Global Existence Results and Uniqueness for Dislocation Equations

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.

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