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.

Keywords: front propagation, non-local equations, dislocation dynamics, mean curvature motion, viscosity solutions, minimizing movements, sets of finite perimeter, currents

@article{COCV_2009__15_1_214_0, author = {Forcadel, Nicolas and Monteillet, Aur\'elien}, title = {Minimizing movements for dislocation dynamics with a mean curvature term}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {214--244}, publisher = {EDP-Sciences}, volume = {15}, number = {1}, year = {2009}, doi = {10.1051/cocv:2008027}, mrnumber = {2488577}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv:2008027/} }

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 PY - 2009 SP - 214 EP - 244 VL - 15 IS - 1 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv:2008027/ DO - 10.1051/cocv:2008027 LA - en ID - COCV_2009__15_1_214_0 ER -

%0 Journal Article %A Forcadel, Nicolas %A Monteillet, Aurélien %T Minimizing movements for dislocation dynamics with a mean curvature term %J ESAIM: Control, Optimisation and Calculus of Variations %D 2009 %P 214-244 %V 15 %N 1 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv:2008027/ %R 10.1051/cocv:2008027 %G en %F COCV_2009__15_1_214_0

Forcadel, Nicolas; Monteillet, Aurélien. Minimizing movements for dislocation dynamics with a mean curvature term. ESAIM: Control, Optimisation and Calculus of Variations, Volume 15 (2009) no. 1, pp. 214-244. doi : 10.1051/cocv:2008027. http://archive.numdam.org/articles/10.1051/cocv:2008027/

[1] Curvature-driven flows: a variational approach. SIAM J. Control Optim. 31 (1993) 387-438. | MR | Zbl

, and ,[2] Existence and uniqueness for dislocation dynamics with nonnegative velocity. Interfaces Free Boundaries 7 (2005) 415-434. | MR | Zbl

, and ,[3] A convergent scheme for a nonlocal Hamilton-Jacobi equation, modeling dislocation dynamics. Num. Math. 104 (2006) 413-572. | MR | Zbl

, , and ,[4] Dislocation dynamics: short time existence and uniqueness of the solution. Arch. Rational Mech. Anal. 85 (2006) 371-414. | MR | Zbl

, , and ,[5] Minimizing movements. Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. 19 (1995) 191-246. | MR | Zbl

,[6] Gradient flows in metric spaces and in the space of probability measures, Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel (2005). | MR | Zbl

, and ,[7] Global existence results and uniqueness for dislocation type equations. SIAM J. Math. Anal. (to appear). | MR | Zbl

, , and ,[8] A simple proof of convergence for an approximation scheme for computing motions by mean curvature. SIAM J. Numer. Anal. 32 (1995) 484-500. | MR | Zbl

and ,[9] Nonlocal first-order hamilton-jacobi equations modelling dislocations dynamics. Comm. Partial Differential Equations 31 (2006) 1191-1208. | MR | Zbl

and ,[10] Front propagation and phase field theory. SIAM J. Control Optim. 31 (1993) 439-469. | MR | Zbl

, and ,[11] Regularity theory for almost minimal currents. Arch. Rational Mech. Anal. 78 (1982) 99-130. | MR | Zbl

,[12] On front propagation problems with nonlocal terms. Adv. Differential Equations 5 (2000) 213-268. | MR | Zbl

,[13] On the energy of a flow arising in shape optimisation. Interfaces Free Bound. (to appear). | MR | Zbl

and ,[14] On the approximation of front propagation problems with nonlocal terms. ESAIM: M2AN 35 (2001) 437-462. | Numdam | MR | Zbl

and ,[15] Measure theory and fine properties of functions, Studies in Advanced Mathematics. CRC Press, Boca Raton, FL (1992). | MR | Zbl

and ,[16] Motion of level sets by mean curvature. II. Trans. Amer. Math. Soc. 330 (1992) 321-332. | MR | Zbl

and ,[17] Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer-Verlag New York Inc., New York (1969). | MR | Zbl

,[18] Dislocations dynamics with a mean curvature term: short time existence and uniqueness. Differential Integral Equations 21 (2008) 285-304. | MR

,[19] Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies 105. Princeton University Press, Princeton, NJ (1983). | MR | Zbl

,[20] Geometric evolution of phase-boundaries, in On the evolution of phase boundaries (Minneapolis, MN, 1990-1991), IMA Vol. Math. Appl. 43, Springer, New York (1992) 51-65. | MR | Zbl

and ,[21] Minimal surfaces and functions of bounded variation, Monographs in Mathematics 80. Birkhäuser Verlag, Basel (1984). | MR | Zbl

,[22] Implicit time discretization for the mean curvature flow equation. Calc. Var. Partial Differential Equations 3 (1995) 253-271. | MR | Zbl

and ,[23] Analytic semigroups and optimal regularity in parabolic problems, Progress in Nonlinear Differential Equations and their Applications 16. Birkhäuser Verlag, Basel (1995). | MR | Zbl

,[24] On a free boundary problem of viscous incompressible flows. Interfaces Free Bound. 9 (2007) 549-589. | MR | Zbl

,[25] Geometric measure theory. A beginner's guide. Academic Press Inc., Boston, MA (1988). | MR | Zbl

,[26] Regularity and singularity estimates on hypersurfaces minimizing parametric elliptic variational integrals. I, II. Acta Math. 139 (1977) 217-265. | MR | Zbl

, and ,[27] Lectures on geometric measure theory, in Proceedings of the Centre for Mathematical Analysis, Vol. 3, Australian National University Centre for Mathematical Analysis, Canberra (1983). | MR | Zbl

,[28] Phase-field theory for FitzHugh-Nagumo-type systems. SIAM J. Math. Anal. 27 (1996) 1341-1359. | MR | Zbl

and ,*Cited by Sources: *