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

Abstract

This paper concerns the time asymptotic behavior toward largerarefaction waves of the solution to general systems of $2\times2$ hyperbolic conservation laws with positive viscositycoefficient $B(u)$$u_t+F(u)_x=(B(u)u_x)_x,\quad u\in R^2,\qquad $ ($*$)$u(0,x)=u_0(x)\rightarrow u_\pm\quad$ as $x\rightarrow\pm\infty.$Assume that the corresponding Riemann problem$u_t+F(u)_x=0,$$ u(0,x)=u^r_0(x)=u_-,\quad x0$can be solved by one rarefaction wave. If $u_0(x)$ in ($*$) is asmall perturbation of an approximate rarefaction wave constructedin Section 2, then we show that the Cauchy problem ($*$) admits aunique 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 thecorresponding global smooth solutions to strong rarefaction waves for$2\times 2$ viscous conservation laws.