Existence and uniqueness for a dislocation model with short-range interactions and varying stress field

Abstract

In this article, we consider a coupled singular parabolic system, describing the dynamics of dislocation densities in a bounded domain. The model takes into consideration the short-range interactions between dislocations, which causes the singularity that appears under the form of dividing by a gradient term. We prove a long time existence and uniqueness under the assumption that the applied stresses on the domain is bounded in space and time. The proof relies on a comparison principle to avoid singularity, and on exponential gradient estimates for the long time existence.

Keywords

• Boundary value problem
• parabolic system
• comparison principle