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In this paper, we study the global in time existence problem for theGroma-Balogh model describing the dynamics of dislocation densities. This modelis a two-dimensional model where the dislocation densities satisfy a system oftransport equations such that the velocity vector field is the shear stress inthe material, solving the equations of elasticity. This shear stress can beexpressed as some Riesz transform of the dislocation densities. The main toolin the proof of this result is the existence of an entropy for this system

Global existence for a system of non-linear and non-local transport equations describing the dynamics of dislocation densities