Nonlinear Stability of Strong Rarefaction Waves for Compressible Navier-Stokes Equations (Mathematical Analysis in Fluid and Gas Dynamics)

Nonlinear

Stability

of

Strong

Rarefaction Waves

for Compressible

Navier-Stokes

Equations

Kenji

Nishihara

School of Political Science

and

Economics

Waseda University,

Tokyo

169-8050,

Japan

Tong Yang

Department

of

Mathematics,

City

University

of Hong

Kong

Tat

Chee

Avenue,

Kowloon,

Hong Kong

Huijiang Zhao

Wuhan Institute

of

Physics and Mathematics

The

Chinese Academy of

Sciences,

Wuhan 430071, P.

R. China

and

School

of Political

Science

and Economics

Waseda

University, Tokyo 169-8050, Japan

1

Introduction and

main

results

Consider the one-dimensional compressible Navier-Stokes equations in the Lagrangian

coordinates

$\{$

$v_{t}-u_{x}=0,$

$u_{t}+p_{x}=(\mu_{v}^{\underline{u}_{\mathrm{a}}})_{x}$

,

$(e+ \frac{u^{2}}{2}),$ $+(up)_{x}=( \kappa_{v}^{\underline{\theta}_{\mathrm{a}}}+\mu\frac{uu}{v})_{\sim}$ ,

(1.1)

where theunknowns$v>0,u$,$\theta>0$

,

_$p,.e$

,

and$s$represent the specific volume, the velocity,

the absolute temperature, the pressure, the internal energy, and the entropy ofthe

gas

respectively. The coefficients of viscosity and heat-conductivity, $\mu$and $\kappa$

, are

assumed to

be positive constants. We assume,

as

is usual in thermodynamics, that by any giventwo

of the five thermodynamical variables, $v,p$,$e$,$\theta$, and

$s$, the remaining three variables

are

expressed.

The second law of thermodynamics asserts that

$flds=de$ $pdv$,

from which, if

we

choose $(v, 0)$, $(v, s)$,

or

$(v, e)$

as

independentvariablesandwrite $(p,e,s)=$

83

Kenji Nishihara, Tong Yang, and Huijiang Zhao

can

deduce that

$\{$

$s_{v}(v, \theta)=p_{\theta}(v, \theta)$,

$s_{\theta}(v, \theta)=\underline{e_{\theta}}\mathrm{L}_{\theta}^{v},\Delta^{\theta}$,

$e_{v}(v, \theta)=\theta p_{\theta}(v, \theta)-p(v, \theta)$,

(1.2)

$\tilde{e}_{v}(v, s)$ _{$=-p(v, \theta)$}, $\tilde{e}_{s}(v, s)=\theta$,

$\tilde{p}_{v}(v, s)$ $=p_{v}(v, \theta)-\frac{\theta(p_{\theta}(v,\theta))^{2}}{e_{\theta}(v,\theta)}$, $\tilde{p}_{s}(v, s)=\frac{\theta pe(v,\theta}{e_{\theta}(v,\theta_{I}’}$

$\tilde{\theta}_{v}(v, s)$ $=-_{e_{\theta}}^{\theta_{\theta}}\ovalbox{\tt\small REJECT}_{v}^{v,\theta},$, $\tilde{\theta}_{s}(v, s)=\overline{e_{\theta}}v\varpi\theta$

$=p_{v}$$(v, \theta)-\frac{\theta(p_{\theta}(v,\theta))^{2}}{e_{\theta}(v,\theta)}$, $\tilde{p}_{s}(v, s)=\frac{\theta pe(v,\theta)}{e_{\theta}(v,\theta)}$

,

$\angle$

,

(1.3)

or

$\{$

$\hat{s}_{e}(v, e)=F\mathrm{l},$ $\hat{s}_{v}(v, e)$ _$=$ $\mathit{1}\mathit{7}j^{J}$

,

$\hat{p}_{e}(v, e)$ $= \frac{p_{\theta}(v,\theta)}{e_{\theta}(v,\theta)}$, $\hat{p}_{v}(v, e)=(p_{v}(v, \theta)-\frac{\theta(p_{\theta}(v,\theta))^{2}}{e_{\theta}(v,\theta)})+\frac{\mathrm{p}(v,\theta)\mathrm{p}_{\theta}(v,\theta)}{e_{\theta}(v,\theta)}$, $\hat{\theta}_{e}(v, e)=\neg e_{\theta}\neg v,\theta 1$, $\hat{\theta}_{v}($

”$e)= \frac{\mathrm{p}(v,\theta)-\theta p_{\theta}(v,\theta)}{e_{\theta}(v,\theta)}$

.

(1.4)

From (1.3) and (1.4),

we

get that

$\tilde{p}_{v}(v, s)$ $=\hat{p}_{v}(v, e)$ $-p(v,\theta)\hat{p}_{e}(v,e)$. (1.5)

What

we

are

interested in this paper is to show that the strong expansion

waves

for

(1.1)

are

nonlinear stable. For this, it is convenient to work with the equations for the

entropy $s$ and the absolute temperature $\theta$, $i.e$

.

$s_{t}=\kappa$ $( \frac{\theta_{x}}{v\theta})_{x}+\kappa\frac{\theta_{x}^{2}}{v\theta^{2}}+\mu\frac{u_{x}^{2}}{v\theta}$ (1.6)

and

$\theta_{t}+\frac{\theta p_{\theta}(v,\theta)}{e_{\theta}(v,\theta)}u_{x}=\frac{\kappa}{e_{\theta}(v,\theta)}(\frac{\theta_{x}}{v})_{x}+\frac{\mu}{e_{\theta}(v,\theta)}\frac{u_{x}^{2}}{v}$

.

(1.7)

In fact, for smooth solutions, equations (l.l)i, $(1.1)_{2}$, $(1.1)_{3}$

are

equivalent to equations

$(1.1)_{1}$

,

$(1.1)_{2}$

,

(1.6)

or

$(1.1)_{1}$

,

$(1.1)_{2}$

,

(1.6). In what follows,

we

will consider $(1.1)_{1}$

,

$(1.1)_{2}$

,

(1.6) with the initial data

$(v,u,\mathrm{s})(t, x)|_{t=0}$ $=(v_{0},u_{0}, s_{0})(x)arrow(v_{\pm},u_{\pm}, s_{1})$ as $xarrow\pm \mathrm{o}\mathrm{o}$. (1.8)

Here $v_{\pm}>0$, $u\pm$, $s_{\pm}$

are

constants. Since

we

will focus

on

the expansion

waves

to (1.1), we

assume

that $s_{+}=s_{-}=\overline{s}$iinthe rest of this paper.

For expansion waves, the right-hand side of (1.1) decays faster than each term

on

the left-hand side. Therefore, the compressible Navier-Stokes equations (1.1) may be

approximated, time- symptotically, by the compressible Euler equations

$\{$

$v_{t}-u_{x}=0,$

$u_{t}+\tilde{p}(v, s)_{x}=0,$

$s_{t}=0.$

There

are

two families of expansion (rarefaction)

waves

for (1.9) which

are

solutions

of the compressible Euler equations (1.9) with Riemanndata $(v_{0}^{R}$,$u_{0}^{R}$

:$s_{0}^{R})(x)$ (cf. [1]),

$(v, u, s)(t, x)|_{t=0}=(v_{0}^{R}$,$u_{0}^{R}$,_{$s_{0}^{R})(x)=\{$}

$(v_{-}, u_{-}, s_{-})$, $x<0,$ $(v_{+}, u_{+}, s_{+})$, $x>0.$

(1.10)

For illustration, we only consider the 1-rarefaction

wave

(

$V^{R}$,$U^{R}$,$S^{R}$

)

_{$(t, x)$}, which is

characterized by

$\{$

5’

$(t, x)$ $=\overline{s}$,

$U^{R}(t_{1}x)-fVR(t,x) \sqrt{-\tilde{p}_{v}(z,\overline{s})}dz=u\pm-\int^{v\pm}\sqrt{-\tilde{p}_{v}(z,\overline{s}}dz$,

$\lambda_{1x}$

(

$V^{R}(t,x)$,$S^{R}(t,x))>0,$ Pw$\{\mathrm{v},$$s)=-\mathrm{q}$

.

(1.11)

The

case

forthe 3-rarefaction

wave

can

be discussed similarly.

Before statingthe main results, we first list the assumptions on the pressure function

$p(v, \theta)$ and the internal energy $e(v, \theta)$ used throughout this paper:

(H2) $Pw\{v,$$\theta)=\frac{\partial p(v,\theta)}{\partial v}<0,$ $\mathrm{e}(\mathrm{v}, \theta)=\frac{\partial e(v,\theta)}{\partial\theta}>0$ and

$(\mathrm{H}_{2})$ $\tilde{p}_{vv}(v, s)$ $= \frac{\partial^{2}\tilde{p}(v,s)}{\partial v^{2}}>0$ and $\tilde{p}(v, s)$ is

convex

with respect to $(v, s)$

.

From (1.3) and (Hi),

we can

deduce that

$\tilde{p}_{v}(v, s)$ $=$Pw{v,$\theta$) $- \frac{\theta(p_{\theta}(v,\theta))^{2}}{e_{\theta}(v,\theta)}<0$, (1.12)

$\{$

$\tilde{e}_{ss}(v, s)$ $=>0\overline{e}_{\theta}\mathrm{T}^{\theta}v,\neg\theta$

’

$\tilde{e}_{vs}(v, s)$ $=\ovalbox{\tt\small REJECT}\theta_{\theta,e_{\theta}’ v}v,’\theta$,

$\tilde{e}_{vv}(v, s)$ $=-p_{v}(v, \theta)+\frac{\theta(\mathrm{p}_{\theta}(v,\theta))^{2}}{\rho_{\Delta}\mathrm{f}r-\theta 1},>0,$

and

$\tilde{e}_{\mathit{8}S}(v, s)\tilde{e}_{vv}(v, s)-(\tilde{e}_{vs}(v, s))^{2}=-\frac{\theta p_{v}(v,\theta)}{e_{\theta}(v,\theta)}>0.$ (1.13)

Equation (1.13) implies that $\tilde{e}(v, s)$ is

convex

with respect to $v$ and $s$

.

Consequently,

$\tilde{e}(v, s)$ $+ \frac{1}{2}u^{2}$ $\mathrm{i}$ astrictly

convex

function of$(v, u, s)$

.

Now

we can

construct the following

normalized entropy $\mathrm{r}$

}

$(\mathrm{v}, u, s;V, U, S)$ around $(V, U, S)(t, x)$, which is the smooth

approxi-mation of the 1-rarefaction

waves

(

$V^{R}$,$U^{R}$

,

$S^{R}$

)

_{$(t, x)$},

V$( \mathrm{v}, s;V, U, S)=(e(v, \theta)+\frac{\mathrm{u}^{2}}{2})-(e(V, \ominus)+\frac{U^{2}}{2})$

95

Kenji Nishihara, Tong Yang, and Huijiang Zhao

Here we have used the fact that $\tilde{e}_{v}$$(v, s)$ $=-p(v, \theta),$ $e\sim s$$(v, s)$ $=\theta$

.

And the approximate

rarefactionwaves $V(t, x)$,$U(t, x)$,$S(t, x)$, and$\ominus(t, x)$ are constructed asfollows (

cf.

[20]).

Given asuitablysmallbut fixed constant$\epsilon$ $>0,$ let$w(t, x)$ be the unique global smooth

solution to the Cauchy problem

$\{$

$w_{t}+ww_{x}=0,$

$w$(t,$x$)$|_{t=0}=w_{0}(x):=$

(1.15) then, $V(t, x)$,$U(t, x)$,$\mathrm{S}(\mathrm{t}, x)$, and $\mathrm{V}(\mathrm{t}, x)$

are

defined by

$\{$

$\lambda_{1}(V(t, x),\overline{s})=-\sqrt{-\tilde{p}_{v}Vt,x,\overline{s}})$$-=$to(t,$x$),

$U(t, x)=u_{\pm}+ \int_{v\pm}^{V(t,x)}\sqrt{-\tilde{p}_{v}(z,\overline{s}}dz$, $S(t, x)=\overline{s}$,

$\Theta(t, x)$ $=\tilde{\theta}(V(t,x),\overline{s})$.

(1-10)

Under the above preparation, for the general

gas,

our

stability result

on

strong

rar-efaction

waves

(

$V^{R}$,$U^{R}$,$S^{R}$

)

_$(t,x)$

can

be stated

as

in the following.

Theorem 1.1 (Local Stability Result for General Gas) Assume that $(V^{R},$ $I^{R}$,

$S^{R})(t, x)$ is the

1-rarefaction

wave solution to the Riemann problem

of

the compressible

Euler equations (1.9), (1.10) and that the initial data $(v_{0}, u_{0}, s_{0})(x)$

of

the compressible

Navier-Stokes equations $($1.$\mathit{1})_{1}$, $($1.$1)\mathrm{i}$, (1-6)

satisfies

$($1.$S)$,

$0<2\underline{V}\leq$w0(x) $V(t, x)\leq \mathit{5}\overline{V}$,

(1.17)

$0<2\underline{\ominus}\leq\theta_{0}(x),$$\ominus(t, x)\leq\frac{1}{2}\overline{\ominus}$

for

all $(t, x)\in \mathrm{R}_{+}\mathrm{x}\mathrm{R}$and

some

positive constants $\underline{V}$,$V,\underline{\Theta}$, and$\Theta$, and

$,.\iota$ .VV $\backslash \cdot$,$\mathrm{w}’ \mathrm{c}$ $\sim\sim+’\backslash -\cdot\cup’\nu \mathrm{u}$ $.\vee\prime\prime y\cdot F..\nu\nu\nu\vee\cdot.\vee’\nu\cdot-\cdot\cdot\nu\cdot\underline{\mathrm{r}}$_’ $\mathrm{r}$

’$\underline{\vee}$, $\ldots\vee\vee’rightarrow\cdot\vee-$

$N(0)=||$$(1^{)}0(x)-V(0, x)$,$u_{0}(x)-U(0, x)$,

so

$(x)-\overline{s})$$||_{H}2(\mathrm{R})$

is sufficiently small Then the Cauchy problem (1.1), (1.8) admits a unique global smooth

solution $(v, u, s)(t, x)$ satisfying

$\lim_{tarrow+\infty}$

s

nug

$1$ $|$

(

$v(t, x)-V^{R}(t, x)$,$\mathrm{V}(\mathrm{t}, x)-U^{R}(t, x)$,$\mathrm{S}(\mathrm{t}, x)-\overline{s}$

)

$||$ $=0.$ (1. 8)

Note that the essential meaning of nonlinear stability of rarefaction

waves

to the

compressible Navier-Stokes equations (1.1), (1.8) in [12], [15], [20], [21], [22] is that if

$(v_{0},u_{0}, s_{0})(x)$ is

a

(small

or

large) perturbation of $(V(0, x)$,$U(0,x)$,$\overline{s})$, the smooth

ap-proximation of the rarefaction

wave

solutions

(

$V^{R}(t, x)$,$U^{R}(t, x),\overline{s}$

)

, then the Cauchy

problem ofthe compressible Navier-Stokes equations (1.1), (1.8) admits

a

unique global

smooth solution $(v, u, s)(t, x)$ whichtendstime-asymptoticallyto $(V^{R}(t, x),$ $U^{R}$(t,_$x$),$\overline{s}$

).

In this sense, the result obtained in Theorem 1.1 does imply the nonlinear stability of

strong rarefaction

waves

for the compressible Navier-Stokes equations. But, due to the

be small, the nonlinear stability result obtainedin Theorem 1.1 is essentially local. Thena

natural questionofimportance and interest is how to get the global stability result which

is for large perturbation. Our secondpurpose is to devote to thisproblem and show that,

forthe ideal polytropic gas, sucha_{global stability result indeed holds for 7} near1 without

the weakness ofthe rarefaction

waves.

Tostate the result precisely,

we

recall that for the

ideal polytropic

gas,

$(p, e)(v, \theta)$ have the following specialconstitutive relations

$\mathrm{p}(\mathrm{v}, =\frac{R\theta}{v}=Av^{-\gamma}\exp(\frac{\gamma-1}{R}s)$, $\mathrm{e}(\mathrm{v}, \theta)=\frac{R\theta}{\gamma-1}$, (1.19)

where $R>0$ is the

gas

constant, $\gamma>1$ the adiabatic constant, and $A$

a

positive constant.

Our secondresult is stated

as

follows.

Theorem 1.2 (Global Stability Result for the Ideal Polytropic Gas)

Assume

that

(

$V^{R}$(t,_$x$), _{$U^{R}(t, x)$},$\overline{s}$

)

is the

1-rarefaction

wave

solution

of

the Riemannproblem

of

the compressible Euler equations (1.9), (1.10) and that ($p,$$\mathrm{e}(\mathrm{v}, \theta)$ satisfy the

constitu-tive relations (1. i9). Then

_for

any $(\mathrm{v}\mathrm{o}(\mathrm{x})-V(0,x),u_{0}(x)-U(0, x),$$(\mathrm{v}\mathrm{o}(\mathrm{x})-\overline{s})\in H^{2}(\mathrm{R})$

satisfying (1.17) and its $H^{1}(\mathrm{R})$

-norm

to be bounded by a constant independent

of

$\frac{1}{\epsilon}$,

the corresponding Cauchy problem (1.1), (1.8) admits

a

unique global smooth solution

$(v,u, s)(t, x)$ satisfying (1.18) provided that

7-1

is sufficiently small.

In the proofof Theorem 1.2, the assumptionthat $\gamma$ is close to 1 is used for obtaining

the a priori assumption $0<\underline{\ominus}<$ 0(ty$\mathrm{x}$)

$<\overline{\ominus}$for _{$(t, x)\in[0, \infty]\mathrm{x}\mathrm{R}$}

so

that

$\theta(t,x)-$

$\Theta(t, x)$ is small. Hence,

one can

image that for the isentropic polytropic

gas,

such

a

smallness assumption

can

be removed and this has been obtained by A. Matsumura and

K. Nishihara in [21], [22] by cleverly introducing another type ofsmooth approximation

of the rarefaction

wave

solution. That is, $w_{0}(x)$ in $(1.15)_{2}$ is replaced by

$w$(t,$x$)$|_{t=0}= \underline{w}_{0}(x)=\frac{\lambda_{1}(v_{-},\overline{s})+\lambda_{1}(v_{+},\overline{s})}{2}+\frac{\lambda_{1}(v_{+},\overline{s})-\lambda_{1}(v_{-},\overline{s})}{2}.K_{q}\int_{0}^{\epsilon x}(1+y^{2})^{-q}$dy,

(1.20)

where $K_{q}>0$ is

a

constant satisfying

$K_{q}/+\infty(1+y^{2})^{-q}dy=1$ (1.21)

for

some

suitably largeconstant $q>0.$

Our thirdpurpose is toshowthe global stabilityresult onstrong rarefactionwaves for

$p$-system with viscosity with

a

generalpressure $p=p(v)$

.

To state this result,

we

recall

that the isentropic compressible Navier-Stokes equations in Lagrangian Coordinates can

be writtenas

$\{$

$v_{t}-u_{x}=0,$

$u_{t}+p(v)_{x}=\mu$$(\begin{array}{l}-u_{\mathrm{A}}v\end{array})x$,

(1.22)

withthe initial data

$(v,u)(t,x)|_{t=0}=(v_{0},u_{0})(x)arrow(v\pm,u\pm)$

as

$xarrow\pm\infty$. (1.23)

Here $v\pm>0$ and $u\pm$

are

given constants

so

that the Riemann problem of the isentropic

compressible Euler equations

$\{$

$v_{t}-u_{x}=0,$ $u_{t}+p(v)_{x}=0,$

87

Kenji Nishihara, Tong Yang, and Huijiang Zhao

with the Riemann data

$(v, u)(t, x)|_{t=0}$ $=(\overline{v}_{0}^{R},\overline{u}0$$)(x)=\{$

$(v_{-}, u_{-})$, $x<0,$ $(v_{+}, u_{+})$, $x>0,$

(1.25) is assumed to admit

a

unique 1-rarefactionwave solution $(\overline{V}^{R},\overline{U}^{R})$ $(t, x)$

.

We only

assume

that$p(v)$ is a positive smooth function for $v>0$ and satisfies

$p’(v)<0,$ $p’(v)>0$ for $v>0.$ (1.26)

Under the above assumptions, we have the followingtheorem.

Theorem 1.3 (Global Stability Result for General Isentropic Gas) Assume that

theRiemannproblem (1.24), (1.25) to thecompressible Eulerequationsadmits

a

unique

1-rarefaction

wave

solution(

$\overline{V}^{R}$

,$\overline{U}^{R}$

)

$(t, x)$ and that $(\overline{V},\overline{U}$

)

$(t, x)$ is

a

smoothapproimation

of

the Riemann solution $(\overline{V}^{R},\overline{U}^{R})(t,x)$ constructed by $\{$

$\overline{V}(t,x)=\lambda_{1}^{-1}\mathrm{w}(\mathrm{t}, x))$, $\mathrm{p}(\mathrm{v})=-\sqrt{-p’(v)}$,

$\overline{U}(t, x)=u\pm+\int_{v\pm}^{\overline{V}(t,x)}\sqrt{-p’s}ds$

.

(1.27)

Here $\overline{w}(t, x)$ is the unique smooth solution to the following Cauchy problem

$\{$

$w_{t}+-$$ww_{x}=0,$

$w(t, x)|_{t=0}=\overline{w}_{0}(x)=$

(1.28)

Then

_for

any$p(v)$ satisfying (1.26) and $(v_{0}(x)-\overline{V}(0, x),$$\mathrm{w}\mathrm{o}(\mathrm{x})-\overline{U}(0, x))\in H^{2}(\mathrm{R})$

sat-isfying $0<2\underline{V}\leq v_{0}(x),\overline{V}$(t,$x$) $\leq\frac{1}{2}\overline{V}$

for

all $(t, x)\in \mathrm{R}_{+}\cross \mathrm{R}$ and

some

positive constants

$\underline{V}$, $\overline{V}$ and with its_{$H^{1}(\mathrm{R})$}

-norm

bounded by

a constant

independent

of

the quantity$\frac{1}{\epsilon}$, the

Cauchy problem (L22), (1.23) admits

a

unique globalsmooth solution $(v, u)(t, x)$ satisfying

$\mathrm{t}arrow \mathrm{z}\mathrm{l}\mathrm{i}$

$\sup_{x\in \mathrm{R}}\{|$

(

$v$$-\overline{V}^{R},u-\overline{U}$

’)

$(t, x)|\}=0.$ (1.29)

Remark 1.1 In [21] and [22], the assumption that$p(v)=v^{-\gamma}(\gamma\geq 1)$ plays

an

essen-tial role in the analysis and it is worth to pointing out that

even

by using their smooth

approimation

_of

the Riemann solutions, their arguments can not be applied to the

case

when $p(v)$

satisfies

only (1.26). However we have assumed that the $H^{1}(\mathrm{R})-nom$

of

the

initial perturbation is bounded by

a

constant independent

_of

$\frac{1}{\epsilon}$ with small

fied

number $\epsilon$ $>0.$ This implies that the data $(v_{0}, u_{0})(x)$

for

(1.23) is initially rather

flat

though

(

$v_{0}(x),\overline{V}(0, x)$,$\mathrm{w}\mathrm{o}(\mathrm{x})-\overline{U}(0, x)$

)

may be large. So,

we

should seek

for

the global solution

and its behavior

_for

any data ($v_{0}$, wo (x) with $||$$(\mathrm{y}\mathrm{o}(\mathrm{x}) -v\pm, u_{0}(x)-u_{\pm})$$||_{H^{1}(\mathrm{R})}\pm$ bounded.

This will be done under

some

additional assumptions on$p(v)$ in Theorem

1.4.

In Theorem 1.1, 1.2, and 1.3,

we

assume

that the solutions to the corresponding Rie

mann

problem of the compressible Euler equations consists of only

one

rarefaction

wave.

the proof of the theorems. To simplify the presentation, we use the isentropic

compress-ible Navier-Stokes equations to explain this Suppose that the solution

(

$\overline{V}^{R},\overline{U}$

’)

$(t, x)$ to

the Riemann problem (1.24), (1.25) consists of

one

1-rarefaction

wave

$(\overline{V}_{1}^{R},$$\overline{U}_{1}^{R})(t, x)$

and

one

2-rarefaction

wave

$(\overline{V}_{2}^{R},\overline{U}_{2}^{R})(t, x)$

.

That is, there exists

a

unique constant

state $(\mathrm{v},\mathrm{u})\in \mathrm{R}^{2}$ such that $(v_{-}, u_{-})$ and $(\mathrm{v},\mathrm{u})$

are

connected by one 1-rarefaction wave $(\overline{V}_{1}^{R},\overline{U}\mathrm{r})$ $(t, x)$, $\mathrm{i}.\mathrm{e}.$, $(\mathrm{v},\mathrm{u})$ _{$\in R_{1}(v_{-}, u-)$}, while $(\mathrm{v},\mathrm{u})$ and $(\mathrm{V}\mathrm{R}, u_{+})$

are

connected by one

2-rarefaction

wave

(

$\overline{V}_{2}^{R}$,$\overline{U}\mathit{2}$

)

$(t, x)$, $i.e.$, $(v_{+}, u_{+})$ $\in R_{2}(\overline{v},\overline{u})$. Here

$R_{1}(v_{-},u_{-})=R_{2}(\overline{v},\overline{u})=\{$ $(v, u)$

$(v, u)$

$u=u_{-}+ \int_{v-}^{v}\sqrt{-\emptyset(s)}ds$, $u\geq u_{-}$ $\mathit{1}$ ,

$u= \overline{u}-\int^{v}\overline{.,}\sqrt{-\mu(s)}ds$, $u\geq\overline{u}\}$

.

(1.30)

Consequently

$(\overline{V}^{R},\overline{U}’)$$(t,x)=(\overline{V}_{1}^{R}(t,x)+\overline{V}$

2

$(t, x)-\overline{v},\overline{U}$

!:(t,

$x$) $+\overline{U}_{2}^{R}(t,x)-\overline u).$ (1.31)

Let $\overline{w}_{i}(t, x)(i=1,2)$ be the unique global smooth solution to the following Cauchy

problem

$\{$

$\overline{w}_{it}+\overline{w}_{i}\overline{w}_{\dot{|}x}=0,$

$\overline{w}_{\dot{l}}(t, x)|_{t=0}=\overline{w}_{i}0(x)=\overline{w}_{*-}$. $\mathrm{S}\overline{w}.\cdot$ $+\overline{w}.\cdot-\overline{w}.\cdot-\tanh(\tilde{2}\mathrm{g}x)$, $i=1,2$,

(1.32)

then,

as

in [20], the smooth approximate solution $(\overline{V},\overline{U})$ $(t, x)$ of $(\overline{V}^{R},\overline{U}^{R})$$(t, x)$ is

con-structed

as

follows:

(

$\overline{V}$

,

$\overline{U}$

)

_{$(t, x)=(\overline{V}_{1}(t, x)+\overline{V}_{2}(t, x)-\overline{v},\overline{U}_{1}(t, x)+\overline{U}_{2}(t, x)-\overline{u})$}

,

(1.33)

where

(

$\overline{V}_{1}$,$\overline{U}_{1}$

)

$(t,x)(\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{p}.$ $(\overline{V}_{2},\overline{U}_{2})$ $(t, x))$ is definedby

$\{$

$\lambda_{1}(\overline{V}_{1}(t, x))=\overline{w}_{1}(t, x)$, $(\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{p}.$ $\lambda_{2}(\overline{V}_{2}(t,x))=$VR(t,$x$)$)$

$\overline{U}_{1}=u_{-}+\int_{v}i^{1}t,x)$$\sqrt{-p’s)}$ds, $($resp. $\overline{U}$

2$(t, x)= \overline{u}-\int_{\overline{v}}’ 2(t,x)\sqrt{-ds}ds)$

(1.34)

and$\overline{w}$

1$(t, x)$ (resp. $\overline{w}$

2($t$,_$x$)) is the solution of (1.32) with$\overline{w}_{1-}=\lambda_{1}(v_{-})$ and$\overline{w}_{1+}=\lambda_{1}(\overline{v})$,

(resp. $\overline{w}_{2-}=$ A2(v) and$\overline{w}_{2+}=\lambda_{2}(v_{+})$).

It is easy todeduce that the smooth functions $(\overline{V},\overline{U})$ $(t, x)$ satisfies the system

$\{$

$V_{t}-U_{x}=0,$

$\overline{U}_{t}+p$$(\overline{V})_{x}=\mathit{9}(\overline{V})_{x}$,

(1.35)

where9$(\overline{V})=p(\overline{V})-p$$(\overline{V}_{1})-p$ $(\overline{V}_{2})+p$$(\mathrm{v})$

.

Hence,

we

onlyneedto control$g(\overline{V}(t, x))_{x}$

suitably in thiscase. Notice thatfromthe propertiesonthe smooth approximation of the

93

Kenji Nishihara, Tong Yang, and Huijiang Zhao

$\int_{0}^{t}||g$$(\overline{V}(\tau))_{x}||_{L^{\mathrm{p}}(\mathrm{R})}d\tau\leq O(1)\epsilon^{-\frac{1}{p}}$. (1.37)

Prom this observation together with the fact that, indeducing

our

mainresults,

we

need

the smallness of$\epsilon$,

a

quantity introduced inthe construction of thesmooth approximation

to the rarefaction wave solutions, to close the energy estimates, it

seems

hopeless to

use

our

method to deal with the nonlinear stability of the superposition of rarefaction

waves

of different families.

We note, however, that $g(\overline{V}(t, x))_{x}$ satisfies the following estimate(c/. [20]): There

exist constants $C>0$,$\alpha>0$ such that for $t\geq 0$,$x\in \mathrm{R}$

$|\mathit{9}(\overline{V}(t,x))_{x}|\leq Ce$$\exp(-\alpha\epsilon(|x|+t))$. (1.37)

Prom (1.37),

we can see

that, like those for the study ofnonlinear stability of travelling

wave

solutions to dissipative hyperbolic systems ofconservationlaws,if

we

givethe smooth

approximation $\overline{V}(t, x)$

a

shift, that is, if

we

let $\vec{V}(t, x)=\overline{V}(t+t_{0}, x)$ with$t_{0}>0$ being

a

suitably chosenfixed constant, then

we

have for $\overline{V}(t, x)$ that

$\int_{0}^{t}||g$$(\vec{V}(\tau))_{x}||_{L^{p}(\mathrm{R})}\leq O(1)\epsilon^{-\frac{1}{p}}\exp(-\alpha\epsilon t_{0})$

.

(1.38)

If

we

let for example $t_{0}=\epsilon^{-2}$, the right-handof (1.38) is controledby $O(1) \epsilon^{-\frac{1}{\mathrm{p}}}\exp(-\frac{\alpha}{\epsilon})$

which can be as small as we wanted ifwe choose $\epsilon$ $>0$ sufficiently small. Consequently,

our method

can

indeed be applied directly to deal with the nonlinear stability of the

superposition ofrarefaction

waves

ofdifferent famih.es provided that

we

approximatethe

rarefaction wave solutions by $V$ $(t, x)(\mathrm{N}\mathrm{o}\mathrm{t}\mathrm{e}$ that in this case, the initialdata $(v_{0}, u_{0})(x)$

ofthe compressible Navier-Stokes equations (1.24) is a perturbation of $(\overline{V},\overline{U})(t_{0},x).)$

In Theorems 1.2 and 1.3,

we

assume

that the $H^{1}$

-norm

of the initial perturbation is

bounded by

a

constant independent of $\frac{1}{\epsilon}$, which is excluded under additional assumption

$\{$

$\mathrm{p}(\mathrm{v})2$$C_{1}^{-1}v^{-1}$, $($ _{$(\mathrm{p}(\mathrm{v})\geq v|p’(v)|=-vp’(v))$} $\geq C_{1}^{-1}$ $(0<v\leq 1)$,

$-p’(v)\geq C_{1}^{-1}v^{-C_{1}}$ $(v\geq 1)$

(1.39)

for arbitrarily fixed constant $C_{1}>2.$ Note that (1.39) derives

$\{$

$C_{1}^{-1}v^{-1}\leq$ p(v) $\leq p(1)v^{-C_{1}}$ _{$(0<v\leq 1)$},

$p(v)\geq p(\infty)f$

$\frac{C_{1}C_{1}-1}{}$ $(v\geq 1)$

.

$v^{1-C_{1}}$

(1.40)

Hence, though (1.40) is not sufficient condition for (1.39), the assumption (1.39), roughly

speaking,

seems

to be reasonable including the typical pressuremodel$p(v)=v^{-\gamma}(\gamma\geq 1)$

.

Then

we

have the final theorem.

Theorem 1.4 Assume that$p(v)$

satisfies

(1.26) and (1.39) and that the solution $(\overline{V}^{R}$,

$\overline{U}^{R}E$,$x)$ to the Riemann problem (1.24), $(1_{-}25)$ is _given by (LSI). Let $(\overline{\overline{V}},\overline{\overline{U}})(t, x)$ be

a smooth approximation

_of

the Riemann solution

(

$\overline{V}^{R}$,

$\overline{U}’ \mathrm{E}$,$x$) constructed by (1.33)-$(\mathit{1}.S\mathit{4})$ with$\overline{w}_{i}\mathrm{o}(x)$ in (1.32) being replaced by

for

$q> \frac{3}{2}$ and $K_{q}$ satisfying (1.21).

Then

_for

any

(

$v_{0}(x)-$$\mathrm{p}(0, x)$,_{$u_{0}(x)-$}$\mathrm{f}\mathrm{f}^{\mathrm{i}}(\mathrm{O}, x))\in H^{2}(\mathrm{R})$ satisfying$0<2\underline{V}\leq v_{0}(x)$,

$\overline{\overline{V}}(t, x)$ $\leq\frac{1}{2}\overline{V}$

for

all $(t, x)\in \mathrm{R}_{+}$

x

$\mathrm{R}$ and some positive constants $\underline{V},$ $\overline{V}_{j}$ the Cauchy

problem (1.22), (1.23) admits

a

uniqueglobal smooth solution ($v$,$u\mathrm{v}$$(t, x)$ satisfying (L29)

Now we outline the main ideas

we

used in proving

our

main results. The main

new

ingredient in

our

analysis is to introduce two quantities $\epsilon$ and $t_{0}$ in the construction of

thesmooth approximation of the rarefaction

wave

solutions to control the possible growth

caused by the nonlinearity of the systems and by the interactions of

waves

from different

families respectively. As to the global stability results, the key point is to get the uniform

lower bound for $v(t, x)$ and our main observation for the isentropic

case

is that if$p(v)$

satisfies (1.26), then we

can

deduce that there exists

a

positive conctant $C_{2}>0$such that

$\Phi(V, z)$ $\geq C_{2}\frac{z^{2}}{z+2V}$

.

(1.41)

Such

an

estimates plays

an

important role in

our

proving Theorem

1.3

and Theorem

1.4.

Here $\Phi(V,z)$ $=p(v)$ $- \int_{V}^{V+z}p(s)$ds.

Remark 1.2 It is worth to pointing out that the large time behavior

of

solutions to the

compressible Navier-Stokes equations (1.1), (1.8) has been studied by many people,

_cf.

[1-24] and the

_references

cited therein. When the initial data $(v0, u_{0}, s_{0})(x)$ is

a

small

perturbation

_of

a non-vacuum

constant state, $i$

.

$e.$, $v_{-}=\mathrm{J}_{\mathrm{H}}$ $>0$,_{$u_{-}=u_{+}$}

,

_{$s_{-}=s_{+}$}

,

quite

perfect results have been obtained,

_cf.

$f\mathit{1}\mathit{0}f$ and [17]. In the

case

when the

far

fields of

the the initial data are different, $i.e.$, $(v_{-}, u_{-}, s_{-})$ $\neq(v_{+}, u_{+}, s_{+})$

,

many interesting results

have been obtained: When the solutions to the corresponding Riemann problem consist in

only shock waves, the nonlinear stability

_of

travelling

wave

solutions has been establishedby

[11], [14], and[19], etc. While, when the solutions to the correspondingRiemannproblem

consist in only

_rarefaction

waves, the correspondingnonlinearstability results

are

obtained

by [12], [15], [21], and[22].

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Differencial’

$nya$ Uravnenija4 (1968), 374380.

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a Hyperbolic-Parabolic Composite, with Applications to the

Equa-tions

_of

Magnetohydrodynamics, Thesis, Kyoto University, 1985.

[11] Kawashima, S. and Matsumura, A., Asymptotic stabilityof travelling wave solutions of

sys-tems for onedimensional gas motion, Commun. Math. Phys. 101 (1985), 97-127.

[12] Kawashima, S., Matsumura, A., andNishihara, K.,Asymptoticbehaviour ofsolutions for the

equations ofaviscousheat-conductivegas, Proc. JapanAcad. Ser. A 62 (1986), 24925,

[13] Ladyzenskaja, 0. A., Solonikov, V. A., and Ural’ceva, N. N., Linear and Quasilinear

Equa-tions

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Parabolic Type, Translations of Mathematical Monographs 23, Amer. Math. Soc,

Provindence, $\mathrm{R}\mathrm{I}$, 1968.

[14] Liu, T.-P., Shockwaves for compressible Navier-Stokes equations arestable, Cornrnun. Pure

Appl. Math. 39 (1986), 565-594.

[15] Liu, T.-P. and Xin, Z.-P., Nonlinear stability ofrarefaction waves for compressible

Navier-Stokes equations, Comm. Math. Phys. 118 (1988), 451-465.

[16] Liu, T.-P. and Xin, Z.-P., Pointwise decay to contact discontinuities for systems of viscous

conservation laws, Asian J. Math. 1 (1997), 3484.

[17] Liu,T.-P. and Zeng,Y.-N.,Largetimebehaviorof solutionsfor generalquasilinear

hyperbolic-parabolicsystems ofconservation laws, $Mem$

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Amer. Math. Soc. 125 (599) (1997), 1-120.

[18] Matsumura, A. and Nishida, T., The initial value problem for the equations of motion of

viscous and heat-conductivegases, J. Math. Kyoto Univ. 26 (1980), 67-104.

[19] Matsumura, A. and Nishihara, K., On the stability of travelling wave solutions of a

one-dimensional modelsystemfor compressible viscousgas, Japan J. Appl. Math. 2 (1985), 17-25.

[20] Matsumura,A. and Nishihara,K.,Asymptotictowardtherarefaction

waves

of the solutions of

aone-dimensional model system for compressible viscousgas, JapanJ. Appl. Math. 3 (1986),

1-13.

[21] Matsumura, A. and Nishihara, K., Global stability of the rarefaction waves of a

one

dimensional model system for compressible viscous gas, Cornrn. Math. Phys. 144 (1992),

325-335.

[22] Matsumura, A. and Nishihara, K., Global asymptotics towardthe rarefaction wavefor

solu-tions of viscous$p$-systemwith boundary effect, Quart. Appl. Math. 58 (2000), 69-83.

[23] Whitham,G., Linear and Nonlinear Waves,Wiley-Interscience, 1974.

[24] Xin, Z.-P.,Asymptoticstabilityofrarefaction

waves

for 2$\mathrm{x}2$viscous hyperbolic conservation

laws, J.

_Differential

Equations73 (1988), 4577.

[25] Xin, Z.-P., Zerodissipation limit to rarefactionwaves forthe onedimensional Navier-Stokes

equationsofcompressible isentropic gases, Comm. Pure Appl. Math. 46 (1993), 621-665.

[9] KaneF, Y., On amodel system of equations of one-dimensional gas motion (in Russian),

Differencial

’

$nya$ Uravnenija 4(1968), 374-380.

[10] Kawashima, S., Systems

_of

a Hyperbolic-Parabolic Composite, with Applications to the

Equa-tions

_of

Magnetohydrodynamics, Thesis, Kyoto University, 1985.

[11] Kawashima, S. and Matsumura, A., Asymptotic stabilityof travelling wave solutions of

sys-tems for one-dimensionalgas motion, Commun. Math. Phys. 101 (1985), 97-127.

[12] Kawashima, S., Matsumura, A., andNishihara, K.,Asymptoticbehaviour ofsolutions for the

equations of aviscous heat-conductivegas, Proc. JapanAcad. Ser. A 62 (1986), 249-252.

[13] Ladyzenskaja, 0. A., Solonikov, V. A., and Ural’ceva, N. N., Linear and Quasilinear

Equa-tions

_of

Pambolic Type, Translations of Mathematical Monographs 23, Amer. Math. Soc,

Provindence, $\mathrm{R}\mathrm{I}$, 1968.

[14] Liu, T.-P., Shockwaves for compressible Navier-Stokes equations arestable, Commun. Pure

Appl. Math. 39 (1986), 565-594.

[15] Liu, T.-P. and Xin, Z.-P., Nonlinear stability ofrarefaction waves for compressible

Navier-Stokes equations, Comm. Math. Phys. 118(1988), 451-465.

[16] Liu, T.-P. and Xin, Z.-P., Pointwise decay to contact discontinuities for systems of viscous

conservation laws, Asian J. Math. 1(1997), 34-84.

[17] Liu,T.-P. and Zeng,Y.-N.,Largetimebehaviorof solutionsfor generalquasilinear

hyperbolic-parabolicsystems ofconservation laws, $Mem$

.

Amer. Math. Soc. 125 (599) (1997), 1-120.

[18] Matsumura, A. and Nishida, T., The initial value problem for the equations of motion of

viscous and heat-conductivegases, J. Math. Kyoto Univ. 26 (1980), 67-104.

[19] Matsumura, A. and Nishihara, K., On the stability of travelling wave solutions of a

one-dimensional modelsystemfor compressible viscousgas, Japan J. Appl. Math. 2(1985), 17-25.

[20] Matsumura,A. and Nishihara,K.,Asymptotictowardtherarefaction

waves

of the solutions of

aone-dimensional model system for compressible viscousgas, JapanJ. Appl. Math. 3 (1986),

1-13.

[21] Matsumura, A. and Nishihara, K., Global stability of the rarefaction waves of a

one-dinensional model system for compressible viscous gas, Comm. Math. Phys. 144 (1992),

325-335.

[22] Matsumura, A. and Nishihara, K., Global asymptotics towardthe rarefaction wavefor

solu-tions of viscous$p$-systemwith boundary effect, Quart. Appl. Math. 58 (2000), 69-83.

[23] Whitham,G., Linear and Nonlinear Waves,Wiley-Interscience, 1974.

[24] Xin, Z.-P.,Asymptoticstabilityofrarefaction

waves

for 2$\mathrm{x}2$viscous hyperbolic conservation

laws, J.

_Differential

Equations73 (1988), 45-77.

[25] Xin, Z.-P., Zerodissipation limit to rarefactionwaves forthe one-dimensional Navier-Stokes