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Existence and uniqueness for dislocation dynamics with nonnegative velocity

Alvarez, Olivier; Cardaliaguet, Pierre; Monneau, Régis (2005), Existence and uniqueness for dislocation dynamics with nonnegative velocity, Interfaces and free boundaries, 7, 4, p. 415-434. http://dx.doi.org/10.4171/IFB/131

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Type
Article accepté pour publication ou publié
Date
2005
Journal name
Interfaces and free boundaries
Volume
7
Number
4
Publisher
European Mathematical Society
Pages
415-434
Publication identifier
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Author(s)
Alvarez, Olivier
Cardaliaguet, Pierre
Monneau, Régis
Abstract (EN)
We study the problem of large time existence of solutions for a mathematical model describing dislocation dynamics in crystals. The mathematical model is a geometric and non local eikonal equation which does not preserve the inclusion. Under the assumption that the dislocation line is expanding, we prove existence and uniqueness of the solution in the framework of discontinuous viscosity solutions. We also show that this solution satisfies some variational properties, which allows to prove that the energy associated to the dislocation dynamics is non increasing.
Subjects / Keywords
Dislocation dynamics; eikonal equation; Hamilton-Jacobi equations; discontinuous viscosity solutions; non-local equations