Minimizing movements for dislocation dynamics with a mean curvature term

Nicolas Forcadel (1) , Aurélien Monteillet (2)

1 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques, Informatique et Calcul Scientifique
2 LM-Brest - Laboratoire de Mathématiques

We prove existence of minimizing movements for the dislocation dynamics evolution law of a propagating front, in which the normal velocity of the front is the sum of a non-local term and a mean curvature term. We prove that any such minimizing movement is a weak solution of this evolution law, in a sense related to viscosity solutions of the corresponding level-set equation. We also prove the consistency of this approach, by showing that any minimizing movement coincides with the smooth evolution as long as the latter exists. In relation with this, we finally prove short time existence and uniqueness of a smooth front evolving according to our law, provided the initial shape is smooth enough.

Domaines: Equations aux dérivées partielles [math.AP]

Nicolas Forcadel, Aurélien Monteillet. Minimizing movements for dislocation dynamics with a mean curvature term. ESAIM: Control, Optimisation and Calculus of Variations, 2009, 15 (1), pp 214-244. ⟨hal-00149406⟩