Vortex motion law for the Schrödinger-Ginzburg-Landau equations

SIAM Journal on Mathematical Analysis

Volumn 34, Issue 6, 2003, Pages 1435-1476

  Spirn, Daniel ^a  

^a BROWN UNIVERSITY   (United States)

Author keywords

Ginzburg Landau theory; Superconductivity; Vortex dynamics

Indexed keywords

EID: 0242511184     PISSN: 00361410     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036141001396667     Document Type: Article

References (36)

1. P. BETHUEL, H. BREZIS, AND F. HELEIN, Ginzburg-Landau Vortices, Birkhäuser Boston, Boston, MA, 1994.
2. F. BETHUEL AND T. RIVIERE, Vortices for a variational problem related to superconductivity, Ann. Inst. H. Póincare Anal. Non Linéaire, 12 (1995), pp. 243-303.
3. H. BREZIS AND T. GALLOUËT, Nonlinear Schrödinger evolution equations, Nonlinear Anal., 4 (1980), pp. 677-681.
4. C. CAROLI AND K. MAKI, Fluctuations of the order parameter in Type-II superconductors. I. Dirty Limit, Phys. Rev., 159 (1967), pp. 306-315.
5. C. CAROLI AND K. MAKI, Fluctuations of the order parameter in Type-II superconductors. II. Pure Limit, Phys. Rev., 159 (1967), pp. 315-326.
6. Y. M. CHEN AND F. H. LIN, Evolution of harmonic maps with Dirichlet boundary conditions, Comm. Anal. Geom., 1 (1993), pp. 327-346.
7. A. CHORIN, Vorticity and Turbulence, Springer-Verlag, New York, 1994.
8. J. E. COLLIANDER AND R. L. JERRARD, Ginzburg-Landau vortices: Weak stability and Schrödinger equation dynamics, J. Anal. Math., 77 (1999), pp. 129-205.
9. M. COMTE AND P. MIRONESCU, The behavior of a Ginzburg-Landau minimizer near its zeroes, Calc. Var. Partial Differential Equations, 4 (1996), pp. 323-340.
10. B. S. DEAVER AND W. M. FAIRBANK, Experimental evidence for quantized flux in superconducting cylinders, Phys. Rev. Lett., 7 (1961), pp. 51-53.
11. R. DOLL AND M. NÄBAUER, Experimental proof of magnetic flux quantization in a superconducting ring, Phys. Rev. Lett., 7 (1961), pp. 43-46.
12. Q. DU Global existence and uniqueness of solutions of the time-dependent Ginzburg-Landau model for superconductivity, Appl. Anal., 53 (1994), pp. 1-17.
13. W. E, Dynamics of vortices in Ginzburg-Landau theories with applications to superconductivity, Phys. D, 77 (1994), pp. 383-404.
14. R. FEYNMAN, Statistical Mechanics W.A. Benjamin, Reading, MA, 1972.
15. V. L. GINZBURG AND L. D. LANDAU, On the theory of superconductivity, in Collected Papers of L. D. Landau, Pergamon Press, New York, 1965, pp. 626-633.
16. S. GUSTAFSON AND D. SPIRN, Vortex Motion Law for the Maxwell-Higgs Equations, preprint.
17. R. L. JERRARD, Lower bounds for generalized Ginzburg-Landau functional, SIAM J. Math. Anal., 30 (1999), pp. 721-746.
18. R. L. JERRARD, personal communication.
19. R. L. JERRARD AND H. M. SONER, Dynamics of Ginzburg-Landau vortices, Arch. Ration. Mech. Anal., 142 (1998), pp. 99-125.
20. F. H. LIN, Some dynamic properties of Ginzburg-Landau vortices, Comm. Pure Appl. Math., 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., 51 (1998), pp. 385-441.
22. F. H. LIN, Vortex dynamics for the nonlinear wave equation, Comm. Pure Appl. Math., 52 (1999), pp. 737-761.
23. F. H. LIN AND Q. DU. Ginzburg-Landau vortices: Dynamics, pinning, and hysteresis, SIAM J. Math. Anal, 28 (1997), pp, 1265-1293.
24. F. H. LIN AND J. X. XIN, On the incompressible fluid limit and the vortex motion law of the nonlinear Schrödinger equation, Comm. Math. Phys., 200 (1999), pp. 249-274.
25. M. MACHIDA AND H. KABURAKI, Direct simulation of the time-dependent Ginzburg-Landau equation for type-II superconducting thin film: Vortex dynamics and V-I characteristics, Phys. Rev. Lett., 71 (1993), pp. 3206-3209.
26. E. MADELUNG, Quantentheorie in hydrodynamischer form, Z. Physik, 40 (1927), p. 322.
27. Y. A. MELNIKOV, Green's Functions in Applied Mathematics, Computational Mechanics Publications, Southampton, 1995.
28. J. NEU, Vortices in complex scalar fields, Phys. D, 43 (1990), pp. 385-406.
29. L. PERES AND J. RUBINSTEIN, Vortex dynamics in U(1) Ginzburg-Landau models, Phys. D, 64 (1993), pp. 299-309.
30. E. SANDIER AND S. SERFATY, Global minimizers for the Ginzburg-Landau functional below the first critical magnetic field, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), pp. 119-145.
31. S. SERFATY, Local minimizers for the Ginzburg-Landau equations near critical magnetic field: Part I, Comm. Cont. Math., 2 (1999). pp. 213-254.
32. S. SERFATY, Local minimizers for the Ginzburg-Landau equations near critical magnetic field: Part II, Comm. Cont. Math., 3 (1999), pp. 295-333.
33. D. SPIRN, Vortex dynamics of the full time-dependent Ginzburg-Landau equations, Comm. Pure Appl. Math., 55 (2002), pp. 537-581.
34. M. STRUWE, On the asymptotic behavior of minimizers of the Ginzburg-Landau model in 2 dimensions, Differential Integral Equations, 7 (1994), pp. 1613-1624.
35. M. TINKHAM, Introduction to Superconductivity McGraw-Hill, New York, 1996.
36. A. TONOMURA, H. KASAI, O. KAMIMURA, T. MATSUDA, K. HARADA, J. SHIMOYAMA, K. KISHIO, AND K. KITAZAWA, Motion of vortices in superconductors, Nature, 397 (1999), pp. 308-309.