Asymptotic stability of the composite wave of rarefaction wave and contact wave to nonlinear viscoelasticity model with non-convex flux

TL;DR

This paper proves the asymptotic stability of a composite wave combining rarefaction and contact waves in a nonlinear viscoelasticity model with non-convex flux, using a novel approach involving coupled diffusion waves and a new weighted Poincaré inequality.

Contribution

It introduces a new method for analyzing stability in systems with non-convex flux, applicable to more general viscoelastic models.

Findings

Established the nonlinear stability of the composite wave.

Developed a new weighted Poincaré inequality for non-convex flux systems.

Demonstrated the method's applicability to general systems.

Abstract

In this paper, we consider the wave propagations of viscoelastic materials, which has been derived by Taiping-Liu to approximate the viscoelastic dynamic system with fading memory (see [T.P.Liu(1988)\cite{LiuTP}]) by the Chapman-Enskog expansion. By constructing a set of linear diffusion waves coupled with the high-order diffusion waves to achieve cancellations to approximate the viscous contact wave well and explicit expressions, the nonlinear stability of the composite wave is obtained by a continuum argument. It emphasis that, the stress function in our paper is a general non-convex function, which leads to several essential differences from strictly hyperbolic systems such as the Euler system. Our method is completely new and can be applied to more general systems and a new weighted Poincar\'e type of inequality is established, which is more challenging compared to the convex case and this inequality plays an important role in studying systems with non-convex flux.