A kinetic formulation of moving fronts and application to dislocations dynamics

Régis Monneau (1)

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

In this article, we consider hypersurfaces moving with normal velocity depending on the time-space coordinates and on the normal to the hypersurface. We naturally define a measure associated to this hypersurface. This measure is defined on a suitable space/unit normal/curvature configuration space. We show that, while the hypersurface stays smooth, then the measure is a solution to a linear transport equation, that we call a kinetic formulation. In the particular case of curves moving in the plane, we get a simple kinetic formulation. With this kinetic formulation in hands, it is then easy to complete the models of dislocations densities that were proposed in the 60's. As a consequence, we therefore propose a closed mean field model for the dynamics of dislocations densities.

Mots clés: kinetic equation; Liouville equation; dislocations densities; transport equation; fronts; normal velocity; curvature; dislocations dynamics; dislocations densities.

Domaines: Equations aux dérivées partielles [math.AP]

Régis Monneau. A kinetic formulation of moving fronts and application to dislocations dynamics. 2006. ⟨hal-00091341⟩