zbMATH — the first resource for mathematics

Vortex filament dynamics for Gross-Pitaevsky type equations. (English) Zbl 1170.35318
Summary: We study solutions of the Gross-Pitaevsky equation and similar equations in \(m\geq 3\) space dimensions in a certain scaling limit, with initial data \(u^\varepsilon_0\) for which the Jacobian \(Ju^\varepsilon_0\) 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 \(\{\Gamma_t\}\) of submanifolds evolves by binomial 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.

35B25 Singular perturbations in context of PDEs
35Q55 NLS equations (nonlinear Schrödinger equations)
Full Text: EuDML
[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. Zbl0585.49030 MR855173 · 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. Zbl1043.35136 MR1655508 · 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. Zbl0933.35177 MR1669387 · Zbl 0933.35177
[5] J. E. Colliander -d R. L. Jerrard, Vortex dynamics for the Ginzburg-Landau-Schrödinger equation, IMRN 1998 (1998), 333-358. Zbl0914.35128 MR1623410 · 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. Zbl0933.35155 MR1753485 · Zbl 0933.35155
[7] W. E, Dynamics of vortices in Ginzburg-Landau theories with applications to superconductivity, Physica D 77 (1994), 383-404. Zbl0814.34039 MR1297726 · Zbl 0814.34039
[8] A. L. Fetter, Vortices in an Imperfect Bose Gas. I. the Condensate, Physical Review 138 (1965), no. 2A, A429-A437. Zbl0127.23103 MR186223 · Zbl 0127.23103
[9] M. Giaquinta - G. Modica - J. Souček, “Cartesian currents in the calculus of variations. I. Cartesian currents”, Springer-Verlag, 1998. Zbl0914.49001 MR1645086 · Zbl 0914.49001
[10] R. L. Jerrard - H. M. Soner, Functions of bounded higher variation, to appear, Calc. Var. MR1911049
[11] R. L. Jerrard - H. M. Soner, The Jacobian and the Ginzburg-Landau energy, to appear, Indiana Univ. Math. Jour. MR1890398 · Zbl 1034.35025
[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. Zbl0946.49033 MR1724082 · 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. Zbl0944.35006 MR1697561 · 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. Zbl0932.35121 MR1491752 · 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. Zbl0920.35145 MR1674000 · 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. Zbl0877.35018 MR1443051 · 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. Zbl1017.82046 MR1750093 · 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. MR1133125
[19] L. M. Pismen - J. Rubinstein, Motion of vortex lines in the Ginzburg-Landau model, Physica D 47 (1991), 353-360. Zbl0728.35090 MR1098255 · 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. Zbl0908.58004 MR1607928 · Zbl 0908.58004
[21] L. Simon, “Lectures on geometric measure theory”, Australian National University, 1984. Zbl0546.49019 MR756417 · Zbl 0546.49019
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.