数理解析研究所講究録
第 1690 巻 2010 年 156-158

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STABILITY OF VISCOUS SHOCK WAVESTAI-PING LIU
To my friend, Professor Kenji Nishihara with admiration

ABSTRACT. The idca of Kenji Nishihara on thc nonlincar stability of shock wavcs for isentropic Navier-Stokes
equations have bccn followcd up by many rcscarchcrs. Wc dcscribe somc of the thinkings that arc motivated by theNishihara mcthod.

1. INTRODUCTION

Consider the hyperbolic-parabolic conservation laws(1).

$u_{t}+f(u)_{\tau}=(B(u)u_{x})_{x}$The most basic example is the isentropic Navier-Stokes equations(2)$\{\begin{array}{l}\rho_{t}+(\rho v)_{x}=0,(\rho\uparrow\{)_{t}+(\rho v^{2}+p(\rho))_{x}=(\kappa\tau_{x})_{x}.\end{array}$$T1_{1}e$viscous shocks, the traveling waves(x–st)of the system have been constructed long time ago. But their stability turned out to be hard to prove, in spite of
the intense interests over the years. Thus the shock wave community was stunned by the paper 6 establishing
their stability under the zero total ma.gs constraint of the perturbation$(\rho, \rho v)(x, t)=\Phi$$[$$(\rho, \rho v)(x, 0)=\Phi(x)+(\overline{\rho},\overline{\rho}\overline{v})(x, 0)$(3)$\int_{-\infty}^{\infty}(\overline{\rho},\overline{\rho}\overline{v})(x, 0)dx=0$$]$,.

There are surprises about the paper: The first being the necessity of the zero total mass assumption (3). Then
there is the simplicity in conception of the energy method in the paper.
Besides drawing on the deep understanding of the energy method for the perturbation of constant states forgeneral system (1) by the Kyoto School, [5], [3], the energy method in [6] makes explicit use of the compressibilityof the shock wave. The compressibility is the direct consequence of the nonlinearity of the system and the second
law of $t_{1}hermodynamics$ , expressed in the form of the positivity of the viscosity coefficient; $\kappa>0$ . The zero total
mass condition makes essential use of another fundamental fact, that the mass and momentum are the conserved
quantities of the system. Thus the two most fundamental properties of the system (2) are used in the paper 6
in clean and effective way. The remaining task for the subsequent researchers is to find ways of removing the zero
total mass condition and the resulting consequences.
$[$$]$

2. ZERO TOTAL MASS ASSUMPTION

Shock waves, being traveling waves of the autonomous system (1), (2), are orbitally stable. Thus a perturbationin general changes the location of the shock ag it evolves in time. The zero total ma.ss assumption implies thatthe shock location does not change time-asymptotically. This allows for the concentration of the analysis on theessential stability mechanism, the compressibility of the shock. The compressibility condition is the considerationof the hyperbolic conservation laws
$u_{t}+f(u)_{x}=0$ .
(4)
The characteristic values
characteristic values$\lambda_{i}(u)$for (4) are the eigenvalues of $f’(u)$ . For isentropic Navier-Stokes equations (2), the$al\cdot e$$\lambda_{1}=v-c,$$\lambda_{2}=v+c$,

This paper is supported in parts by NSC Grant $9t$ ) $- 2628- M- 001-|)11$ and NSF Grant DMS-0709248.

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TAI-PING LIU

where is the sound speed, $c^{2}=p’(\rho)$ . An i-shock is a consequence of the compressibility ofthe polytropic gases$c$$p(\rho)=0\rho^{\gamma},$$\gamma>0$$\lambda_{i}$.

For gases including,

there is the genuinely nonlinear property, [8], when$\frac{d^{2}}{d\tau^{2}}p\neq 0,$$\tau\equiv 1\rho$.

This implies that an i-shock $\Phi(x-st)$ is strongly compressible in that$\frac{d}{dx}\lambda_{i}(\Phi(x))<0$.

In particular$\lambda(\Phi(-\infty))<s<\lambda_{i}(\Phi(\infty))$.

This inequality has the geometric meaning in that the i-waves propagate toward the i-shock
which is the basic
reason of the stability of the shock. One can thus use the weighted energy method to obtain the decay rate, [4].
The zero total mass as sumption precludes many of the other wave phenomena, chief among them is the nonlinear
coupling of waves pertaining to distinct characteristic families. On the other hand, around the shock wave, thecompressibility property is the main mechanism for its stability. There is the need to locate the shock wave throughwave tracing. The tracing is done by the local conservation laws in place of the zero total mass assumption. With
such a wave tracing setting the weighted energy method is appropriate for estimates around the shock, [11], [15].
Thus the energy method in [6] has been generalized to general situations.
$\Phi$

When the shock wave is not pertaining to a genuinely nonlinear characteristic field, the compressibility property
is weakened, and the stability analysis is largely open. There is an interesting result for scalar conservation laws, [7].

The tracing requires atthe next section.

$1ea_{A}st$some more local, pointwise estimates of the solutions, a topic to be described in

3. $GREEN’ S$ FUNCTION APPROACH
Pointwise estimates aim at describing the coupling of waves pertaining to different characteristic field for generalsystems. The study of the Green’s functions is essential for the quantitative study. For system (1) linearizedaround a constant state, there is now an explicit expression for the Green’s function, starting with the isentropic
Navier-Stokes equation by Yanni Zeng, [16], and extended to general systems with physical viscosity, [12]. The
idea of inversing the Fourier transfdrm using the complex analytic method in [16] is a key step in the constructionof the Green’s function. With Green’s function explicitly constructed, one can then use the Duhamel’s principle tostudy the quantitative properties of the solutions. The Green’s function captures the basic two properties of thewave behavior: Because physical systems are not uniformly parabolic, but hyperbolic-parabolic, the -function at
the initial data propagates and decays into the Green’s function at later time. This so also for the dissipative,
non-parabolic, systems such as the Boltzmann equation, [10]. That the viscosity matrix is not a diagonal matrix
induces rich wave $co$ upling phenomena, [13].
$\delta$$is$

The study of shock stability, initiated in [6] for the most basic physical system (2), is so far complete only forsystems with artificial viscosity$u_{f}+f(u)_{x}=u_{\tau x}$,[9], [14]. In [14], it is shown that the shock is nonlinearly stable when both the shock and the perturbation areweak, though their relative strengths are allowed to vary. The study for physical viscosity is being completed by
the same authors. one notes in the passing that the important zero dissipative limit has been studied only forsystems with artificial viscosity, [1].

In [2], Kenji Nishihara and his coauthor did a detailed and very useful study of the Green’s function for theBurgers equation linearized around a rarefaction wave. This work reveals the strong hyperbolic linear dissipation
for the rarefaction wave in addition to the usual sublinear parabolic dissipation. Shih-Hsien Yu and the author arefollowing this line of research and completing a work on rarefaction waves for the system.

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STABILITY OF VISCOUS SHOCK WAVES$R.EFF_{\gamma}RF_{1}NCES$[1] Bianchini, Stcfano, Brcssan, Alberto: Vanishing viscosity solutions of nonlinear hyperbolic systcms. Ann. of Math. (2) 161 (2005),
no. 1, 223-342.
[2] Hattori, Youiclii, Nishihara, Kcnji A notc on thc stability of thc rarcfaction wave of thc Burgcrs cquation. Japan J. Indust.
Appl. Math. 8 (1991), no. 1, 85-96.
[3] Kawashima, Shuichi Large-time behaviour of solutions to hypcrbolic-parabolic systems of conscrvation laws and applications.
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[4] Kawashima, Shuichi; Matsumura, Akitaka Asymptotic stabibty of traveling wave solutions of systcms for one-dimensional gas
motion. Comm. Math. Phys. 101 (1985), no. 1, 97-127.
[5] Matsumura, Akitaka; Nishida, Takaaki The initial valuc problem for thc equations of motion of viscous and heat-conductivc
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[6] Matsumura, Akitaka; Nishihara, Kcnji. On the stability of travclling wavc solutions of a one-dimcnsional modcl system forcomprcssible viscous gas. Japan J. ppl. Math. 2 (1985), no. 1, 17-25.
[7] Matsumura, Akitaka; Nishihara, Kcnji Asymptotic stability of traveling waves for scalar viscous conservation laws with nonconvcx nonlinearity. Comm. Math. Phys. 165 (1994), no. 1, 83-96.
[8] Lax, P. D. Hyperbolic systcms of conservation laws. II. Comm. Purc Appl. Math. 10 (1957), 537-566.
[9] Liu, Tai-Ping : Pointwisc convergence to shock waves for viscous conscrvation laws. Comm. Pure Appl. Math. 50 (1997), no. 11,$\Lambda$$\cdot$1113-1182.

[10] Liu, Tai-Ping, Yu, Shih-Hsicn Grccn’s function of Boltzmann cquation, 3-D wavcs. Bull. Inst. Math. Acad. Sin. (N.S.) 1 (2006),
no. 1, 1-78.
[11] Liu, Tai-Ping; Yu, Shih-Hsien Viscous shock wave tracing, local conscrvation laws, and pointwisc cstimatcs, Bull. Inst. Math.
Acad. Sinica $(NS)4$ (2009), no. 3, 235-298.
[12] Liu, Tai-Ping, Zeng, Yanni Largc time bchavior of solutions for gcncral quasilincar hyperbolic-parabolic systems of conservation
laws. Mcm. Amer. Math. Soc. 125 (1997), no. 599, viii $+120$ pp.
Green’s function for hypcrbolic-parabolic systcm. Acta Mathematica Scicntia, 29(B) (6) (2009),[13] Liu, Tai-Ping; Zeng, Yanni

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Liu, Tai-Liu, Zeng, Yanni Time-asymptotic behavior of wave propagation around a viscous shock profile. Comm. Math. Phys.
290 (2009), no. 1, 23-82.
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over a Boltzniann shock profile. J A M S. (To appear.)[15] Yu, Shih-llsien Noiiliinear wave
asymptotic behavior of compressible, isentropic, viscous 1-D flow. Comm. Pure Appl. Math. 47 (1994), no. 8,[16] Zeng, Yanni[14$|$$P^{r()}1$$0b^{\prime\iota tion}$$L^{1}$

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