Dynamics of dislocation densities in a bounded channel. Part I: smooth solutions to a singular coupled parabolic system.

Hassan Ibrahim (1) , Mustapha Jazar (2) , Régis Monneau (1)

1 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques, Informatique et Calcul Scientifique
2 Mathematics Department

We study a coupled system of two parabolic equations in one space dimension. This system is singular because of the presence of one term with the inverse of the gradient of the solution. Our system describes an approximate model of the dynamics of dislocation densities in a bounded channel submitted to an exterior applied stress. The system of equations is written on a bounded interval and requires a special attention to the boundary layer. The proof of existence and uniqueness is done under the use of two main tools: a certain comparison principle on the gradient of the solution, and a Kozono-Taniuchi parabolic type inequality.

Domaines: Equations aux dérivées partielles [math.AP]

Hassan Ibrahim, Mustapha Jazar, Régis Monneau. Dynamics of dislocation densities in a bounded channel. Part I: smooth solutions to a singular coupled parabolic system.. Communications on Pure and Applied Mathematics, 2010, 9 (3), pp.703-719. ⟨10.3934/cpaa.2010.9.703⟩. ⟨hal-00281487v3⟩