Global structure of solutions toward the rarefaction waves for the Cauchy problem of the scalar conservation law with nonlinear viscosity

Abstract

In this paper, we investigate the global structure of solutions to the Cauchy problem for the scalar viscous conservation law where the far field states are prescribed. Especially, we deal with the case when the viscous/diffusive flux $\sigma(v) \sim |\, v \,|^p$ is of non-Newtonian type (i.e., $p>0$), including a pseudo-plastic case (i.e., $p<0$). When the corresponding Riemann problem for the hyperbolic part admits a Riemann solution which consists of single rarefaction wave, under a condition on nonlinearity of the viscosity, it has been recently proved by Matsumura-Yoshida \cite{matsumura-yoshida'} that the solution of the Cauchy problem tends toward the rarefaction wave as time goes to infinity for the case $p>3/7$ without any smallness conditions. The new ingredients we obtained are the extension to the stability results in \cite{matsumura-yoshida'} to the case $p > 1/3$ (also without any smallness conditions), and furthermore their precise time-decay estimates.