Nonlocal First-Order Hamilton-Jacobi Equations Modelling Dislocations Dynamics

Details Nonlocal First-Order Hamilton-Jacobi Equations Modelling Dislocations Dynamics Nonlocal First-Order Hamilton-Jacobi Equations Modelling Dislocations Dynamics

2006-03-24

arxiv.org/abs/math/0603570v1

Comm. Partial Differential Equations 31, 7-9 (2006) 1191-1208

Mathematics

Analysis of PDEs

Scientific paper

10.1080/03605300500361446

We study nonlocal first-order equations arising in the theory of dislocations. We prove the existence and uniqueness of the solutions of these equations in the case of positive and negative velocities, under suitable regularity assumptions on the initial data and the velocity. These results are based on new $L^1$-type estimates on the viscosity solutions of first-order Hamilton-Jacobi Equations appearing in the so-called ``level-sets approach''. Our work is inspired by and simplifies a recent work of Alvarez, Cardaliaguet and Monneau.

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