Existence and uniqueness for dislocation dynamics with nonnegative velocity

Abstract

We study the problem of large time existence of solutions for a mathematical model describing dislocation dynamics in crystals. The mathematical model is a geometric and nonlocal eikonal equation which does not preserve the inclusion. Under the assumption that the dislocation line is expanding, we prove existence and uniqueness of the solution in the framework of discontinuous viscosity solutions. We also show that this solution satisfies some variational properties, which allows us to prove that the energy associated to the dislocation dynamics is nonincreasing. © European Mathematical Society 2005.