Uniqueness and existence of spirals moving by forced mean curvature motion
Monneau, Régis; Imbert, Cyril; Forcadel, Nicolas (2012), Uniqueness and existence of spirals moving by forced mean curvature motion, Interfaces and Free Boundaries, 14, 3, p. 365-400. http://dx.doi.org/10.4171/IFB/285
TypeArticle accepté pour publication ou publié
Lien vers un document non conservé dans cette basehttp://arxiv.org/abs/1002.0326v3
Nom de la revueInterfaces and Free Boundaries
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Résumé (EN)In this paper, we study the motion of spirals by mean curvature type motion in the (two dimensional) plane. Our motivation comes from dislocation dynamics; in this context, spirals appear when a screw dislocation line reaches the surface of a crystal. The first main result of this paper is a comparison principle for the corresponding parabolic quasi-linear equation. As far as motion of spirals are concerned, the novelty and originality of our setting and results come from the fact that, first, the singularity generated by the attached end point of spirals is taken into account for the first time, and second, spirals are studied in the whole space. Our second main result states that the Cauchy problem is well-posed in the class of sub-linear weak (viscosity) solutions. We also explain how to get the existence of smooth solutions when initial data satisfy an additional compatibility condition.
Mots-clésmean curvature motion; viscosity solutions; quasi-linear parabolic equation; comparison principle; motion of interfaces; spirals
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