Existence and uniqueness for a viscoelastic Kelvin-Voigt model with nonconvex stored energy

TL;DR

This paper proves the existence and uniqueness of solutions for a nonlinear viscoelastic Kelvin-Voigt model with nonconvex stored energy, and establishes energy conservation and regularity results.

Contribution

It introduces a framework for weak solutions with nonconvex stored energies satisfying an Andrews-Ball condition, including existence, uniqueness, and energy conservation.

Findings

Existence of weak solutions for superquadratic energies.

Uniqueness of weak solutions in two dimensions.

Energy conservation and global regularity under mild growth conditions.

Abstract

We consider nonlinear viscoelastic materials of Kelvin-Voigt type with stored energies satisfying an Andrews-Ball condition, allowing for non convexity in a compact set. Existence of weak solutions with deformation gradients in $H^1$ is established for energies of any superquadratic growth. In two space dimensions, weak solutions notably turn out to be unique in this class. Conservation of energy for weak solutions in two and three dimensions, as well as global regularity for smooth initial data in two dimensions are established under additional mild restrictions on the growth of the stored energy.