WEAK SOLUTIONS FOR DISLOCATION TYPE EQUATIONS

Olivier Ley (1, 2)

1 LMPT - Laboratoire de Mathématiques et Physique Théorique
2 IRMAR - Institut de Recherche Mathématique de Rennes

We describe recent results obtained by G. Barles, P. Cardaliaguet, R. Monneau and the author recently. They are concerned with nonlocal Eikonal equations arising in the study of the dynamics of dislocation lines in crystals. These equations are nonlocal but also non monotone. We use a notion of weak solution to provide solutions for all time. Then, we discuss the link between these weak solutions and the classical viscosity solutions, and state some uniqueness results in particular cases. A counter-example to uniqueness is given.

Mots clés: Nonlocal Hamilton-Jacobi Equations; dislocation dynamics; level-set approach; lower-bound gradient estimate; viscosity solutions; $L^1$-dependence in time.; $L^1$-dependence in time

Domaines: Equations aux dérivées partielles [math.AP]

Olivier Ley. WEAK SOLUTIONS FOR DISLOCATION TYPE EQUATIONS. International Conference for the 25th Anniversary of Viscosity Solutions, Jun 2007, Tokyo, Japan. pp.117-132. ⟨hal-00243982⟩