数学物理学报(英文版) ›› 2005, Vol. 25 ›› Issue (1): 119-129.

• 论文 • 上一篇    下一篇

ASYMPTOTIC STABILITY OF RAREFACTION WAVE FOR GENERALIZED BURGERS EQUATION

徐艳玲,蒋咪娜   

  1. Department of Mathematics and Information Science, Basic Sciences,
    Huazhong Agricultural University,Department of Mathematical and Statistical Sciences, Central China Normal University,
  • 出版日期:2005-01-20 发布日期:2005-01-20
  • 基金资助:

    . The research was supported by three grants from the Na-
    tional Natural Science Foundation of China (10171037), the scientific research systems of Huazhong Agricultural
    University and Younger Science Foundation (10401021)

ASYMPTOTIC STABILITY OF RAREFACTION WAVE FOR GENERALIZED BURGERS EQUATION

 XU Yan-Ling, JIANG Mai-Na   

  • Online:2005-01-20 Published:2005-01-20
  • Supported by:

    . The research was supported by three grants from the Na-
    tional Natural Science Foundation of China (10171037), the scientific research systems of Huazhong Agricultural
    University and Younger Science Foundation (10401021)

PDF (PC)

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摘要：

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.

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

中图分类号: 

  • 35K65
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