Vortex motion and phase-vortex interaction in dissipative Ginzburg-Landau dynamics
Journées équations aux dérivées partielles (2004), article no. 10, 12 p.

We discuss the asymptotics of the parabolic Ginzburg-Landau equation in dimension N2. Our only asumption on the initial datum is a natural energy bound. Compared to the case of “well-prepared” initial datum, this induces possible new energy modes which we analyze, and in particular their mutual interaction. The two dimensional case is qualitatively different and requires a separate treatment.

Nous étudions l’équation de Ginzburg-Landau parabolique sur l’espace tout entier, plus particulièrement lorsqu’une des échelles caractéristiques tend vers zéro. Notre seule hypothèse sur la donnée initiale est une borne naturelle sur l’énergie. En comparaison avec le cas des données préparées, notre hypothèse laisse place à de nouveaux phénomènes, en particulier la présence de différents modes pour l’énergie, dont nous étudions l’interaction. Le cas de la dimension 2 d’espace est qualitativement différent et requiert une analyse séparée.

DOI: 10.5802/jedp.10
Bethuel, F. 1; Orlandi, G. 2; Smets, D. 3

1 Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, 4 place Jussieu BC 187, 75252 Paris, France & Institut Universitaire de France.
2 Dipartimento di Informatica, Università di Verona, Strada le Grazie, 37134 Verona, Italy.
3 Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, 4 place Jussieu BC 187, 75252 Paris, France.
@article{JEDP_2004____A10_0,
     author = {Bethuel, F. and Orlandi, G. and Smets, D.},
     title = {Vortex motion and phase-vortex interaction in dissipative {Ginzburg-Landau} dynamics},
     journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     eid = {10},
     pages = {1--12},
     publisher = {Groupement de recherche 2434 du CNRS},
     year = {2004},
     doi = {10.5802/jedp.10},
     zbl = {1067.35031},
     mrnumber = {2135365},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/jedp.10/}
}
TY  - JOUR
AU  - Bethuel, F.
AU  - Orlandi, G.
AU  - Smets, D.
TI  - Vortex motion and phase-vortex interaction in dissipative Ginzburg-Landau dynamics
JO  - Journées équations aux dérivées partielles
PY  - 2004
SP  - 1
EP  - 12
PB  - Groupement de recherche 2434 du CNRS
UR  - http://www.numdam.org/articles/10.5802/jedp.10/
DO  - 10.5802/jedp.10
LA  - en
ID  - JEDP_2004____A10_0
ER  - 
%0 Journal Article
%A Bethuel, F.
%A Orlandi, G.
%A Smets, D.
%T Vortex motion and phase-vortex interaction in dissipative Ginzburg-Landau dynamics
%J Journées équations aux dérivées partielles
%D 2004
%P 1-12
%I Groupement de recherche 2434 du CNRS
%U http://www.numdam.org/articles/10.5802/jedp.10/
%R 10.5802/jedp.10
%G en
%F JEDP_2004____A10_0
Bethuel, F.; Orlandi, G.; Smets, D. Vortex motion and phase-vortex interaction in dissipative Ginzburg-Landau dynamics. Journées équations aux dérivées partielles (2004), article  no. 10, 12 p. doi : 10.5802/jedp.10. http://www.numdam.org/articles/10.5802/jedp.10/

[1] G. Alberti, S. Baldo and G. Orlandi, Variational convergence for functionals of Ginzburg-Landau type, Indiana Math. Journal, submitted. | Zbl

[2] L. Ambrosio and M. Soner, A measure theoretic approach to higher codimension mean curvature flow, Ann. Sc. Norm. Sup. Pisa, Cl. Sci. 25 (1997), 27-49. | Numdam | MR | Zbl

[3] P. Baumann, C-N. Chen, D. Phillips, P. Sternberg, Vortex annihilation in nonlinear heat flow for Ginzburg-Landau systems, Eur. J. Appl. Math. 6 (1995), 115–126. | MR | Zbl

[4] F. Bethuel, G. Orlandi and D. Smets, Vortex rings for the Gross-Pitaevskii equation, Jour. Eur. Math. Soc. 6 (2004), 17-94. | MR | Zbl

[5] F. Bethuel, G. Orlandi and D. Smets, Convergence of the parabolic Ginzburg-Landau equation to motion by mean curvature, Annals of Math., to appear. | MR | Zbl

[6] F. Bethuel, G. Orlandi and D. Smets, Collisions and phase-vortex interactions in dissipative Ginzburg-Landau dynamics, preprint. | Zbl

[7] K. Brakke, The motion of a surface by its mean curvature, Princeton University Press, 1978. | MR | Zbl

[8] L. Bronsard and R.V. Kohn, Motion by mean curvature as the singular limit of Ginzburg-Landau dynamics, J. Differential Equations 90 (1991), 211-237. | MR | Zbl

[9] W. E, Dynamics of vortices in Ginzburg-Landau theories with applications to superconductivity, Phys. D 77 (1994), no. 4, 383-404. | MR | Zbl

[10] G. Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom. 20 (1984), 237-266. | MR | Zbl

[11] G. Huisken, Asymptotic behavior for singularities of the mean curvature flow, J. Differential Geom. 31 (1990), 285-299. | MR | Zbl

[12] T. Ilmanen, Convergence of the Allen-Cahn equation to Brakke’s motion by mean curvature, J. Differential Geom. 38 (1993), 417-461. | MR | Zbl

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

[14] R.L. Jerrard and H.M. Soner, Dynamics of Ginzburg-Landau vortices, Arch. Rational Mech. Anal. 142 (1998), 99-125. | MR | Zbl

[15] R.L. Jerrard and H.M. Soner, Scaling limits and regularity results for a class of Ginzburg-Landau systems, Ann. Inst. H. Poincaré Anal. Non Linéaire 16 (1999), 423-466. | Numdam | MR | Zbl

[16] R.L. Jerrard and H.M. Soner, The Jacobian and the Ginzburg-Landau energy, Calc. Var. PDE 14 (2002), 151-191. | MR | Zbl

[17] F.H. Lin, Some dynamical properties of Ginzburg-Landau vortices, Comm. Pure Appl. Math. 49 (1996), 323–359. | MR | Zbl

[18] F.H. Lin, Complex Ginzburg-Landau equations and dynamics of vortices, filaments, and codimension-2 submanifolds, Comm. Pure Appl. Math. 51 (1998), 385-441. | MR | Zbl

[19] J.C. Neu, Vortices in complex scalar fields, Phys. D 43 (1990), no.2-3, 385-406. | MR | Zbl

[20] L.M. Pismen and J. Rubinstein, Motion of vortex lines in the Ginzburg-Landau model, Phys. D 47 (1991), 353-360. | MR | Zbl

[21] J. Rubinstein and P. Sternberg, On the slow motion of vortices in the Ginzburg-Landau heat-flow, SIAM J. Appl. Math. 26 (1995), 1452-1466. | MR | Zbl

[22] E. Sandier and S. Serfaty, Gamma-convergence of gradient flows with applications to Ginzburg-Landau, Comm. Pure App. Math., to appear. | MR | Zbl

[23] S. Serfaty, Vortex Collision and Energy Dissipation Rates in the Ginzburg-Landau Heat Flow, in preparation.

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

[25] D. Spirn, Vortex dynamics of the full time-dependent Ginzburg-Landau equations, Comm. Pure Appl. Math. 55 (2002), no. 5, 537-581. | MR | Zbl

Cited by Sources: