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

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  • 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.

    Mathematics Subject Classification: 35L65, 35L60.


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