Existence and uniqueness for a nonlinear parabolic/Hamilton-Jacobi coupled system describing the dynamics of dislocation densities

Hassan Ibrahim (1)

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

We study a mathematical model describing the dynamics of dislocation densities in crystals. This model is expressed as a one-dimensional system of a parabolic equation and a first order Hamilton-Jacobi equation that are coupled together. We show the existence and uniqueness of a viscosity solution among those assuming a lower-bound on their gradient for all time including the initial data. Moreover, we show the existence of a viscosity solution when we have no such restriction on the initial data. We also state a result of existence and uniqueness of an entropy solution of the system obtained by spatial derivation. The uniqueness of this entropy solution holds in the class of ``bounded from below'' solutions. In order to prove these results, we use a relation between scalar conservation laws and Hamilton-Jacobi equations, mainly to get some gradient estimates. This study takes place on $\R$, and on a bounded domain with suitable boundary conditions.

Mots clés: Hamilton-Jacobi equations; scalar conservation laws; viscosity solutions; entropy solutions; dynamics of dislocation densities

Domaines: Equations aux dérivées partielles [math.AP]

Hassan Ibrahim. Existence and uniqueness for a nonlinear parabolic/Hamilton-Jacobi coupled system describing the dynamics of dislocation densities. 2007. ⟨hal-00138531⟩