Weak Solutions for Dislocation Type Equations

TL;DR

This paper studies nonlocal, non-monotone Eikonal equations modeling dislocation lines in crystals, introducing weak solutions to ensure existence over time and exploring their relation to classical solutions.

Contribution

It introduces a weak solution framework for nonlocal dislocation equations, addressing existence, uniqueness, and the connection to viscosity solutions.

Findings

Weak solutions exist for all time

Counter-example shows non-uniqueness in some cases

Links between weak and viscosity solutions are established

Abstract

We describe recent results obtained by G. Barles, P. Cardaliaguet, R. Monneau and the author recently. They are concerned with nonlocal Eikonal equations arising in the study of the dynamics of dislocation lines in crystals. These equations are nonlocal but also non monotone. We use a notion of weak solution to provide solutions for all time. Then, we discuss the link between these weak solutions and the classical viscosity solutions, and state some uniqueness results in particular cases. A counter-example to uniqueness is given.