Nonlinear stability of rarefaction waves for a hyperbolic system with Cattaneo's law

Abstract

<p style="text-indent:20px;">This paper is concerned with the time-asymptotically nonlinear stability of rarefaction waves to the Cauchy problem and the initial-boundary value problem in the half space with impermeable wall boundary condition for a scalar conservation laws with an artificial heat flux satisfying Cattaneo's law. In our results, although the <inline-formula><tex-math id="M1">\begin{document}$ L^2\cap L^\infty- $\end{document}</tex-math></inline-formula>norm of the initial perturbation is assumed to be small, the <inline-formula><tex-math id="M2">\begin{document}$ H^1- $\end{document}</tex-math></inline-formula>norm of the first order derivative of the initial perturbation with respect to the spatial variable can indeed be large. Moreover the far fields of the artificial heat flux can be different. Our analysis is based on the <inline-formula><tex-math id="M3">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula> energy method.</p>