Stability of Large-Amplitude Viscous Shock Under Periodic Perturbation for 1-d Viscoelasticity with Non-Convex Constitutive Relations

TL;DR

This paper proves the stability of large-amplitude viscous shocks under periodic perturbations in 1D viscoelasticity with non-convex stress relations, extending previous results to more general conditions without amplitude restrictions.

Contribution

It extends stability analysis of viscous shocks to large amplitudes and non-convex constitutive relations, handling space-periodic perturbations without amplitude restrictions.

Findings

Solutions converge to a shifted viscous shock profile.

No amplitude restrictions on shock waves are needed.

The method involves decomposing large shocks into small shocks with specialized transforms.

Abstract

This paper investigates the large-time behavior of the viscous shock profile for the one-dimensional system of viscoelasticity, subject to initial perturbations that approach space-periodic functions at far fields. We specifically address the case with non-convex constitutive stress relations and non-degenerate Lax's shock. Under the assumptions of suitably small initial perturbations satisfying a zero-mass type condition, we prove that the solution of the system converges to a viscous shock profile with a shift, which is partially determined by the space-periodic perturbation. Notably, our result imposes no amplitude restrictions on the viscous shock waves. This work extends the result of Kawashima-Matsumura (\textit{Commun. Pure Appl. Math.} \textbf{47}, 1994) by simultaneously handling both large-amplitude shocks and space-periodic perturbations, while also generalizing the result of Huang-Yuan (\textit{Commun. Math. Phys.} \textbf{387}, 2021) by allowing for a non-convex constitutive relation. The key ingredient of proof is decomposing the large-amplitude shock wave into small-amplitude shocks and, for each, introducing suitable transform and weight functions to counteract the adverse effects of non-convex constitutive relations encountered during weighted energy estimates on the system in effective velocity and deformation gradient variables.