1}^{l}\frac{1}{p_{j}^{*}}=\frac{1}{2}\,.

Then Hölder’s inequality and (5.3) give

\displaystyle\|L\|_{L^{2}}^{2}\leq C\Pi_{j=1}^{l}\|\partial_{x}^{\beta_{j}}\psi\|_{L^{p_{j}^{*}}}^{2}\leq(C\epsilon_{2})^{2(l-1)}\|\psi\|_{H^{k+1}}^{2}\leq C\epsilon_{2}\sum_{j=0}^{k}D_{j}\,.

2) case $k+1-|\beta_{l}|=m$ and $d\geq 2$ . Then $k=m$ and $|\beta_{j}|=1$ for any $1\leq j\leq l$ . By (5.2), we have

\displaystyle\|\partial_{x}^{\beta_{l}}\psi\|_{L^{p}}\leq C\|\psi\|_{H^{m+1}}\,,

for any $2\leq p<\infty:=p_{l}$ . Since $1/p_{l}=0$ and $l-1\leq m$ ,

\displaystyle\sum_{j=1}^{l}\frac{1}{p_{j}}\leq\frac{(d-2m+2)m}{2d}\leq\frac{d+1}{4d}<\frac{1}{2}\;\text{ and }\;\frac{l}{2}\geq\frac{1}{2}\,.

Hence, there exist $p_{1}^{*},...,p_{l}^{*}$ such that

\displaystyle 2\leq p_{j}^{*}<p_{j}\;\text{ and }\;\sum_{j=1}^{l}\frac{1}{p_{j}^{*}}=\frac{1}{2}\,.

Then Hölder’s inequality and (5.3) give

\displaystyle\|L\|_{L^{2}}^{2}\leq C\Pi_{j=1}^{l}\|\partial_{x}^{\beta_{j}}\psi\|_{L^{p_{j}^{*}}}^{2}\leq(C\epsilon_{2})^{2(l-1)}\|\psi\|_{H^{m+1}}^{2}\leq C\epsilon_{2}\sum_{j=0}^{m}D_{j}\,.

3) case $k+1-|\beta_{l}|=m$ and $d=1$ . Then $k=m=1$ , $l=2$ , and $|\beta_{1}|=|\beta_{2}|=1$ . Hölder’s inequality, Gagliardo-Nirenberg inequality and (3.1) give

\displaystyle\|L\|_{L^{2}}^{2}

\displaystyle\leq C\|\partial_{x}\psi\|_{L^{2}}^{2}\|\partial_{x}\psi\|_{L^{\infty}}^{2}\leq C\epsilon_{2}\|\psi\|_{H^{2}}^{2}\leq C\epsilon_{2}(D_{1}+D_{0})\,.

∎

5.2. $H^{m}$ estimates

We prove the $H^{m}$ estimates by induction.

Lemma 5.2.

For any $U_{-}\in\mathbb{R}^{n}$ , there exist $\delta_{0},\epsilon_{2},C,\Lambda_{S_{1}},\Lambda_{W_{n}},\tilde{C}_{1},\tilde{C}_{n}>0$ such that the following is true.

Assume Assumption 3.2. Then for any $t\in[0,T]$ ,

(5.4)			$\displaystyle\\|U(t,\cdot)-\tilde{U}(t,\cdot)\\|_{H^{m}(\mathbb{R}\times{\mathbb{T}^{d-1}})}^{2}+\int_{0}^{t}\Bigl{(}\sum_{k=0}^{m}D_{k}(U)+G_{S_{1}}(U)+G_{W_{n}}(U)+{Y}\Bigl{)}\,ds$
(5.4)			$\displaystyle\leq C\\|U_{0}-\tilde{U}_{0}\\|_{H^{m}(\mathbb{R}\times{\mathbb{T}^{d-1}})}^{2}+CE^{2}\,,$

and for any $t\in[0,T]$ and any $\beta\in\mathbb{N}^{d}$ such that $1\leq|\beta|\leq m$ ,

(5.5)			$\displaystyle\int_{0}^{t}\Big{\|}\frac{d}{dt}\int\big{\|}\partial_{x}^{\beta}(U-\tilde{U})\big{\|}^{2}\,dx\Big{\|}\,ds$
(5.5)			$\displaystyle\leq C\int_{0}^{t}\Bigl{(}\sum_{k=0}^{m}D_{k}(U)+G_{S_{1}}(U)+G_{W_{n}}(U)+Y\Bigl{)}\,ds+CE^{2}\,,$

where $m$ is defined in (1.8) and $D_{k},G_{S_{1}},G_{W_{n}},Y,E$ are defined in (3.5).

Proof.

We will prove the lemma by induction. Recall the $L^{2}$ estimates shown in Lemma 4.5. Let $1\leq k\leq m$ . We assume the inequality is true for $k-1$ , i.e.,

\displaystyle\begin{split}&\|U(t,\cdot)-\tilde{U}(t,\cdot)\|_{H^{k-1}(\mathbb{R}\times{\mathbb{T}^{d-1}})}^{2}+\int_{0}^{t}\Bigl{(}\sum_{j=0}^{k-1}D_{j}(U)+G_{S_{1}}(U)+G_{W_{n}}(U)+Y\Bigl{)}\,ds\\ &\leq C\|U_{0}-\tilde{U}_{0}\|_{H^{k-1}(\mathbb{R}\times{\mathbb{T}^{d-1}})}^{2}+CE^{2}\,.\end{split}

We want to show the inequality is true for $k$ , i.e.,

(5.6)

\displaystyle\begin{split}&\|U(t,\cdot)-\tilde{U}(t,\cdot)\|_{H^{k}(\mathbb{R}\times{\mathbb{T}^{d-1}})}^{2}+\int_{0}^{t}\Bigl{(}\sum_{j=0}^{k}D_{j}(U)+G_{S_{1}}(U)+G_{W_{n}}(U)+Y\Bigl{)}\,ds\\ &\leq C\|U_{0}-\tilde{U}_{0}\|_{H^{k}(\mathbb{R}\times{\mathbb{T}^{d-1}})}^{2}+CE^{2}\,.\end{split}

Note the constants $C>0$ in this proof depend on $B_{j},f,\eta,U_{-},k$ , but the dependency on $k$ does not matter as $m$ is fixed and finite.

Recall the definition of $\psi$ in (4.1). Let $\alpha_{k}\in\mathbb{N}^{d}$ be such that $|\alpha_{k}|=k$ . We take $\partial_{x}^{\alpha_{k}}$ on both sides of (2.17) and get

	$\displaystyle\partial_{t}\partial_{x}^{\alpha_{k}}\psi+\partial_{x}^{{\alpha_{k}}}\partial_{x_{1}}\bigl{(}f(U)-f(\tilde{U})\bigl{)}+\sum_{j=2}^{d}\partial_{x}^{{\alpha_{k}}}\partial_{x_{j}}\bigl{(}g_{j}(U)-g_{j}(\tilde{U})\bigl{)}$
	$\displaystyle=\sum_{j=1}^{d}\partial_{x}^{\alpha_{k}}\partial_{x_{j}}\bigl{(}B_{j}(U)\partial_{x_{j}}\eta^{\prime}(U)-B_{j}(\tilde{U})\partial_{x_{j}}\eta^{\prime}(\tilde{U})\bigl{)}-\partial_{x}^{\alpha_{k}}Z-\partial_{x}^{\alpha_{k}}E_{1}-\partial_{x}^{\alpha_{k}}E_{2}\,.$

Then we multiply both sides by $\partial_{x}^{\alpha_{k}}\psi$ and take integration w.r.t. $x$ . Recall (1.17). By integration by parts, we have

		$\displaystyle\frac{d}{dt}\int\frac{\|\partial_{x}^{\alpha_{k}}\psi\|^{2}}{2}\,dx+\sum_{j=1}^{d}\int\partial_{x}^{\alpha_{k}}\bigl{(}B_{j}(U)\eta^{\prime\prime}(U)\partial_{x_{j}}U-B_{j}(\tilde{U})\eta^{\prime\prime}(\tilde{U})\partial_{x_{j}}\tilde{U}\bigl{)}\,\partial_{x}^{\alpha_{k}+e_{j}}\psi\,dx$
		$\displaystyle=\int\partial_{x}^{\alpha_{k}}\bigl{(}f(U)-f(\tilde{U})\bigl{)}\,\partial_{x}^{\alpha_{k}+e_{1}}\psi\,dx+\sum_{j=2}^{d}\int\partial_{x}^{\alpha_{k}}\bigl{(}g_{j}(U)-g_{j}(\tilde{U})\bigl{)}\,\partial_{x}^{\alpha_{k}+e_{j}}\psi\,dx$
		$\displaystyle\quad-\int\partial_{x}^{\alpha_{k}}Z\,\partial_{x}^{\alpha_{k}}\psi\,dx-\int\partial_{x}^{\alpha_{k}}E_{1}\,\partial_{x}^{\alpha_{k}}\psi\,dx-\int\partial_{x}^{\alpha_{k}}E_{2}\,\partial_{x}^{\alpha_{k}}\psi\,dx\,,$

where $e_{j}\in\mathbb{N}^{d}$ is the multi-index whose j-th component is $1$ and all other components are 0.

Let $0<\tau<1$ to be chosen later in (5.8). Recall (1.17). By Young’s inequality and integration by parts, we have

\displaystyle\frac{d}{dt}\int\frac{|\partial_{x}^{\alpha_{k}}\psi|^{2}}{2}\,dx+I_{1}\leq\tau D_{k}+CD_{k-1}+\frac{C}{\tau}(I_{2}+I_{3}+I_{4}+I_{5})\,,

where

	$\displaystyle I_{1}$	$\displaystyle=\sum_{j=1}^{d}\int\partial_{x}^{\alpha_{k}}\bigl{(}B_{j}(U)\eta^{\prime\prime}(U)\partial_{x_{j}}U-B_{j}(\tilde{U})\eta^{\prime\prime}(\tilde{U})\partial_{x_{j}}\tilde{U}\bigl{)}\,\partial_{x}^{\alpha_{k}+e_{j}}\psi\,dx\,,$
	$\displaystyle I_{2}$	$\displaystyle=\int\big{\|}\partial_{x}^{\alpha_{k}}\bigl{(}f(U)-f(\tilde{U})\bigl{)}\big{\|}^{2}\,dx+\sum_{j=2}^{d}\int\big{\|}\partial_{x}^{\alpha_{k}}\bigl{(}g_{j}(U)-g_{j}(\tilde{U})\bigl{)}\big{\|}^{2}\,dx\,,$
	$\displaystyle I_{3}$	$\displaystyle=Y\,,$
	$\displaystyle I_{4}$	$\displaystyle=\int\|\partial_{x}^{\alpha_{k}}E_{1}\|^{2}\,dx\,,$
	$\displaystyle I_{5}$	$\displaystyle=\begin{cases}\int\|\partial_{x}^{\alpha_{k}-e_{1}}E_{2}\|^{2}\,dx\,,\;\textup{if }(\alpha_{k})_{1}\textup{ is odd and }W_{n}=R_{n}\,,\\ \int\|\partial_{x}^{\alpha_{k}}E_{2}\|^{2}\,dx\,,\;\textup{otherwise.}\end{cases}$

Lemma 4.2 and Lemma 5.1 give

	$\displaystyle\|I_{1}-\tilde{I}_{1}\|$	$\displaystyle\leq(\tau+\frac{C}{\tau}\epsilon_{2})D_{k}+\frac{C}{\tau}(G_{S_{1}}+G_{W_{n}}+\sum_{j=0}^{k-1}D_{j})\,,$
	$\displaystyle I_{2}$	$\displaystyle\leq C\epsilon_{2}D_{k}+C(G_{S_{1}}+G_{W_{n}}+\sum_{j=0}^{k-1}D_{j})\,,$

where

\displaystyle\tilde{I}_{1}=\sum_{j=1}^{d}\int\bigl{(}B_{j}(U)\eta^{\prime\prime}(U)\partial_{x}^{\alpha_{k}+e_{j}}U-B_{j}(\tilde{U})\eta^{\prime\prime}(\tilde{U})\partial_{x}^{\alpha_{k}+e_{j}}\tilde{U}\bigl{)}\,\partial_{x}^{\alpha_{k}+e_{j}}\psi\,dx\,.

We first evaluate $\tilde{I}_{1}$ . We have

\tilde{I}_{1}=\tilde{D}_{k}+\tilde{I}_{11}+\tilde{I}_{12}\,,

where

	$\displaystyle\tilde{D}_{k}$	$\displaystyle=\sum_{j=1}^{d}\int B_{j}(U_{-})\eta^{\prime\prime}(U_{-})\partial_{x}^{\alpha_{k}+e_{j}}\psi\,\partial_{x}^{\alpha_{k}+e_{j}}\psi\,dx\,,$
	$\displaystyle\tilde{I}_{11}$	$\displaystyle=\sum_{j=1}^{d}\int\bigl{(}B_{j}(U)\eta^{\prime\prime}(U)-B_{j}(U_{-})\eta^{\prime\prime}(U_{-})\bigl{)}\partial_{x}^{\alpha_{k}+e_{j}}\psi\,\partial_{x}^{\alpha_{k}+e_{j}}\psi\,dx\,,$
	$\displaystyle\tilde{I}_{12}$	$\displaystyle=\sum_{j=1}^{d}\int\bigl{(}B_{j}(U)\eta^{\prime\prime}(U)-B_{j}(\tilde{U})\eta^{\prime\prime}(\tilde{U})\bigl{)}\partial_{x}^{\alpha_{k}+e_{j}}\tilde{U}\,\partial_{x}^{\alpha_{k}+e_{j}}\psi\,dx\,.$

As $B_{j}$ are positive definite and $\eta^{\prime\prime}$ is strictly convex,

\displaystyle\tilde{D}_{k}\geq C\sum_{j=1}^{d}\|\partial_{x}^{\alpha_{k}+e_{j}}\psi\|_{L^{2}}^{2}\,.

By (3.2) and Young’s inequality, we get

\displaystyle|\tilde{I}_{11}|+|\tilde{I}_{12}|\leq C(\epsilon_{2}+\delta_{0})D_{k}+{C}(G_{S_{1}}+G_{W_{n}})\,.

Therefore, we have

\displaystyle|I_{1}-\tilde{D}_{k}|\leq\bigl{(}\tau+\frac{C}{\tau}(\epsilon_{2}+\delta_{0})\bigl{)}D_{k}+\frac{C}{\tau}(G_{S_{1}}+G_{W_{n}}+\sum_{j=0}^{k-1}D_{j})\,.

If $W_{n}=S_{n}$ , then Lemma 4.2 gives

	$\displaystyle I_{4}+I_{5}$	$\displaystyle\leq C\Bigl{(}\big{\\|}\|S_{n}^{\mathbf{X}_{n}}-U_{m}\|\|\partial_{x_{1}}S_{1}^{\mathbf{X}_{1}}\|\big{\\|}_{L^{2}}^{2}+\big{\\|}\|S_{1}^{\mathbf{X}_{1}}-U_{m}\|\|\partial_{x_{1}}S_{n}^{\mathbf{X}_{n}}\|\big{\\|}_{L^{2}}^{2}$
		$\displaystyle\quad+\big{\\|}\|\partial_{x_{1}}S_{n}^{\mathbf{X}_{n}}\|\|\partial_{x_{1}}S_{1}^{\mathbf{X}_{1}}\|\big{\\|}_{L^{2}}^{2}\Bigl{)}$
		$\displaystyle=:\mathcal{E}_{S_{n}}\,.$

If $W_{n}=R_{n}$ , then Lemma 4.2 gives

	$\displaystyle I_{4}+I_{5}$	$\displaystyle\leq C\Bigl{(}\\|\partial_{x_{1}}^{2j_{*}}R_{n}\\|_{L^{2}}^{2}+\\|\partial_{x_{1}}R_{n}\\|_{L^{4}}^{4}+\big{\\|}\|\partial_{x_{1}}R_{n}\|\|\partial_{x_{1}}S_{n}^{\mathbf{X}_{n}}\|\big{\\|}_{L^{2}}^{2}$
		$\displaystyle\quad+\big{\\|}\|R_{n}-U_{m}\|\|\partial_{x_{1}}S_{n}^{\mathbf{X}_{n}}\|\big{\\|}_{L^{2}}^{2}+\big{\\|}\|S_{n}^{\mathbf{X}_{n}}-U_{m}\|\|\partial_{x_{1}}R_{n}\|\big{\\|}_{L^{2}}^{2}\Bigl{)}$
		$\displaystyle=:\mathcal{E}_{R_{n}}\,,$

for some $j_{*}\in\mathbb{N}^{*}$ .

We get

(5.7)		$\displaystyle\frac{d}{dt}\int\frac{\|\partial_{x_{1}}^{\alpha_{k}}\psi\|^{2}}{2}\,dx+C\sum_{j=1}^{d}\\|\partial_{x}^{\alpha_{k}+e_{j}}\psi\\|_{L^{2}}^{2}\leq\bigl{(}2\tau+\frac{C}{\tau}(\epsilon_{2}+\delta_{0})\bigl{)}D_{k}$
(5.7)		$\displaystyle+\frac{C}{\tau}(G_{S_{1}}+G_{W_{n}}+\sum_{j=0}^{k-1}D_{j}+Y+\mathcal{E}_{W_{n}})\,.$

Let

\displaystyle\mathcal{A}_{k}=\bigl{\{}\alpha\in\mathbb{N}^{d}:|\alpha|=k\bigl{\}}\,.

As $\alpha_{k}$ is arbitrary, we get

	$\displaystyle\frac{d}{dt}\sum_{\alpha\in\mathcal{A}_{k}}\int\frac{\|\partial_{x_{1}}^{\alpha}\psi\|^{2}}{2}\,dx+C_{1}D_{k}\leq\bigl{(}2\|\mathcal{A}_{k}\|\tau+\frac{C_{2}}{\tau}(\epsilon_{2}+\delta_{0})\bigl{)}D_{k}$
	$\displaystyle+\frac{C}{\tau}(G_{S_{1}}+G_{W_{n}}+\sum_{j=0}^{k-1}D_{j}+Y+\mathcal{E}_{W_{n}})\,,$

where $C_{1},C_{2}>0$ are constants that depend only on $B_{j},f,\eta,U_{-},k$ . We take

(5.8)

\tau=\textup{min}\bigl{\{}\frac{C_{1}}{6|\mathcal{A}_{k}|},1\bigl{\}}\,\textup{ and }\,\epsilon_{2},\delta_{0}\leq\frac{C_{1}\tau}{6C_{2}}\,.

Finally, we take integration over time. By (2.11) and Lemma 4.1, we get the estimates (5.6). Hence, we prove (5.4). The inequality (5.7) also gives

\displaystyle\big{|}\frac{d}{dt}\int{|\partial_{x_{1}}^{\alpha_{k}}\psi|^{2}}\,dx\big{|}\leq{C}(G_{S_{1}}+G_{W_{n}}+\sum_{j=0}^{k}D_{j}+Y+\mathcal{E}_{W_{n}})\,.

By taking integration over time, (2.11) and Lemma 4.1 give (5.5). ∎

6. Appendix

6.1. Proof of Proposition 1.2

The entropy of the 3-D barotropic Brenner-Navier-Stokes equations is

\displaystyle\eta=\rho\frac{|u|^{2}}{2}+Q\,.

We let

\displaystyle U=\begin{pmatrix}\rho\\ \rho u\end{pmatrix}\,.

We rewrite the 3-D barotropic Brenner-Navier-Stokes equations (1.4) w.r.t. $U$ :

\begin{cases}\partial_{t}\rho+div(\rho u)=\Delta Q^{\prime}\,,\\ \partial_{t}\rho u+div(\rho u\otimes u)+\nabla\rho^{\gamma}=\nu\Delta u+div(u\otimes\nabla Q^{\prime})\,.\end{cases}

\displaystyle\eta^{\prime}(U)=\begin{pmatrix}Q^{\prime}-\frac{|u|^{2}}{2}\\ u\end{pmatrix}\,,

we get

\displaystyle\Delta Q^{\prime}=\Delta\eta_{1}^{\prime}+\Delta\frac{|u|^{2}}{2}\,,

and for any $1\leq j\leq 3$ ,

	$\displaystyle div(u_{j}\otimes\nabla Q^{\prime})$	$\displaystyle=\sum_{k=1}^{3}\partial_{k}(u_{j}\partial_{k}Q^{\prime})$
		$\displaystyle=\sum_{k=1}^{3}\partial_{k}(u_{j}\partial_{k}\eta_{1}^{\prime})+\partial_{k}(u_{j}\partial_{k}\frac{\|u\|^{2}}{2})\,.$

Hence, for any $1\leq j\leq 3$ ,

\displaystyle B_{j}(U)=\begin{pmatrix}1&u_{1}&u_{2}&u_{3}\\ u_{1}&u_{1}^{2}+\nu&u_{1}u_{2}&u_{1}u_{3}\\ u_{2}&u_{1}u_{2}&u_{2}^{2}+\nu&u_{2}u_{3}\\ u_{3}&u_{1}u_{3}&u_{2}u_{3}&u_{3}^{2}+\nu\end{pmatrix}\,.

Since the determinant of $B_{j}$ is $\nu^{3}$ , $B_{j}$ is positive definite.

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	$\displaystyle\|\mathbf{S}_{i}({x_{1}})-U_{L}\|$	$\displaystyle\leq C\delta_{S_{i}}e^{-C\delta_{S_{i}}\|{x_{1}}\|},\quad\forall{x_{1}}<0\,,$
	$\displaystyle\|\mathbf{S}_{i}({x_{1}})-U_{R}\|$	$\displaystyle\leq C\delta_{S_{i}}e^{-C\delta_{S_{i}}\|{x_{1}}\|},\quad\forall{x_{1}}>0\,,$
	$\displaystyle\|\partial_{x_{1}}\mathbf{S}_{i}\|$	$\displaystyle\leq C\delta_{S_{i}}^{2}e^{-C\delta_{S_{i}}\|{x_{1}}\|},\quad\forall{x_{1}}\in\mathbb{R}\,,$
	$\displaystyle\|\partial_{x_{1}}^{j}\mathbf{S}_{i}\|$	$\displaystyle\leq C_{j}\delta_{S_{i}}\|\partial_{x_{1}}\mathbf{S}_{i}\|,\quad\forall{x_{1}}\in\mathbb{R}\,,\;\forall j\geq 2\,.$

	$\displaystyle\\|\partial_{x_{1}}^{j}R_{n}\\|_{L^{p}(\mathbb{R}\times\mathbb{T}^{d-1})}$	$\displaystyle\leq C_{p,j}\,\textup{min}\bigl{\{}\delta_{R_{n}},\delta_{R_{n}}^{1/p}(1+t)^{-1+1/p}\bigl{\}}\,,\;\forall p\in[1,\infty]\,,$
	$\displaystyle\\|\partial_{x_{1}}^{2j}R_{n}\\|_{L^{p}(\mathbb{R}\times\mathbb{T}^{d-1})}$	$\displaystyle\leq C_{p,j}\,\textup{min}\bigl{\{}\delta_{R_{n}},(1+t)^{-1}\bigl{\}}\,,\;\forall p\in[1,\infty)\,,$
	$\displaystyle\|\partial_{x_{1}}^{j}R_{n}\|$	$\displaystyle\leq C_{j}\|\partial_{x_{1}}R_{n}\|\,,$

	$\displaystyle\|R_{n}(t,x)-U_{m}\|$	$\displaystyle\leq C\delta_{R_{n}}e^{-2\|x_{1}-\lambda_{n}(U_{m})t\|}\,,\quad\forall x_{1}\leq\lambda_{n}(U_{m})t\,,$
	$\displaystyle\|\partial_{x_{1}}R_{n}(t,x)\|$	$\displaystyle\leq C\delta_{R_{n}}e^{-2\|x_{1}-\lambda_{n}(U_{m})t\|}\,,\quad\forall x_{1}\leq\lambda_{n}(U_{m})t\,,$
	$\displaystyle\|R_{n}(t,x)-U_{+}\|$	$\displaystyle\leq C\delta_{R_{n}}e^{-2\|x_{1}-\lambda_{n}(U_{+})t\|}\,,\quad\forall x_{1}\geq\lambda_{n}(U_{+})t\,,$
	$\displaystyle\|\partial_{x_{1}}R_{n}(t,x)\|$	$\displaystyle\leq C\delta_{R_{n}}e^{-2\|x_{1}-\lambda_{n}(U_{+})t\|}\,,\quad\forall x_{1}\geq\lambda_{n}(U_{+})t\,.$

	$\displaystyle\|S_{1}^{\mathbf{X}_{1}}-U_{m}\|$	$\displaystyle\leq C\delta_{S_{1}}e^{-C\delta_{S_{1}}\|x_{1}-{\sigma}_{1}t+\mathbf{X}_{1}(t)\|}$
		$\displaystyle\leq C\delta_{S_{1}}\textup{exp}\bigl{(}\frac{-C\delta_{S_{1}}\|x_{1}-{\sigma}_{1}t+\mathbf{X}_{1}(t)\|}{2}\bigl{)}\textup{exp}\bigl{(}\frac{-C\delta_{S_{1}}\Delta t}{8}\bigl{)}\,,$
	$\displaystyle\|\partial_{x_{1}}S_{1}^{\mathbf{X}_{1}}\|$	$\displaystyle\leq C\delta_{S_{1}}^{2}e^{-C\delta_{S_{1}}\|x_{1}-{\sigma}_{1}t+\mathbf{X}_{1}(t)\|}$
		$\displaystyle\leq C\delta_{S_{1}}^{2}\textup{exp}\bigl{(}\frac{-C\delta_{S_{1}}\|x_{1}-{\sigma}_{1}t+\mathbf{X}_{1}(t)\|}{2}\bigl{)}\textup{exp}\bigl{(}\frac{-C\delta_{S_{1}}\Delta t}{8}\bigl{)}\,.$

	$\displaystyle\|S_{n}^{\mathbf{X}_{n}}-U_{m}\|$	$\displaystyle\leq C\delta_{S_{n}}e^{-C\delta_{S_{n}}\|x_{1}-\sigma_{n}t+\mathbf{X}_{n}(t)\|}$
		$\displaystyle\leq C\delta_{S_{n}}\textup{exp}\bigl{(}\frac{-C\delta_{S_{n}}\|x_{1}-\sigma_{n}t+\mathbf{X}_{n}(t)\|}{2}\bigl{)}\textup{exp}\bigl{(}\frac{-C\delta_{S_{n}}\Delta t}{8}\bigl{)}\,,$
	$\displaystyle\|\partial_{x_{1}}S_{n}^{\mathbf{X}_{n}}\|$	$\displaystyle\leq C\delta_{S_{n}}^{2}e^{-C\delta_{S_{n}}\|x_{1}-\sigma_{n}t+\mathbf{X}_{n}(t)\|}$
		$\displaystyle\leq C\delta_{S_{n}}^{2}\textup{exp}\bigl{(}\frac{-C\delta_{S_{n}}\|x_{1}-\sigma_{n}t+\mathbf{X}_{n}(t)\|}{2}\bigl{)}\textup{exp}\bigl{(}\frac{-C\delta_{S_{n}}\Delta t}{8}\bigl{)}\,.$

Time-asymptotic stability of composite weak planar waves for a general n×nn\times n multi-D viscous system

Abstract.

1. Introduction

Theorem 1.1.

Proposition 1.2.

Lemma 1.3.

2. Preliminaries

2.1. Viscous shock wave

Lemma 2.1.

2.2. Construction of approximate rarefaction wave

Lemma 2.2.

2.3. Construction of weight functions, shift functions and the superposition wave

3. Proof of Theorem 1.1

3.1. Local-in-time estimates and a priori estimates

Proposition 3.1.

Assumption 3.2.

Proposition 3.3.

3.2. Global existence and estimates

3.3. Time-asymptotic behavior

4. Energy estimate

4.1. Wave interaction estimates

Lemma 4.1.

Proof.

4.2. Higher derivatives estimates

Lemma 4.2.

Proof.

4.3. Relative entropy method

Lemma 4.3.

Proof.

4.4. Poincaré type inequality

Lemma 4.4.

Proof.

4.5. L2L^{2} estimates

Lemma 4.5.

Proof.

5. Proof of Prop 3.3

5.1. Some estimates

Lemma 5.1.

Proof.

5.2. HmH^{m} estimates

Lemma 5.2.

Proof.

6. Appendix

6.1. Proof of Proposition 1.2

References

Time-asymptotic stability of composite weak planar waves for a general $n\times n$ multi-D viscous system

4.5. $L^{2}$ estimates

5.2. $H^{m}$ estimates