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

Marco Cannone (1) , Ahmad El Hajj (2) , Regis Monneau (3) , Francis Ribaud (1)

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
2 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans
3 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques, Informatique et Calcul Scientifique

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

Mots clés: dynamics of dislocation densities.; Cauchy's problem; system of non-linear transport equations; system of non-local transport equations; system of hyperbolic equations; entropy; Riesz transform; Zygmund space; dynamics of dislocation densities

Domaines: Physique mathématique [math-ph]; Physique mathématique [math-ph]

Marco Cannone, Ahmad El Hajj, Regis Monneau, Francis Ribaud. Global existence for a system of non-linear and non-local transport equations describing the dynamics of dislocation densities. Archive for Rational Mechanics and Analysis, 2010, 196 (1), pp.71-96. ⟨10.1007/s00205-009-0235-8⟩. ⟨hal-00319937⟩