Global existence for a system of non-linear and non-local transport equations describing the dynamics of dislocation densities

TL;DR

This paper proves the global existence of solutions for a two-dimensional non-linear, non-local transport system modeling dislocation densities, using entropy methods to handle the shear stress expressed via Riesz transforms.

Contribution

It establishes the global in time existence of solutions for a complex dislocation dynamics model with non-local interactions, a novel result in this context.

Findings

Global existence of solutions proved

Entropy method applied to non-local transport system

Model captures dislocation dynamics in materials

Abstract

In this paper, we study the global in time existence problem for the Groma-Balogh model describing the dynamics of dislocation densities. This model is a two-dimensional model where the dislocation densities satisfy a system of transport equations such that the velocity vector field is the shear stress in the material, solving the equations of elasticity. This shear stress can be expressed as some Riesz transform of the dislocation densities. The main tool in the proof of this result is the existence of an entropy for this system