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

We study a strongly coupled system 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 and require special attention to the boundary layer. For this system, we prove a result of 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. For this regularized system, a result of global existence and uniqueness of a solution was given in a previous work of the authors. We show some uniform bounds on this solution which uses in particular an entropy estimate for the densities