数理解析研究所講究録
第 1782 巻 2012 年 150-157

150

Asymptotic stability of stationary waves forsymmetric hyperbolic-parabolic system
in half space
九州大学数理東京工業大学情報理工l2

中村 徹 (TOHRU NAKAMURA) 1
西畑 伸也 (SHINYA NISHIBATA) 2

Faculty of Mathematics, Kyushu University,
Fukuoka 819-0395, Japan

Deparrment of Mathematical and Computing Sciences,
Tokyo Institute of Technology
Tokyo 152-8552, Japan

Abstract: In the present paper, we consider a large-time behavior of solutions tothe symmetric hyperbolic-parabolic system in the half space. Our main concern is toshow existence and asymptotic stability of the stationary solution (boundary layer
solution) under the situation where all of characteristics are non-positive. We firstly
prove the existence of the stationary solution by assuming that a boundary strengthis sufficiently small. Especially, in the case where one eigenvalue of Jacobian matrix
appeared in a stationary problem becomes zero, we assume that the characteristics
field corresponding to the zero eigenvalue is genuine nonlinear in order to show theexistence of a degenerate stationary solution with the aid of a center manifold theory.
We next prove that a nondegenerate stationary solution is time asymptotically stablewith a small initial perturbation. The key to proof is to derive the uniform a priori
estimates by using the energy method. To obtain a priori estimates, we use theenergy method in half space developed by Matsumura and Nishida as well as thestability condition of Shizuta-Kawashima type.

1

Introduction

This article is a survey of the paper [10] on large-time behavior of solutions to a,system of viscous conservation laws over one-dimensional half space$\mathbb{R}_{+}:=(0, \infty)$$f^{0}(u)_{t}+f(u)_{x}=(G(u)u_{x})_{x}$,$x\in \mathbb{R}_{+},$$t>0$ .

(1.1)

Here $u=u(t, x)$ is an unknown m-vector function taking values in an open convex
set
and $f(u)$ are smooth m-vector functions; $G(u)$ is a smooth $m\cross m$real matrix function. We assume that $f^{0}(u)$ has no singularity, i.e., $\det D_{u}f^{0}(u)\neq 0$. It is also assumed that $G(u)$ is a non-negative matrix given by aholds forform$\mathcal{O}\in \mathbb{R}^{m};f^{0}(u)$$u\in \mathcal{O}$$G(u)=(\begin{array}{ll}0 00 G_{2}(u)\end{array})$

where $G_{2}(u)$ is an, wherefor$u\in \mathcal{O}$real matrix function and uniformly positive definite
is a positive integer satisfying $0<m_{2}<m$ . Thus the$m_{2}\cross m_{2}$$m_{2}$,

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system (1.1) consists of$m_{1}$-hyperbolic equations and$m_{2}$-parabolic equations where$m_{1}:=m-m_{2}$ .

We assume that the system (1.1) has a strictly convex entropy $\eta=\eta(z)$ satisfying(i)is a strictly convex scalar function, i.e., the Hessian matrix
positive definite for.
$\eta(z)$$D_{z}^{2}\eta(z)$is$z\in f^{0}(\mathcal{O})$(ii) There exists a smooth scalar function $q(u)$ (entropy flux) such that $D_{u}q(u)=$$D_{z}\eta(f^{0}(u))D_{u}f(u)$ .

(iii) The matrix $B(u):=\tau_{D_{u}f^{0}(u)D_{z}^{2}\eta(f^{0}(u))G(u)}$ is real symmetric and nonnegative.
Then the system (1.1) is deduced to the symmetric system(1.2)$A^{0}(u)u_{t}+A(u)u_{x}=B(u)u_{xx}+g(u, u_{x})$ ,where $A^{0}(u)$ is a real symmetric and positive matrix, $A(u)$ is a real symmetric matrix,$B(u)$ is a real symmetric and non-negative matrix, and $g(u, u_{x})$ is non-linear terms
satisfying $|g(u, u_{x})|\leq C|u_{x}|^{2}$ . Moreover, under suitable conditions as in [4], thesystem (1.2) is rewritten to the decomposed form
$A_{1}^{0}(u)v_{t}+A_{11}(u)v_{x}+A_{12}(u)w_{x}=g_{1}(u, w_{x})$(1.3a)(1.3b),$A_{2}^{0}(u)w_{t}+A_{21}(u)v_{x}+A_{22}(u)w_{x}=B_{2}(u)w_{xx}+g_{2}(u, u_{x})$,are unknown
and-vector functions respectively, given by$u=\tau(v, w)$ . In the system (1.3),
andare real symmetric and positive
matrices; $A_{ij}(u)(i,j=1,2)$ are real matrices satisfying
where

$v$

and

$w$

$m_{2}$$m_{1^{-}}$$A_{1}^{0}(u)$$A_{2}^{0}(u)$$A(u)=(\begin{array}{ll}A_{11}(u) A_{12}(u)A_{21}(u) A_{22}(u)\end{array})$,

and $A(u)$ is symmetric, i.e., $A_{11}(u)$ and $A_{22}(u)$ are symmetric and $A_{21}(u)=TA_{12}(u)$ ;$B_{2}(u)$ is a real symmetric positive matrix; $g_{1}(u, w_{x})$ andare non-linear
terms. For system (1.3), we put the following condition.
$g_{2}(u, u_{x})$[Al] The matrix $A_{11}(u)$ is negative and $A(u)$ is non-positive for$u\in \mathcal{O}$.

We prescribe the initial and boundary conditions for (1.2) as
(14)
(15)

$u(0, x)=u_{0}(x)=T(v_{0}, w_{0})(x)$ ,$w(t, 0)=w_{b}$ ,whereis a constant. Notice that the problem $(1.3)-(1.5)$ is well-posedsince the boundary condition for is not necessary due to the condition $A_{11}(u)<0$ .
We assume that a spatial asymptotic state of the initial data is a constant:$w_{b}\in \mathbb{R}^{m_{2}}$$v$$\lim_{xarrow\infty}u_{0}(x)=u_{+}=T(v_{+}, w_{+})$, i.e.,$\lim_{xarrow\infty}(v_{0}, w_{0})(x)=(v_{+}, w_{+})$.

Related results. For the heat-conductive model of compressible viscous gases in, Matsumura and Nishida in [7] show the asymptotic stability of a constant state(or a stationary solution corresponding to an external potential force) and establish$\mathbb{R}^{3}$

152, Umeda,a technical energy method. For the system (1.1) in the full space
Kawashima and Shizuta in [14] consider a sufficient condition, introduced in Section
3 as the condition [K], which guarantee a dissipative structure of the system (1.1)
and show the asymptotic stability of the constant state. Shizuta and Kawashima in[13] show an equivalence of the condition [K] and the condition [SK] introduced in
Section 3.
For a barotropic model of compressible viscous gases in half space, Kawashima,
Nishibata and Zhu in [6] consider an outflow problem, where a negative Dirichlet data for the velocity is imposed, and show the existence and the asymptotic
stability of boundary layer solutions. The generalization of this problem to a multiis considered by Kagei and Kawashima in
dimensional half space[2]. For the heat-conductive model, Kawashima, Nakamura, Nishibata and Zhu [5]prove the existence and the asymptotic stability of boundary layer solutions for theoutflow problem. For the inflow problem, the barotropic model is considered in [9]and the heat-conductive model is considered in [1, 11, 12].
The main purpose of the present paper is to show the existence and the asymptotic stability of boundary layer solutions in the half space
which covers the
6].
[5,
problemresults for the outflow$\mathbb{R}^{n}$$\mathbb{R}_{+}^{n}=\mathbb{R}_{+}\cross \mathbb{R}^{n-1}$$\mathbb{R}_{+}$

Notations. For 1 $\leq p\leq\infty,$
denotes a standard Lebesgue space overequipped with a norm
. For a non-negative integer
denotesan s-th order Sobolev space overin the. Noticesense with a norm
.
.
that
and
I
I . For a function $f=f(u),$ $D_{u}f(u)$denotes a Fr\’echet derivative of with respect to . Especially, in the case of
and $f(u)=(f_{1}, \ldots, f_{m})(u)\in \mathbb{R}^{m}$ , the Fr\’echet derivative
is anmatrix.
$If(\mathbb{R}_{+})$$\Vert\cdot\Vert_{L^{p}}$$\mathbb{R}_{+}$$s,$$H^{0}(\mathbb{R}_{+})=L^{2}(\mathbb{R}_{+})$$\Vert$$H^{s}(\mathbb{R}_{+})$$L^{2}$$\mathbb{R}_{+}$$H^{0}=\Vert$$\Vert\cdot\Vert_{H^{S}}$$L^{2}$$f$$u$$u=(u_{1}, \ldots, u_{7l})\in \mathbb{R}^{n}$$D_{u}f=( \frac{\partial f_{i}}{\partial u_{j}})_{ij}$2$m\cross n$

Stationary solution

The stationary solution
dent of . Thus$t$is defined as a solution to (1.1) indepensatisfies equations$\tilde{u}(x)=T(\tilde{v},\tilde{w})(x)$$\tilde{u}=\tau(\tilde{v},\tilde{w})$$f(\tilde{u})_{x}=(G(\tilde{u})\tilde{u}_{x})_{x}$, i.e.,$\{\begin{array}{l}f_{1}(\tilde{v},\tilde{w})_{x}=0,f_{2}(\tilde{v},\tilde{w})_{x}=(G_{2}(\tilde{u})\tilde{w}_{x})_{x},\end{array}$(2.1)

where $f=^{T}(f_{1}, f_{2})$ . The boundary conditions are prescribed as$\tilde{w}(0)=w_{b}$,$\lim_{xarrow\infty}\tilde{u}(x)=u_{+}$

Integrating the first equation in (2.1) over$(x, \infty)$[A2] $\det D_{v}f_{1}(v_{+}, w_{+})\neq 0$ .

$\tilde{v}$(2.2), we have$f_{1}(\tilde{v},\tilde{w})=f_{1}(v_{+}, w_{+})$

We solve this equation with respect todo this, we assume.

.

by using the implicit function theorem. To

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Then there exists $V=V(\tilde{w})$ satisfying
and $V(w_{+})=v_{+}$ .
$\mu_{j}(w)(j=1,\ldots,m_{2})$$:=G_{2}(u_{+})^{-1}D_{w}H(w)$,Letbe eigenvalues of the matrix$r_{j}(w)$$H(w)$$:=f_{2}(V(w),
w)$
where
, and letbe corresponding eigenvectors. We asissume that the eigenvalues
are distinct and the first eigenvalue
non-positive. Namely, We assume$f_{1}(V(\tilde{w}),\tilde{w})=f_{1}(v_{+}, w_{+})$$\tilde{A}(w)$$\mu_{1}(w_{+})$$\mu_{j}(w)$[A3] Eigenvalues of$\tilde{A}(w)$are distinct, i.e., $\mu_{1}(w)>\mu_{2}(w)>\cdots>\mu_{m_{2}}(w)$ .

[A4] $\mu_{1}(w_{+})\leq 0$ .
Under the above assumptions, we solve the boundary value problem (2.1) and(2.2).

Theorem 2.1. Assume that $[A2]-[A4]$ hold and that $\delta:=|w_{b}-w_{+}|$ is sufficiently
small.
(i) (Non-degenerate case) For the case $of\mu_{1}(w_{+})<0$ , there exists a unique smooth
solutionto (2.1) and (2.2) satisfying$\tilde{u}(x)$$|\partial_{x}^{k}(\tilde{u}(x)-u_{+})|\leq Ce^{-cx}$

for $k=0,1,$$\ldots$.

(ii) (Degenerate case) For the case of $\mu_{1}(w_{+})=0$ , there exists a certain region
such that if $w_{b}\in M$ and $D_{w}\mu_{1}(w_{+})\cdot r_{1}(w_{+})\neq 0$ , then there exists
a unique smooth solution
satisfying$\mathcal{M}\subset \mathbb{R}^{m_{2}}$$\tilde{u}(x)$$| \partial_{x}^{k}(\tilde{u}(x)-u_{+})|\leq C\frac{\delta^{k+1}}{(1+\delta x)^{k+1}}+Ce^{-cx}$

Proof. Integrating the second equation in (2.1) over$V(\tilde{w})$for $k=0,1,$$(x, \infty)$$\ldots$.

and substituting$\tilde{v}=$in the resultant equation, we have

(2.3)$\tilde{w}_{x}=G_{2}(\tilde{w})^{-1}(f_{2}(V(\tilde{w}),\tilde{w})-f_{2}(v_{+}, w_{+}))$$= \tilde{A}(w_{+})(\tilde{w}-w_{+})+\frac{1}{2}G_{2}(w_{+})^{-1}D_{w}^{2}H(\tilde{w})(\tilde{w}-w_{+})^{2}+O(|\tilde{w}|^{3})$

.

(2.4)

The non-degenerate case can be proved easily because the condition $\mu_{1}(w_{+})<0$
yields that the equilibrium $w+$ of the system (2.3) is asymptotically stable. In
the degenerate case, we diagonalize the system (2.4) by employing a new unknown
functiondefined by$\tilde{z}(x)=(\tilde{z}_{1}, \ldots,\tilde{z}_{m_{2}})(x)$2 $:=P^{-1}(\tilde{w}-w_{+})$ ,We have the equation for$\tilde{z}$$P:=(r_{1}(w_{+}), \ldots, r_{m_{2}}(w_{+}))$

as$\tilde{z}_{1x}=h_{1}(\tilde{z})$,$\tilde{z}_{kx}=\mu_{k}(w_{+})\tilde{z}_{k}+h_{k}($wheresatisfies$h_{k}(\tilde{z})$

.

を

$)$

for $k=2,$$\ldots,$$m_{2}$,

is a nonlinear term. By a straightforward computation, we see that$h_{1}( \tilde{z})=\frac{1}{2}D_{w}\mu_{1}(w_{+})\cdot r_{1}(w_{+})\tilde{z}_{1}^{2}+O(|\tilde{z}|^{3})$$h_{1}$.

Therefore, using the fact that $\mu_{k}(w_{+})<0(k=2, \ldots, m_{2})$ and the center manifold
theorem, we obtain the result in Theorem 2.1-(ii).
$\square$154

Stability of stationary solution

3

In this section, we summarize the stability result of the non-degenerate stationary
solution, of which existence is shown in Theorem l-(i). We also show a brief outlineof a proof of a priori estimates. To do this, we have to assume a condition whichguarantee a dissipative structure of the system. This kind of dissipative structure
was studied mainly by Kawashima in $1980’ s$ , and the following condition was imposed in [3, 14].

[K] There exists an $m\cross m$ real matrix $K$ such that $KA^{0}(u_{+})$ is skew-symmetric
and $[KA(u_{+})]+B(u_{+})$ is positive definite, where $[A]$ $:=(A+\tau A)/2$ is asymmetric part of a matrix .
$A$

Shizuta and Kawashima in [13] prove the equivalence of the condition [K] and thefollowing condition [SK].
[SK] Let $\lambda A^{0}(u_{+})\phi=A(u_{+})\phi$ and $B(u_{+})\phi=0$ for$\phi=0$ .

$\lambda\in \mathbb{R}$and$\phi\in \mathbb{R}^{m}$. ThenKawashima proved the asymptotic stability of a constant state for the full space
problem under the condition [K] (or [SK]) in his doctor thesis [3]. The main purposeof the present paper is to show the asymptotic stability of the boundary layer solutionin half space under the condition [SK].
Theorem 3.1. Let
be a non-degenemte stationary solution shown in Theorem
l-(i). Assume that the condition [SK] (or [K]) holds. Then there exists a positive
constant such that if$\tilde{u}(x)$$\epsilon_{1}$$\Vert u_{0}-\tilde{u}\Vert_{H^{2}}+\delta\leq\epsilon_{1}$,the problem (1.3), (1.4) and (1.5) has a unique solution $u(t, x)$ globally in timesatisfying$u-\tilde{u}\in C([0, \infty), H^{2}(\mathbb{R}_{+}))$

Moreover the solution$u$.

converges to the stationary solution :$\tilde{u}$$\lim_{tarrow\infty}\Vert u(t)-\tilde{u}\Vert_{L^{\infty}}=0$.

The crucial point of a proof of Theorem 3.1 is to obtain a uniform a priori estimate$:=(v, w)-(\tilde{v},\tilde{w})$ be aof a perturbation from the stationary solution. Let
perturbation from the stationary solution. Then we have the equation foras$(\varphi, \psi)$$(\varphi, \psi)$$A_{1}^{0}(u)\varphi_{t}+A_{11}(u)\varphi_{x}+A_{12}(u)\psi_{x}=\tilde{g}_{1}$,$A_{2}^{0}(u)\psi_{t}+A_{21}(u)\varphi_{x}+A_{22}(u)\psi_{x}=B_{2}(u)\psi_{xx}+\tilde{g}_{2}$where$\tilde{g}_{1}$andprescribed as$\tilde{g}_{2}$,(3.la)(3.lb)

are non-linear terms. The initial and the boundary conditions are$(\varphi, \psi)(0, x)=(\varphi_{0}, \psi_{0}):=(v_{0}, w_{0})-(\tilde{v},\tilde{w})$$\psi(t, 0)=0$

.

,

(3.2)
(3.3)

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To summarize the a priori estimate, we define an energy norm $N(t)$ and a dissipative
norm $D(t)$ by$N(t):= \sup_{0\leq\tau\leq t}\Vert(\varphi, \psi)(\tau)\Vert_{H^{2}}$,.

$D(t)^{2}:= \int_{0}^{t}(\Vert\varphi_{x}(\tau)\Vert_{H^{1}}^{2}+\Vert\psi_{x}(\tau)\Vert_{H^{2}}^{2})d\tau$

Proposition 3.2. Let
be a solution to $(3.1)-(3.3)$ for acertain $T>0$ . Then there exists a positive constantsuch that $lfN(t)+\delta\leq\epsilon_{1}$ ,the solution satisfies$(\varphi, \psi)\in C([0, T];H^{2}(\mathbb{R}_{+}))$$\epsilon_{1}$$\Vert(\varphi, \psi)(t)\Vert_{H^{2}}^{2}+\int_{0}^{t}(\Vert\varphi_{x}(\tau)\Vert_{H^{1}}^{2}+\Vert\psi_{x}(\tau)\Vert_{H^{2}}^{2})d\tau\leq C\Vert(\varphi_{0}, \psi_{0})\Vert_{H^{2}}^{2}$

.

(3.4)

Proof. the proof of Proposition 3.2 is divided into several steps.

In this paper,
we only show the brief derivation of estimates for the solution up to first orderderivatives. The estimate for the second order estimate can be obtained similarly.
Step 1. Firstly, we obtain a lower order estimate of$(\varphi, \psi)$:$\Vert(\varphi, \psi)(t)\Vert_{L^{2}}^{2}+\int_{0}^{t}(\Vert\psi_{x}(\tau)\Vert_{L^{2}}^{2}+|\varphi(\tau, 0)|^{2})d\tau$$\leq C\Vert(\varphi_{0}, \psi_{0})\Vert_{L^{2}}^{2}+C\delta\int_{0}^{t}\Vert\varphi_{x}(\tau)\Vert_{L^{2}}^{2}d\tau$To get the estimate (3.5), we employ an energy form$\mathcal{E}$

.

(3.5)

defined by$\mathcal{E}:=\eta(f^{0}(u))-\eta(f^{0}(\tilde{u}))-D_{z}\eta(f^{0}(\tilde{u}))(f^{0}(u)-f^{0}(\tilde{u}))$.

Note that, if $N(t)$ is sufficiently small, the energy form is equivalent tobecause the Hessian matrix
is positive. From a direct computation, we see thatsatisfies$|(\varphi, \psi)|^{2}$$\mathcal{E}$$D_{z}^{2}\eta$$\mathcal{E}$$\mathcal{E}_{t}+\mathcal{F}_{x}+\langle B_{2}(u)\psi_{x},$ $\psi_{x}\}=\mathcal{B}_{x}+\mathcal{R}$

(3.6)

,

$\mathcal{F}:=q(u)-q(\tilde{u})-D_{z}\eta(f^{0}(\tilde{u}))(f(u)-f(\tilde{u}))$,,$\mathcal{B}:=(D_{z}\eta(f^{0}(u))-D_{z}\eta(f^{0}(\tilde{u})))(G(u)u_{x}-G(\tilde{u})\tilde{u}_{x})$.
whereis a remainder term satisfying
Integrating (3.6) overand using the assumption $A_{11}<0$ in [Al], we
get the estimate (3.5). Notice that we also utilize the Poincar\’e type inequality tocontrol remainder terms .
$|\mathcal{R}|\leq C|\tilde{u}_{x}|(|(\varphi, \psi)|^{2}+|(\varphi, \psi)||(\varphi_{x}, \psi_{x})|)$$\mathcal{R}$$(0, T)\cross \mathbb{R}+$$\mathcal{R}$

Step 2. Next we obtain estimates for first order derivatives$\varphi_{x}$. Namely we get$\Vert\varphi_{x}(t)\Vert_{L^{2}}^{2}+\int_{0}^{t}(\Vert^{T}A_{12}\varphi_{x}(\tau)\Vert_{L^{2}}^{2}+|\varphi_{x}(\tau, 0)|^{2})d\tau$$\leq C\Vert(\varphi_{0}, \psi_{0})\Vert_{H^{1}}^{2}+\int_{0}^{t}(\epsilon\Vert\varphi_{x}(\tau)\Vert_{L^{2}}^{2}+C_{\epsilon}\Vert\psi_{x}(\tau)\Vert_{L^{2}}^{2})d\tau+CN(t)D(t)^{2}$

(3.7)

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by using Matsumura-Nishida’s energy method in half space developed in [8], whereis a positive constant depending on . Preis an arbitrary positive constant andand integrate the resultant
cisely, we computeequality to get (3.7).
$C_{\epsilon}$$\epsilon$$\langle\partial_{x}(3.1a),$$\epsilon$$\varphi_{x}\rangle+\langle B_{2}^{-1}(3.1b),$

Step 3. Next we obtain the estimate for$\psi_{x}$$\tau A_{12}\varphi_{x}\}$:$\Vert\psi_{x}(t)\Vert_{L^{2}}^{2}+\int_{0}^{t}\Vert\psi_{xx}(\tau)\Vert_{L^{2}}^{2}d\tau$$\leq C\Vert(\varphi_{0}, \psi_{0})\Vert_{H^{1}}^{2}+C\int_{0}^{t}\Vert^{T}A_{12}\varphi_{x}(\tau)\Vert_{L^{2}}^{2}d\tau+CN(t)D(t)^{2}$,

which can be obtained by computing an inner product $\{(3.1b), -\psi_{xx}\}$ .

Step 4. Finally, we obtain the dissipative estimate foras follows:$\varphi_{x}$by using the condition$\overline{[K]}$$\int_{0}^{t}\Vert\varphi_{x}(\tau)\Vert_{L^{2}}^{2}d\tau\leq C\Vert(\varphi_{0}, \psi_{0})\Vert_{H^{1}}^{2}+C\Vert(\varphi_{x}, \psi_{x})(t)\Vert_{L^{2}}^{2}$$+C \int_{0}^{t}(\Vert\psi_{xx}(\tau)\Vert_{L^{2}}^{2}+|\varphi_{x}(\tau, 0)|^{2})d\tau+CN(t)D(t)^{2}$. (3.8)Combining the estimates from Step 1 to Step 4, we obtain the estimate up to
the first derivatives as$\Vert(\varphi, \psi)(t)\Vert_{H^{1}}^{2}+\int_{0}^{t}(\Vert\varphi_{x}(\tau)\Vert_{L^{2}}^{2}+\Vert\psi_{x}(\tau)\Vert_{H^{1}}^{2})d\tau\leq C\Vert(\varphi_{0}, \psi_{0})\Vert_{H^{1}}^{2}+CN(t)D(t)^{2}$.

By a similar computation, we get the estimate for the second order derivatives.
Combining these estimates, we obtain the desired estimate (3.4).

$\square$

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