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Minimizing movements for dislocation dynamics with a mean curvature term

Published online by Cambridge University Press:  23 January 2009

Nicolas Forcadel  and
Aurélien Monteillet
Nicolas Forcadel
Affiliation:
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
Aurélien Monteillet
Affiliation:
Université de Bretagne Occidentale, UFR Sciences et Techniques, 6 av. Le Gorgeu, BP 809, 29285 Brest, France. aurelien.monteillet@univ-brest.fr
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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.

Keywords

Front propagationnon-local equationsdislocation dynamicsmean curvature motionviscosity solutionsminimizing movementssets of finite perimetercurrents
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 15 , Issue 1 , January 2009 , pp. 214 - 244
Copyright
© EDP Sciences, SMAI, 2008

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