GLOBAL EXISTENCE RESULTS AND UNIQUENESS FOR DISLOCATION EQUATIONS

Guy Barles (1) , Pierre Cardaliaguet (2) , Olivier Ley (1, 3) , Regis Monneau (4)

1 LMPT - Laboratoire de Mathématiques et Physique Théorique
2 LM-Brest - Laboratoire de Mathématiques
3 IRMAR - Institut de Recherche Mathématique de Rennes
4 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques, Informatique et Calcul Scientifique

We are interested in nonlocal Eikonal Equations arising in the study of the dynamics of dislocations lines in crystals. For these nonlocal but also non monotone equations, only the existence and uniqueness of Lipschitz and local-in-time solutions were available in some particular cases. In this paper, we propose a definition of weak solutions for which we are able to prove the existence for all time. Then we discuss the uniqueness of such solutions in several situations, both in the monotone and non monotone case.

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

Domaines: Equations aux dérivées partielles [math.AP]

Guy Barles, Pierre Cardaliaguet, Olivier Ley, Regis Monneau. GLOBAL EXISTENCE RESULTS AND UNIQUENESS FOR DISLOCATION EQUATIONS. SIAM Journal on Mathematical Analysis, 2008, 40 (1), pp.44-69. ⟨10.1137/070682083⟩. ⟨hal-00129352v2⟩