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Minimizing movements for dislocation dynamics with a mean curvature term

Published online by Cambridge University Press:  23 January 2009

Nicolas Forcadel  and
Aurélien Monteillet
Nicolas Forcadel
Affiliation:
CERMICS, École des Ponts, Paris Tech, 6 et 8 avenue Blaise Pascal, Cité Descartes, Champs-sur-Marne, 77455 Marne-la-Vallée Cedex 2, France. [email protected]
Aurélien Monteillet
Affiliation:
Université de Bretagne Occidentale, UFR Sciences et Techniques, 6 av. Le Gorgeu, BP 809, 29285 Brest, France. [email protected]
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Abstract

We prove existence of minimizing movements for thedislocation 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 viscositysolutions of the corresponding level-set equation. We also prove theconsistency of this approach, by showing that any minimizing movementcoincides with the smooth evolution as long as the latter exists. Inrelation with this, we finally prove short time existence and uniqueness of a smoothfront evolving according to our law, provided the initial shape issmooth enough.

Keywords

Front propagationnon-local equationsdislocation dynamicsmean curvature motionviscosity solutionsminimizing movementssets of finite perimetercurrents
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 15 , Issue 1 , January 2009 , pp. 214 - 244
Copyright
© EDP Sciences, SMAI, 2008

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References

F. Almgren, J.E. Taylor and L. Wang, Curvature-driven flows: a variational approach. SIAM J. Control Optim. 31 (1993) 387–438.
Alvarez, O., Cardaliaguet, P. and Monneau, R., Existence and uniqueness for dislocation dynamics with nonnegative velocity. Interfaces Free Boundaries 7 (2005) 415–434. CrossRef
Alvarez, O., Carlini, E., Monneau, R. and Rouy, E., A convergent scheme for a nonlocal Hamilton-Jacobi equation, modeling dislocation dynamics. Num. Math. 104 (2006) 413–572. CrossRef
Alvarez, O., Hoch, P., Le Bouar, Y. and Monneau, R., Dislocation dynamics: short time existence and uniqueness of the solution. Arch. Rational Mech. Anal. 85 (2006) 371–414.
Ambrosio, L., Minimizing movements. Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. 19 (1995) 191–246.
L. Ambrosio, N. Gigli and G. Savaré, Gradient flows in metric spaces and in the space of probability measures, Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel (2005).
G. Barles, P. Cardaliaguet, O. Ley and R. Monneau, Global existence results and uniqueness for dislocation type equations. SIAM J. Math. Anal. (to appear).
Barles, G. and Georgelin, C., A simple proof of convergence for an approximation scheme for computing motions by mean curvature. SIAM J. Numer. Anal. 32 (1995) 484–500. CrossRef
Barles, G. and Nonlocal, O. Ley first-order hamilton-jacobi equations modelling dislocations dynamics. Comm. Partial Differential Equations 31 (2006) 1191–1208. CrossRef
Barles, G., Soner, H.M. and Souganidis, P.E., Front propagation and phase field theory. SIAM J. Control Optim. 31 (1993) 439–469. CrossRef
Bombieri, E., Regularity theory for almost minimal currents. Arch. Rational Mech. Anal. 78 (1982) 99–130. CrossRef
Cardaliaguet, P., On front propagation problems with nonlocal terms. Adv. Differential Equations 5 (2000) 213–268.
P. Cardaliaguet and O. Ley, On the energy of a flow arising in shape optimisation. Interfaces Free Bound. (to appear).
Cardaliaguet, P. and Pasquignon, D., On the approximation of front propagation problems with nonlocal terms. ESAIM: M2AN 35 (2001) 437–462. CrossRef
L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics. CRC Press, Boca Raton, FL (1992).
Evans, L.C. and Spruck, J., Motion of level sets by mean curvature. II. Trans. Amer. Math. Soc. 330 (1992) 321–332. CrossRef
H. Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer-Verlag New York Inc., New York (1969).
Forcadel, N., Dislocations dynamics with a mean curvature term: short time existence and uniqueness. Differential Integral Equations 21 (2008) 285–304.
M. Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies 105. Princeton University Press, Princeton, NJ (1983).
Y. Giga and S. Goto, 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.
E. Giusti, Minimal surfaces and functions of bounded variation, Monographs in Mathematics 80. Birkhäuser Verlag, Basel (1984).
Luckhaus, S. and Sturzenhecker, T., Implicit time discretization for the mean curvature flow equation. Calc. Var. Partial Differential Equations 3 (1995) 253–271. CrossRef
A. Lunardi, Analytic semigroups and optimal regularity in parabolic problems, Progress in Nonlinear Differential Equations and their Applications 16. Birkhäuser Verlag, Basel (1995).
Maekawa, Y., On a free boundary problem of viscous incompressible flows. Interfaces Free Bound. 9 (2007) 549–589.
F. Morgan, Geometric measure theory. A beginner's guide. Academic Press Inc., Boston, MA (1988).
R. Schoen, L. Simon and F.J. Almgren, Jr., Regularity and singularity estimates on hypersurfaces minimizing parametric elliptic variational integrals. I, II. Acta Math. 139 (1977) 217–265.
L. Simon, Lectures on geometric measure theory, in Proceedings of the Centre for Mathematical Analysis, Vol. 3, Australian National University Centre for Mathematical Analysis, Canberra (1983).
P. Soravia and P.E. Souganidis, Phase-field theory for FitzHugh-Nagumo-type systems. SIAM J. Math. Anal. 27 (1996) 1341–1359.