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.
Classification : 53C44, 49Q15, 49L25, 28A75, 58A25
Mots clés : 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://www.numdam.org/item/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, Tome 15 (2009) no. 1, pp. 214-244. doi : 10.1051/cocv:2008027. http://www.numdam.org/item/COCV_2009__15_1_214_0/
[1] , and , Curvature-driven flows: a variational approach. SIAM J. Control Optim. 31 (1993) 387-438. | MR 1205983 | Zbl 0783.35002
[2] , and , Existence and uniqueness for dislocation dynamics with nonnegative velocity. Interfaces Free Boundaries 7 (2005) 415-434. | MR 2191694 | Zbl 1099.35148
[3] , , and , A convergent scheme for a nonlocal Hamilton-Jacobi equation, modeling dislocation dynamics. Num. Math. 104 (2006) 413-572. | MR 2249672 | Zbl 1103.65100
[4] , , and , Dislocation dynamics: short time existence and uniqueness of the solution. Arch. Rational Mech. Anal. 85 (2006) 371-414. | MR 2231781 | Zbl 1158.74335
[5] , Minimizing movements. Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. 19 (1995) 191-246. | MR 1387558 | Zbl 0957.49029
[6] , and , Gradient flows in metric spaces and in the space of probability measures, Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel (2005). | MR 2129498 | Zbl 1090.35002
[7] , , and , Global existence results and uniqueness for dislocation type equations. SIAM J. Math. Anal. (to appear). | MR 2403312 | Zbl 1158.49029
[8] and , A simple proof of convergence for an approximation scheme for computing motions by mean curvature. SIAM J. Numer. Anal. 32 (1995) 484-500. | MR 1324298 | Zbl 0831.65138
[9] and , Nonlocal first-order hamilton-jacobi equations modelling dislocations dynamics. Comm. Partial Differential Equations 31 (2006) 1191-1208. | MR 2254611 | Zbl 1109.35026
[10] , and , Front propagation and phase field theory. SIAM J. Control Optim. 31 (1993) 439-469. | MR 1205984 | Zbl 0785.35049
[11] , Regularity theory for almost minimal currents. Arch. Rational Mech. Anal. 78 (1982) 99-130. | MR 648941 | Zbl 0485.49024
[12] , On front propagation problems with nonlocal terms. Adv. Differential Equations 5 (2000) 213-268. | MR 1734542 | Zbl 1029.53081
[13] and , On the energy of a flow arising in shape optimisation. Interfaces Free Bound. (to appear). | MR 2453130 | Zbl 1159.35076
[14] and , On the approximation of front propagation problems with nonlocal terms. ESAIM: M2AN 35 (2001) 437-462. | Numdam | MR 1837079 | Zbl 0992.65097
[15] and , Measure theory and fine properties of functions, Studies in Advanced Mathematics. CRC Press, Boca Raton, FL (1992). | MR 1158660 | Zbl 0804.28001
[16] and , Motion of level sets by mean curvature. II. Trans. Amer. Math. Soc. 330 (1992) 321-332. | MR 1068927 | Zbl 0776.53005
[17] , Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer-Verlag New York Inc., New York (1969). | MR 257325 | Zbl 0176.00801
[18] , Dislocations dynamics with a mean curvature term: short time existence and uniqueness. Differential Integral Equations 21 (2008) 285-304. | MR 2484010
[19] , Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies 105. Princeton University Press, Princeton, NJ (1983). | MR 717034 | Zbl 0516.49003
[20] and , 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 1226914 | Zbl 0771.35027
[21] , Minimal surfaces and functions of bounded variation, Monographs in Mathematics 80. Birkhäuser Verlag, Basel (1984). | MR 775682 | Zbl 0545.49018
[22] and , Implicit time discretization for the mean curvature flow equation. Calc. Var. Partial Differential Equations 3 (1995) 253-271. | MR 1386964 | Zbl 0821.35003
[23] , Analytic semigroups and optimal regularity in parabolic problems, Progress in Nonlinear Differential Equations and their Applications 16. Birkhäuser Verlag, Basel (1995). | MR 1329547 | Zbl 0816.35001
[24] , On a free boundary problem of viscous incompressible flows. Interfaces Free Bound. 9 (2007) 549-589. | MR 2358216 | Zbl 1132.76303
[25] , Geometric measure theory. A beginner's guide. Academic Press Inc., Boston, MA (1988). | MR 933756 | Zbl 0671.49043
[26] , and , Regularity and singularity estimates on hypersurfaces minimizing parametric elliptic variational integrals. I, II. Acta Math. 139 (1977) 217-265. | MR 467476 | Zbl 0386.49030
[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 756417 | Zbl 0546.49019
[28] and , Phase-field theory for FitzHugh-Nagumo-type systems. SIAM J. Math. Anal. 27 (1996) 1341-1359. | MR 1402444 | Zbl 0863.35047
