Dynamics of Dislocation Densities in a Bounded Channel. Part II: Existence of Weak Solutions to a Singular Hamilton–Jacobi/Parabolic Strongly Coupled System

Abstract

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. Our system is a natural extension of that studied in [16] where the applied stress was set to be zero. 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. For this regularized system, a result of global existence and uniqueness of a solution has been given in [17]. We show some uniform bounds on this solution which uses in particular an entropy estimate for the densities.

Derivation and study of dynamical models of dislocation densities

“…(Uniqueness for the one-dimensional submodel, [4]) Assume that the initial data ρ ± 0 is Lipschitz non-decreasing and that for some L > 0, the functions y → ρ ± 0 (y) − Ly are 1 − periodic. Then there exists a unique viscosity solution (ρ + , ρ − ) to the system (14), (11). Moreover this solution is globally Lipschitz in space and time.…”

Formal derivation and existence result of an approximate model on dislocation densities

“…This model is written as a coupled system of nonlinear parabolic equations on a bounded domain in one‐space dimension. Several variants of the GCZ models have been treated in the work of [] where particular assumptions on the exterior stress field have been involved. In fact, in their work, they only considered either constant or bounded space‐time dependent stresses.…”