  We study a strongly coupled system consisting 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 with Dirichlet
boundary conditions and require special attention to the boundary. We prove a
result of global 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. We show some uniform bounds on this solution which uses
in particular an entropy estimate for the densities.
