Document ESAIM: COCV
DOI: 10.1051/cocv:2008027
Minimizing movements for dislocation dynamics with a mean curvature term
Nicolas Forcadel1 and Aurélien Monteillet21 CERMICS, École des Ponts, Paris Tech, 6 et 8 avenue Blaise Pascal, Cité Descartes, Champs-sur-Marne, 77455 Marne-la-Vallée Cedex 2, France; forcadel@cermics.enpc.fr
2 Université de Bretagne Occidentale, UFR Sciences et Techniques, 6 av. Le Gorgeu, BP 809, 29285 Brest, France; aurelien.monteillet@univ-brest.fr
(Received May 25, 2007. Revised December 20, 2007. Published online March 28, 2008.)
Abstract
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.
Mathematics Subject Classification. 53C44, 49Q15, 49L25, 28A75, 58A25
Key words: Front propagation, non-local equations, dislocation dynamics, mean curvature motion, viscosity solutions, minimizing movements, sets of finite perimeter, currents
© EDP Sciences, SMAI 2008