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In this paper, we study the model of Groma and Balogh describing the dynamicsof dislocation densities. This is a two-dimensional model where the dislocationdensities satisfy a system of two transport equations. The velocity vectorfield is the shear stress in the material solving the equations of elasticity.This shear stress can be related to Riesz transforms of the dislocationdensities. Basing on some commutator estimates type, we show thatthis model hasa 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 theH\"older-Zygmund space.

Short time existence and uniqueness in H\"older spaces for the 2D dynamics of dislocation densities