A mathematical model which described dislocation dynamics in crystals was studied. A single dislocation line moving in its slip plane was considered. The normal velocity was given by the Peach-Koehler force created by the dislocation line itself. The mathematical model was an eikonal equation with a velocity which was a non-local quantity depending on the whole shape of the dislocation line. The special case where the dislocation line was assumed to be a graph or a closed loop was studied. The existence and uniqueness of a solution for short times was proved within the framework of discontinuous viscosity solutions for Hamilton–Jacobi equations. Physical explanations and a formal derivation of the mathematical model were also given. Finally, numerical results based upon a level-sets formulation of the problem were presented. These results showed that there was no general inclusion principle for this model.

Dislocation Dynamics - Short-Time Existence and Uniqueness of the Solution. O.Alvarez, P.Hoch, Y.Le Bouar, R.Monneau: Archive for Rational Mechanics and Analysis, 2006, 181[3], 449-504