Existence and uniqueness for dislocation dynamics with nonnegative velocity

We study the problem of large time existence of solutions for a mathe- matical model describing dislocation dynamics in crystals. The mathemat- ical model is a geometric and non local 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 frame- work of discontinuous viscosity solutions. We also show that this solution satisfles some variational properties, which allows to prove that the energy associated to the dislocation dynamics is non increasing.