The global temporal existence problem for the Groma-Balogh model describing the dynamics of dislocation densities was studied. This model was a two-dimensional model, wherein the dislocation densities satisfied a system of transport equations such that the velocity vector field was the shear stress in the material, solving the equations of elasticity. This shear stress could be expressed as a Riesz transform of the dislocation densities. The main tool in the proof of the result was the existence of an entropy term for this system.

Global Existence for a System of Non-Linear and Non-Local Transport Equations Describing the Dynamics of Dislocation Densities. M.Cannone, A.El Hajj, R.Monneau, F.Ribaud: Archive for Rational Mechanics and Analysis, 2010, 196[1], 71-96