Dislocation Dynamics: a Non-local Moving Boundary

In this article, we present briefly a mathematical study of the dynamics of line defects called dislocations, in crystals. The mathematical model is an eikonal equation describing the motion of the dislocation line with a velocity which is a non-local function of the whole shape of the dislocation. We present some partial existence and uniqueness results. Finally, we also show that the self-dynamics of a dislocation line at large scale is asymptotically described by an anisotropic mean curvature motion.