Vortex filament dynamics for Gross-Pitaevsky type equations

Jerrard, Robert L.

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 1 (2002) no. 4, pp. 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.

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

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

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

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

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

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

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

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

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