ASYMPTOTIC STABILITY OF RAREFACTION WAVE FOR GENERALIZED BURGERS EQUATION

Abstract

Abstract:

This paper is concerned with the stability of
the rarefaction wave for the Burgers equation \\

$\left\{\begin{array}{l} u_t+f(u)_x=\mu t^{\alpha}u_{xx}, \ \ \ \mu >0,\ \ x \in {\bf R},\ \ t > 0,\\ u|_{t=0}=u_0(x) \rightarrow u_{\pm},\ \ \ x \rightarrow{\pm}{\infty}, \end{array} \right. \eqno({\rm I})$
where

$0\leq {\alpha}<{\frac{1}{4q}}$ (

$q$
is determined by

$(2.2)$ ). Roughly
speaking, under the assumption that
$u_- to the Cauchy problem (I), also find
the solution

$u(x,t)$ to the Cauchy problem (I)
satisfying

$\sup\limits_{x\in {\bf R}}|u(x,t)-u^R(x/t)| \rightarrow 0$
as

$t \rightarrow \infty$ , where

$u^R(x/t)$ is the rarefaction wave of
the non-viscous Burgers equation

$u_t+f(u)_x=0$ with Riemann initial
data
$    u(x,0)=\left\{\begin{array}{l}
        u_-, \ \ x<0, \\
        u_+, \ \ x>0.
\end{array}
\right.

Key words: Burgers equation;rarefaction wave;the method of successive approximation;
maximum principle;a priori estimate;stability

CLC Number:

35K65

XU Yan-Ling, JIANG Mai-Na. ASYMPTOTIC STABILITY OF RAREFACTION WAVE FOR GENERALIZED BURGERS EQUATION[J].Acta mathematica scientia,Series B, 2005, 25(1): 119-129.

Trendmd

References

[1] Crighton D G. Model equation of nonlinear acoustics. A Rev Fluid Mech, 1979, 11: 11-33

[2] Crighton D G, Scott J F. Asymptotic solutions of model equations in nonlinear acoustics. Phil Trans R

Soc Lond, 1979, 292A: 101-134

[3] Ding Xiaxi, Wang Jinghua. Global solutions for a semilinear parabolic system. Acta Mathematica Scientia,

1983, 3(3): 397-414

[4] Goodman J. Nonlinear asymptotic stability of viscous shock profile for conservation laws. Arch Rat Mech

Anal, 1986, 95: 325-344

[5] Harabetian E. Rarefaction and large time behavior for parabolic equation and monotone schmes. Comm

Math Phys, 1998, 14: 527-536

[6] Hattori Y, Nishihara K. A note on the stability of the rarefaction wave of the Burgers equation. Japan J

Indust Appl Math, 1991, 8: 85-96

[7] Hopf E. The partial differenatial equation ut + uux = µuxx. Comn Pure Appl Math, 1950, 3: 201-230

[8] Ito K. Asymptotic decay toward the planar rarefaction waves of solution for viscous conservation laws in

several space dimension. Math Models Methods Appl Sci, 1996, 6: 315-338

[9] Il’in A M, Oleinik O A. Asymptotic behavior of the solution of the Cauchy problem for certain quasilinear

equations for large time. Math USSR Sb, 1960, 51: 191-216

[10] Kawashima S, Matsumura A. Asymptotic stability of traveling wave soluyions of systems for one-simentinal

gas motion. Commun Math Phys, 1985, 101: 97-127

[11] Liu T P. Nonlinear stability of shock waves for viscous conservation laws. Mem AMS, 1985, 328: 1-108

[12] Liu T P, Matsumura A, Nishihara K. Behavior of solutions for the Burgers equation with boundary

corresponding to rarefaction waves. SIAM J Math Anal, 1998, 29: 293-308

[13] Matsumura A, Nishihara K. Asymptotic toward the rarefaction waves of the solution of Burgers equation

with nonlinear degenerate viscosity. Nonlinear Analysis, Theory, Methods & Applications, 1994, 23: 605-

614

[14] Matsumura A, Nishihara K. Global stability of rarefaction waves of a one-dimention model system for

compressible viscious gas. Comm Math Phys, 1992, 144: 325-335

No.1 Xu & Jiang: ASYMPTOTIC STABILITY OF RAREFACTION WAVE 129

[15] Matsumura A, Nishhara K. Asymptotic toward the rarefaction waves of the solution of a gas one-dimension

model system for compressible viscious gas. Jaoan J Appl Math, 1986, 3: 1-13

[16] Nishihara K. Asymptotic behaviors of solutions to viscous conservation laws via the L2-energy method.

Lectures note deliverd in Summer School in Fudan University, Shanghai, China, 1999

[17] Nishihara K. A note on the stability of travelling wave solution of Burgers equation. Japan Appl Math,

1985, 2: 27-35

[18] Nishikawa M, Nishihara K. Asymptotic toward the planar rarefaction wave for viscous conservation law in

space dimentions. Trans AMS, 2000, 352: 1203-1215

[19] Smoller J. Shock Waves and Reaction-Diffusion Equation. New York, Berlin: Springer-Verlag, 1983

[20] Scott J F. The long time asymptotic of solution to the generalized Burgers eqution. Proc Roy Soc Lond,

1981, 373A: 443-456

[21] Sauchder P L. Nonlinear Diffusive Waves. New York: Cambridge University Press, 1987

[22] Wang Jinhua, Zhang Hui. A new viscous regulation of the Riemann problem for Burgers equation. J

Differential Equations, 2000, 13(3): 253-263

[23] Wang Z A. Large Behavior of Solution for the Generalized Korteweg-de Vries-Burgers Equation.
[Master’s

thesis]. Central China Normal Univerity, April 2001

[24] Whithman G. Linear and Nonlinear Waves. Wiley-Interscience, 1974

[25] Zhu C J. Quaslinear Hyperbolic Systems with Dissipation Effects.
[Ph.D thesis]. City University of Hong

Kong, Feburary 1999

[26] Zhu C J. Asymptotic behavior of solutions for p-system with relaxation. J Differential Equations, 2002,

180 : 273-306

[27] Zhao Huijiang, Zhu Changjiang, Yu Zhong. Existence and convergence of solutions to a singular perturbed

higher order partial differential equation. Nonlinear Analysis, TMA, 1995, 24: 1435-1455

[28] Zhang Hui. Existence of weak solutions for a degenerate generalized Burgers equation with large initial

data. Acta Mathematica Scientia, 2002, 22B(2): 241-248