Short time existence and uniqueness in Hölder spaces for the 2D dynamics of dislocation densities

Abstract: In this paper, we study the model of Groma and Balogh describing the dynamics of dislocation densities. This is a two-dimensional model where the dislocation densities satisfy a system of two transport equations. The velocity vector field is the shear stress in the material solving the equations of elasticity. This shear stress can be related to Riesz transforms of the dislocation densities. Basing on some commutator estimates type, we show thatthis model has a unique local-in-time solution corresponding to any initial datum in the space $C^r(\R^2)\cap L^p(\R^2)$ for $r>1$ and $1<p<+\infty$, where $C^r(\R^2)$ is the H\"older-Zygmund space.