Global Existence Results and Uniqueness for Dislocation Equations

Barles, Guy; Cardaliaguet, Pierre; Ley, Olivier; Monneau, Régis

Barles, Guy; Cardaliaguet, Pierre; Ley, Olivier; Monneau, Régis (2008), Global Existence Results and Uniqueness for Dislocation Equations, SIAM Journal on Mathematical Analysis, 40, 1, p. 44-69. http://dx.doi.org/10.1137/070682083

Type

Article accepté pour publication ou publié

External document link

http://hal.archives-ouvertes.fr/hal-00129352/fr/

Date

2008

Journal name

SIAM Journal on Mathematical Analysis

Volume

Number

Publisher

SIAM

Pages

44-69

Abstract (EN)

We are interested in nonlocal eikonal equations arising in the study of the dynamics of dislocation lines in crystals. For these nonlocal but also nonmonotone 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 the nonmonotone case.

Subjects / Keywords

nonlocal Hamilton–Jacobi equations; dislocation dynamics; nonlocal front propagation; level-set approach; geometrical properties; lower-bound gradient estimate; viscosity solutions; eikonal equation; L1-dependence in time