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.

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