Global asymptotics toward rarefaction waves for solutions of the scalar conservation law with nonlinear viscosity

TL;DR

This paper studies the long-term behavior of solutions to a scalar conservation law with nonlinear viscosity, showing convergence to rarefaction waves in the non-Newtonian viscosity case without smallness assumptions.

Contribution

It establishes the global asymptotic convergence to rarefaction waves for scalar viscous conservation laws with nonlinear, including pseudo-plastic, viscosity.

Findings

Solutions tend toward rarefaction waves over time.

Convergence holds without smallness restrictions.

Applicable to non-Newtonian viscosity cases.

Abstract

In this paper, we investigate the asymptotic behavior 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 viscosity is of non-Newtonian type, including a pseudo-plastic case. 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 is proved that the solution of the Cauchy problem tends toward the rarefaction wave as time goes to infinity, without any smallness conditions.