Dynamics of dislocation densities in a bounded channel. Part II: existence of weak solutions to a singular Hamilton-Jacobi/parabolic strongly coupled system

TL;DR

This paper proves the global existence of weak solutions for a coupled system of parabolic and singular Hamilton-Jacobi equations modeling dislocation densities under stress, using regularization and entropy estimates.

Contribution

It establishes the existence of weak solutions for a complex coupled system with boundary conditions, advancing understanding of dislocation dynamics in materials.

Findings

Proved global existence of weak solutions.

Developed a regularization and limiting process.

Derived uniform bounds using entropy estimates.

Abstract

We study a strongly coupled system consisting of a parabolic equation and a singular Hamilton-Jacobi equation in one space dimension. This system describes the dynamics of dislocation densities in a material submitted to an exterior applied stress. The equations are written on a bounded interval with Dirichlet boundary conditions and require special attention to the boundary. We prove a result of global existence of a solution. The method of the proof consists in considering first a parabolic regularization of the full system, and then passing to the limit. We show some uniform bounds on this solution which uses in particular an entropy estimate for the densities.