Existence and stability of viscoelastic shock profiles

TL;DR

This paper studies the existence and stability of viscoelastic shock profiles in planar models, extending previous work to compressible cases and providing a rigorous stability theory with numerical verification methods.

Contribution

It introduces a comprehensive stability framework for viscoelastic shock profiles, including compressible cases and numerical Evans function computations.

Findings

Spectral stability implies linear and nonlinear stability.

Verification of stability for small-amplitude Lax shocks.

Numerical methods for large-amplitude and nonclassical shocks.

Abstract

We investigate existence and stability of viscoelastic shock profiles for a class of planar models including the incompressible shear case studied by Antman and Malek-Madani. We establish that the resulting equations fall into the class of symmetrizable hyperbolic--parabolic systems, hence spectral stability implies linearized and nonlinear stability with sharp rates of decay. The new contributions are treatment of the compressible case, formulation of a rigorous nonlinear stability theory, including verification of stability of small-amplitude Lax shocks, and the systematic incorporation in our investigations of numerical Evans function computations determining stability of large-amplitude and or nonclassical type shock profiles.