Asymptotic Limits for Strain-Gradient Viscoelasticity with Nonconvex Energy

TL;DR

This paper analyzes the asymptotic behavior of a higher-order gradient viscoelastic system with nonconvex energy, focusing on limits as viscosity and dispersion vanish, including convergence rates in two dimensions.

Contribution

It establishes asymptotic limits for strain-gradient viscoelasticity with nonconvex energy, providing convergence rates in 2D for the zero-viscosity case.

Findings

Asymptotic limits for viscosity and dispersion parameters.

Convergence rate in 2D for zero-viscosity limit.

Analogies with Navier-Stokes zero-viscosity results.

Abstract

We consider the system of viscoelasticity with higher-order gradients and nonconvex energy in several space dimensions. We establish the asymptotic limits when the viscosity $\nu\rightarrow 0$ or when the dispersion coefficient $\delta \rightarrow 0$. For the latter problem, it is worth noting that, for the case of two space dimensions, we also establish a rate of convergence. This result bears analogies to a result of Chemin \cite{jean1996remark} on the rate of convergence of the zero-viscosity limit for the two-dimensional Navier-Stokes equations with bounded vorticity.