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

TL;DR

This paper studies the global behavior of solutions to a scalar viscous conservation law with nonlinear viscosity, extending previous stability results to a broader range of nonlinearity parameters and providing detailed decay estimates.

Contribution

It extends the stability analysis of solutions toward rarefaction waves to the case p > 1/3 and offers precise decay rate estimates, broadening the understanding of nonlinear viscous conservation laws.

Findings

Extended stability results to p > 1/3 without smallness conditions.

Proved solutions tend toward rarefaction waves for broader nonlinearity range.

Provided detailed time-decay estimates for solutions.

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.