Existence and Uniqueness of Continuous Solution for a Non-local Coupled System Modeling the Dynamics of Dislocation Densities

Abstract

In this paper, we study a non-local coupled system that arises in the theory of dislocations densities dynamics. Based on a new gradient entropy estimate in $$L \log L$$ space, we prove the global existence of a continuous solution. The approach is made by adding a viscosity term and passing to the limit for vanishing viscosity. A comparison principle with respect to time is used for proving uniqueness of the solution for the local problem.