February  2005, 12(2): 251-282. doi: 10.3934/dcds.2005.12.251

Asymptotics toward strong rarefaction waves for $2\times 2$ systems of viscous conservation laws


Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong


School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

Received  August 2003 Revised  July 2004 Published  December 2004

This paper concerns the time asymptotic behavior toward large rarefaction waves of the solution to general systems of $2\times 2$ hyperbolic conservation laws with positive viscosity coefficient $B(u)$

$u_t+F(u)_x=(B(u)u_x)_x,\quad u\in R^2,\qquad $ ($*$)

$u(0,x)=u_0(x)\rightarrow u_\pm\quad$ as $x\rightarrow \pm\infty.$

Assume that the corresponding Riemann problem


$ u(0,x)=u^r_0(x)=u_-,\quad x<0, and u_+,\quad x>0$

can be solved by one rarefaction wave. If $u_0(x)$ in ($*$) is a small perturbation of an approximate rarefaction wave constructed in Section 2, then we show that the Cauchy problem ($*$) admits a unique global smooth solution $u(t,x)$ which tends to $ u^r(t,x)$ as the $t$ tends to infinity. Here, we do not require $|u_+ - u_-|$ to be small and thus show the convergence of the corresponding global smooth solutions to strong rarefaction waves for $2\times 2$ viscous conservation laws.

Citation: Tong Yang, Huijiang Zhao. Asymptotics toward strong rarefaction waves for $2\times 2$ systems of viscous conservation laws. Discrete and Continuous Dynamical Systems, 2005, 12 (2) : 251-282. doi: 10.3934/dcds.2005.12.251

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