A transport formulation of moving fronts and application to dislocation dynamics

Abstract

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 transport formulation. In the particular case of curves moving in the plane, we get a simple transport formulation. With this transport 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.