Minimizing movements for dislocation dynamics with a mean curvature term

Nicolas Forcadel; Aurélien Monteillet

ESAIM: Control, Optimisation and Calculus of Variations (2009)

  • Volume: 15, Issue: 1, page 214-244
  • ISSN: 1292-8119

Abstract

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We prove existence of minimizing movements for the dislocation dynamics evolution law of a propagating front, in which the normal velocity of the front is the sum of a non-local term and a mean curvature term. We prove that any such minimizing movement is a weak solution of this evolution law, in a sense related to viscosity solutions of the corresponding level-set equation. We also prove the consistency of this approach, by showing that any minimizing movement coincides with the smooth evolution as long as the latter exists. In relation with this, we finally prove short time existence and uniqueness of a smooth front evolving according to our law, provided the initial shape is smooth enough.

How to cite

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Forcadel, Nicolas, and Monteillet, Aurélien. "Minimizing movements for dislocation dynamics with a mean curvature term." ESAIM: Control, Optimisation and Calculus of Variations 15.1 (2009): 214-244. <http://eudml.org/doc/250605>.

@article{Forcadel2009,
abstract = { We prove existence of minimizing movements for the dislocation dynamics evolution law of a propagating front, in which the normal velocity of the front is the sum of a non-local term and a mean curvature term. We prove that any such minimizing movement is a weak solution of this evolution law, in a sense related to viscosity solutions of the corresponding level-set equation. We also prove the consistency of this approach, by showing that any minimizing movement coincides with the smooth evolution as long as the latter exists. In relation with this, we finally prove short time existence and uniqueness of a smooth front evolving according to our law, provided the initial shape is smooth enough. },
author = {Forcadel, Nicolas, Monteillet, Aurélien},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Front propagation; non-local equations; dislocation dynamics; mean curvature motion; viscosity solutions; minimizing movements; sets of finite perimeter; currents; front propagation},
language = {eng},
month = {1},
number = {1},
pages = {214-244},
publisher = {EDP Sciences},
title = {Minimizing movements for dislocation dynamics with a mean curvature term},
url = {http://eudml.org/doc/250605},
volume = {15},
year = {2009},
}

TY - JOUR
AU - Forcadel, Nicolas
AU - Monteillet, Aurélien
TI - Minimizing movements for dislocation dynamics with a mean curvature term
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2009/1//
PB - EDP Sciences
VL - 15
IS - 1
SP - 214
EP - 244
AB - We prove existence of minimizing movements for the dislocation dynamics evolution law of a propagating front, in which the normal velocity of the front is the sum of a non-local term and a mean curvature term. We prove that any such minimizing movement is a weak solution of this evolution law, in a sense related to viscosity solutions of the corresponding level-set equation. We also prove the consistency of this approach, by showing that any minimizing movement coincides with the smooth evolution as long as the latter exists. In relation with this, we finally prove short time existence and uniqueness of a smooth front evolving according to our law, provided the initial shape is smooth enough.
LA - eng
KW - Front propagation; non-local equations; dislocation dynamics; mean curvature motion; viscosity solutions; minimizing movements; sets of finite perimeter; currents; front propagation
UR - http://eudml.org/doc/250605
ER -

References

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