Comptes Rendus
Partial Differential Equations
Convergence of the parabolic Ginzburg–Landau equation to motion by mean curvature
[Convergence de l'équation de Ginzburg–Landau parabolique vers un mouvement par courbure moyenne]
Comptes Rendus. Mathématique, Volume 336 (2003) no. 9, pp. 719-723.

Nous présentons de nouveaux résultats concernant l'étude asymptotique du flot de la chaleur pour l'énergie de Ginzburg–Landau. En particulier, nous montrons que, lorsque le paramètre ε tend vers 0, la vorticité évolue selon un mouvement par courbure moyenne, dans un sens faible introduit par Brakke. Notre seule hypothèse concerne une borne naturelle portant sur l'énergie de la condition initiale. Dans certains cas, nous montrons également la convergence vers un mouvement par courbure moyenne dans un sens plus fort dû à Ilmanen.

We present some new results for the asymptotic behavior of the complex parabolic Ginzburg–Landau equation. In particular, we establish that, as the parameter ε tends to 0, vorticity evolves according to motion by mean curvature in Brakke's weak formulation. The only assumption we make is a natural energy bound on the initial data. In some cases, we also prove convergence to enhanced motion in the sense of Ilmanen.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(03)00167-5

Fabrice Bethuel 1, 2 ; Giandomenico Orlandi 3 ; Didier Smets 1

1 Laboratoire Jacques-Louis Lions, Université de Paris 6, 4, place Jussieu, BC 187, 75252 Paris, France
2 Institut universitaire de France
3 Dipartimento di Informatica, Università di Verona, Strada le Grazie, 37134 Verona, Italy
@article{CRMATH_2003__336_9_719_0,
     author = {Fabrice Bethuel and Giandomenico Orlandi and Didier Smets},
     title = {Convergence of the parabolic {Ginzburg{\textendash}Landau} equation to motion by mean curvature},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {719--723},
     publisher = {Elsevier},
     volume = {336},
     number = {9},
     year = {2003},
     doi = {10.1016/S1631-073X(03)00167-5},
     language = {en},
}
TY  - JOUR
AU  - Fabrice Bethuel
AU  - Giandomenico Orlandi
AU  - Didier Smets
TI  - Convergence of the parabolic Ginzburg–Landau equation to motion by mean curvature
JO  - Comptes Rendus. Mathématique
PY  - 2003
SP  - 719
EP  - 723
VL  - 336
IS  - 9
PB  - Elsevier
DO  - 10.1016/S1631-073X(03)00167-5
LA  - en
ID  - CRMATH_2003__336_9_719_0
ER  - 
%0 Journal Article
%A Fabrice Bethuel
%A Giandomenico Orlandi
%A Didier Smets
%T Convergence of the parabolic Ginzburg–Landau equation to motion by mean curvature
%J Comptes Rendus. Mathématique
%D 2003
%P 719-723
%V 336
%N 9
%I Elsevier
%R 10.1016/S1631-073X(03)00167-5
%G en
%F CRMATH_2003__336_9_719_0
Fabrice Bethuel; Giandomenico Orlandi; Didier Smets. Convergence of the parabolic Ginzburg–Landau equation to motion by mean curvature. Comptes Rendus. Mathématique, Volume 336 (2003) no. 9, pp. 719-723. doi : 10.1016/S1631-073X(03)00167-5. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00167-5/

[1] L. Ambrosio; M. Soner A measure theoretic approach to higher codimension mean curvature flow, Ann. Scoula Norm. Sup. Pisa Cl. Sci., Volume 25 (1997), pp. 27-49

[2] F. Bethuel; J. Bourgain; H. Brezis; G. Orlandi W1,p estimates for solutions to the Ginzburg–Landau functional with boundary data in H1/2, C. R. Acad. Sci. Paris, Ser. I, Volume 333 (2001), pp. 1-8

[3] F. Bethuel; H. Brezis; F. Hélein Ginzburg–Landau Vortices, Birkhäuser, Boston, 1994

[4] F. Bethuel; H. Brezis; G. Orlandi Asymptotics for the Ginzburg–Landau equation in arbitrary dimensions, J. Funct. Anal., Volume 186 (2001), pp. 432-520 Erratum 188 (2002) 548–549

[5] F. Bethuel; G. Orlandi Uniform estimates for the parabolic Ginzburg–Landau equation, ESAIM: Control Optim. Calc. Var., Volume 8 (2002), pp. 219-238

[6] F. Bethuel, G. Orlandi, D. Smets, Vortex rings for the Gross–Pitaevskii equation, J. European Math. Soc., to appear

[7] F. Bethuel, G. Orlandi, D. Smets, Convergence of the parabolic Ginzburg–Landau equation to motion by mean curvature, Preprint

[8] F. Bethuel; T. Rivière A minimization problem related to superconductivity, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 12 (1995), pp. 243-303

[9] J. Bourgain; H. Brezis; P. Mironescu On the structure of the Sobolev space H1/2 with values into the circle, C. R. Acad. Sci. Paris, Ser. I, Volume 331 (2000), pp. 119-124 (detailed paper to appear)

[10] K. Brakke The Motion of a Surface by its Mean Curvature, Princeton University Press, 1978

[11] L. Bronsard; R.V. Kohn Motion by mean curvature as the singular limit of Ginzburg–Landau dynamics, J. Differential Equations, Volume 90 (1991), pp. 211-237

[12] X. Chen Generation and propagation of interfaces for reaction–diffusion equations, J. Differential Equations, Volume 96 (1992), pp. 116-141

[13] P. de Mottoni; M. Schatzman Development of interfaces in N , Proc. Roy. Soc. Edinburgh Sect. A, Volume 116 (1990), pp. 207-220

[14] L.C. Evans; H.M. Soner; P.E. Souganidis Phase transitions and generalized motion by mean curvature, Comm. Pure Appl. Math., Volume 45 (1992), pp. 1097-1123

[15] T. Ilmanen Convergence of the Allen–Cahn equation to Brakke's motion by mean curvature, J. Differential Geom., Volume 38 (1993), pp. 417-461

[16] T. Ilmanen Elliptic regularization and partial regularity for motion by mean curvature, Mem. Amer. Math. Soc., Volume 108 (1994) no. 520

[17] R.L. Jerrard; H.M. Soner Dynamics of Ginzburg–Landau vortices, Arch. Rational Mech. Anal., Volume 142 (1998), pp. 99-125

[18] R.L. Jerrard; H.M. Soner Scaling limits and regularity results for a class of Ginzburg–Landau systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 16 (1999), pp. 423-466

[19] R.L. Jerrard; H.M. Soner The Jacobian and the Ginzburg–Landau energy, Calc. Var. Partial Differential Equations, Volume 14 (2002), pp. 151-191

[20] F.H. Lin Some dynamical properties of Ginzburg–Landau vortices, Comm. Pure Appl. Math., Volume 49 (1996), pp. 323-359

[21] F.H. Lin Complex Ginzburg–Landau equations and dynamics of vortices, filaments, and codimension-2 submanifolds, Comm. Pure Appl. Math., Volume 51 (1998), pp. 385-441

[22] F.H. Lin; T. Rivière Complex Ginzburg–Landau equation in high dimension and codimension 2 area minimizing currents, J. European Math. Soc., Volume 1 (1999), pp. 237-311 (Erratum)

[23] F.H. Lin; T. Rivière A quantization property for static Ginzburg–Landau vortices, Comm. Pure Appl. Math., Volume 54 (2001), pp. 206-228

[24] F.H. Lin; T. Rivière A quantization property for moving line vortices, Comm. Pure Appl. Math., Volume 54 (2001), pp. 826-850

[25] L.M. Pismen; J. Rubinstein Motion of vortex lines in the Ginzburg–Landau model, Physica D, Volume 47 (1991), pp. 353-360

[26] T. Rivière Line vortices in the U(1) Higgs model, ESAIM: Control Optim. Calc. Var., Volume 1 (1996), pp. 77-167

[27] H.M. Soner Ginzburg–Landau equation and motion by mean curvature. I. Convergence; II. Development of the initial interface, J. Geom. Anal., Volume 7 (1997) no. 3, pp. 437-475 (477–491)

[28] M. Struwe On the asymptotic behavior of the Ginzburg–Landau model in 2-dimensions, J. Differential Equation, Volume 7 (1994), pp. 1613-1624 Erratum 8 (1995) 224

[29] C. Wang, On moving Ginzburg–Landau filament vortices, Max-Planck-Institut Leipzig, Preprint

Cité par Sources :

Commentaires - Politique