HAL: hal-00452241, version 3

arXiv: 1002.0326

DOI: 10.4171/IFB/285

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Interfaces and Free Boundaries 14, 3 (2012) 365-400

Available versions:

v1 (2010-02-01)

v2 (2010-07-12)

v3 (2011-07-16)

Uniqueness and existence of spirals moving by forced mean curvature motion

Nicolas Forcadel 1, Cyril Imbert 1, Régis Monneau 2

(2012)

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.

1:	CEntre de REcherches en MAthématiques de la DEcision (CEREMADE)
	CNRS : UMR7534 – Université Paris IX - Paris Dauphine
2:	Centre d'Enseignement et de Recherche en Mathématiques, Informatique et Calcul Scientifique (CERMICS)
	INRIA – Ecole des Ponts ParisTech

Subject

:

Mathematics/Analysis of PDEs

Keyword(s): spirals – motion of interfaces – comparison principle – quasi-linear parabolic equation – viscosity solution – mean curvature motion

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From: Cyril Imbert <>

Submitted on: Thursday, 14 July 2011 17:40:38

Updated on: Friday, 23 November 2012 22:06:29