Vortex filament dynamics for Gross-Pitaevsky type equations
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 1 (2002) no. 4, p. 733-768

We study solutions of the Gross-Pitaevsky equation and similar equations in $m\ge 3$ space dimensions in a certain scaling limit, with initial data ${u}_{0}^{ϵ}$ for which the jacobian $J{u}_{0}^{ϵ}$ concentrates around an (oriented) rectifiable $m-2$ dimensional set, say ${\Gamma }_{0}$, of finite measure. It is widely conjectured that under these conditions, the jacobian at later times $t>0$ continues to concentrate around some codimension $2$ submanifold, say ${\Gamma }_{t}$, and that the family $\left\{{\Gamma }_{t}\right\}$ of submanifolds evolves by binormal mean curvature flow. We prove this conjecture when ${\Gamma }_{0}$ is a round $m-2$-dimensional sphere with multiplicity $1$. We also prove a number of partial results for more general inital data.

Classification:  35B25,  35Q55
@article{ASNSP_2002_5_1_4_733_0,
author = {Jerrard, Robert L.},
title = {Vortex filament dynamics for Gross-Pitaevsky type equations},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
publisher = {Scuola normale superiore},
volume = {Ser. 5, 1},
number = {4},
year = {2002},
pages = {733-768},
zbl = {pre02217020},
mrnumber = {1991001},
language = {en},
url = {http://www.numdam.org/item/ASNSP_2002_5_1_4_733_0}
}

Jerrard, Robert L. Vortex filament dynamics for Gross-Pitaevsky type equations. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 1 (2002) no. 4, pp. 733-768. http://www.numdam.org/item/ASNSP_2002_5_1_4_733_0/

[1] G. Alberti - S. Baldo - G. Orlandi, in preparation.

[2] F. Almgren, Optimal isoperimetric inequalities, Indiana Univ. Math. J. 35 (1986), no. 3, 451-547. | MR 855173 | Zbl 0585.49030

[3] L. Ambrosio - H. M. Soner, A measure theoretic approach to higher codimension mean curvature flow, Ann. Scuola Norm. Sup. Pisa Cl. Sci (4) 25 (1997), 27-48. | Numdam | MR 1655508 | Zbl 1043.35136

[4] F. Bethuel - J.-C. Saut, Travelling waves for the Gross-Pitaevsky equation i, Ann. Inst. H. Poincaré Phys. Théor. 70 (1999), no. 2, 147-238. | Numdam | MR 1669387 | Zbl 0933.35177

[5] J. E. Colliander -d R. L. Jerrard, Vortex dynamics for the Ginzburg-Landau-Schrödinger equation, IMRN 1998 (1998), 333-358. | MR 1623410 | Zbl 0914.35128

[6] J. E. Colliander -d R. L. Jerrard, Ginzburg-Landau vortices: weak stability and Schrödinger equation dynamics, Journal d'Analyse Mathematique 77 (1999), 129-205. | MR 1753485 | Zbl 0933.35155

[7] W. E, Dynamics of vortices in Ginzburg-Landau theories with applications to superconductivity, Physica D 77 (1994), 383-404. | MR 1297726 | Zbl 0814.34039

[8] A. L. Fetter, Vortices in an Imperfect Bose Gas. I. the Condensate, Physical Review 138 (1965), no. 2A, A429-A437. | MR 186223 | Zbl 0127.23103

[9] M. Giaquinta - G. Modica - J. Souček, “Cartesian currents in the calculus of variations. I. Cartesian currents”, Springer-Verlag, 1998. | MR 1645086 | Zbl 0914.49001

[10] R. L. Jerrard - H. M. Soner, Functions of bounded higher variation, to appear, Calc. Var. | MR 1911049

[11] R. L. Jerrard - H. M. Soner, The Jacobian and the Ginzburg-Landau energy, to appear, Indiana Univ. Math. Jour. | MR 1890398

[12] R. L. Jerrard - H. M. Soner, Rectifiability of the distributional Jacobian for a class of functions, C.R. Acad. Sci. Paris, Série I 329 (1999), 683-688. | MR 1724082 | Zbl 0946.49033

[13] 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 16 (1999), no. 4, 423-466. | Numdam | MR 1697561 | Zbl 0944.35006

[14] F. H. Lin, Complex Ginzburg-Landau equations and dynamics of vortices, filaments, and codimension 2 submanifolds, Comm. Pure Appl. Math. 51 (1998), no. 4, 385-441. | MR 1491752 | Zbl 0932.35121

[15] F. H. Lin - J. X. Xin, On the incompressible fluid limit and the vortex law of motion of the nonlinear Schrödinger equation, Comm. Math. Phys. 200 (1999), 249-274. | MR 1674000 | Zbl 0920.35145

[16] T. C. Lin, On the stability of the radial solution to the Ginzburg-Landau equation, Comm. Partial Differential Equations 22 (1997), no. 3-4, 619-632. | MR 1443051 | Zbl 0877.35018

[17] T. C. Lin, Rigorous and generalized derivation of vortex line dynamics in superfluids and superconductors, SIAM J. Appl. Math. 60 (2000), no. 3, 1099-1110. | MR 1750093 | Zbl 1017.82046

[18] F. Lund, Defect dynamics for the nonlinear Schrödinger equation derived from a variational principle, Physics Letters A 159 (1991), 245-251. | MR 1133125

[19] L. M. Pismen - J. Rubinstein, Motion of vortex lines in the Ginzburg-Landau model, Physica D 47 (1991), 353-360. | MR 1098255 | Zbl 0728.35090

[20] E. Sandier, Lower bounds for the energy of unit vector fields and applications, J. Funct. Anal. 152 (1998), no. 2, 379-403. | MR 1607928 | Zbl 0908.58004

[21] L. Simon, “Lectures on geometric measure theory”, Australian National University, 1984. | MR 756417 | Zbl 0546.49019