On the well-posedness of a 2D nonlinear and nonlocal system arising from the dislocation dynamics

TL;DR

This paper establishes local and global well-posedness results for a 2D nonlinear, nonlocal dislocation dynamics model, providing mathematical validation for the system's solvability under certain conditions.

Contribution

It proves the local existence and uniqueness of strong solutions and extends to global well-posedness in dissipative cases for the first time.

Findings

Local well-posedness of the system

Global well-posedness in dissipative cases

Application of nonlocal maximum principle

Abstract

In this paper we consider a 2D nonlinear and nonlocal model describing the dynamics of the dislocation densities. We prove the local well-posedness of strong solution to this system in the suitable functional framework, and we show the global well-posedness for some dissipative cases by the method of nonlocal maximum principle.