Motion of concentration sets in Ginzburg-Landau equations

 Access to full text

 Full (PDF)

1. [1] Alberti ( G. ) , Baldo ( S.) and Orlandi ( G.), Variational convergence for functionals of Ginzburg-Landau type, preprint 2002. 
2. [2] Alweida ( L. ) and Bethuel ( F.), Topological methods for the Ginzburg-Landau equation, J. Math Pures Appl.11, p. 1-49 (1998). Zbl0904.35023MR1617594
3. [3] Ambrosio ( L.) and Soner ( M.), A measure theoretic approach to higher codimension mean curvature flow, Ann. Sc. Norm. Sup. Pisa, Cl. Sci.25, p. 27-49 (1997). Zbl1043.35136MR1655508
4. [4] Bethuel ( F. ), Bourgain ( J.), Brezis ( H.) and Orlandi ( G.), W1,p estimates for solutions to the Ginzburg-Landau functional with boundary data in H1/2 , C. R. Acad. Sci. Paris (1) 333, p. 1-8 (2001). Zbl1080.35020MR1881236
5. [5] Bethuel ( F. ) , Brezis ( H.) and Hélein ( F.), Ginzburg-Landau vortices, Birkhäuser, Boston, 1994. Zbl0802.35142MR1269538
6. [6] Bethuel ( F. ), Brezis ( H.) and Orlandi ( G.), Asymptotics for the Ginzburg-Landau equation in arbitrary dimensions, J. Funct. Anal.186 (2001), p. 432-520. Zbl1077.35047MR1864830
7. Erratum188, p. 548-549 (2002). MR1883416
8. [7] Bethuel ( F. ) and Orlandi ( G.), Uniform estimates for the parabolic Ginzburg-Landau equation, ESAIM, C.O.C.V8, p. 219-238 (2002). Zbl1078.35013MR1932951
9. [8] Bethuel ( F. ), Orlandi ( G.) and Smets ( D.), Vortex rings for the Gross-Pitaevskii equation, J. Eur. Math. Soc., to appear. Zbl1091.35085
10. [9] Bethuel ( F. ), Orlandi ( G.) and Smets ( D.), Convergence of the parabolic Ginzburg-Landau equation to motion by mean curvature, C. R. Acad. Sci. Paris (I) 336, (2003) and preprint Lab. J.L. Lions (2003). Zbl1031.35026MR1988309
11. [10] Bethuel ( F.) and Rivière ( T.), A minimization problem related to superconductivity, Annales IHP, Analyse Non Linéaire12, p. 243-303 (1995). Zbl0842.35119
12. [11] Bethuel ( F. ) and Saut ( J.C.), Travelling waves for the Gross-Pitaevskii equation, Annales IHP, Phys. Théor.70, (1999). Zbl0933.35177MR1669387
13. [12] Bourgain ( J. ) , Brezis ( H.) and Mironescu ( P.), On the structure of the Sobolev space H1/2 with values into the circle, C.R. Acad. Sci. Paris (I) 331, p. 119-124, (2000) and detailed paper in preparation. Zbl0970.35069MR1781527
14. [13] Brakke ( K.) , The motion of a surface by its mean curvature, Princeton University Press (1978). Zbl0386.53047MR485012
15. [14] Brezis ( H. ) and Mironescu ( P.), Sur une conjecture de E. De Giorgi relative à l'énergie de Ginzburg-Landau, C. R. Acad. Sci. Paris Sér. I Math.319, p. 167-170 (1994). Zbl0805.49004MR1288397
16. [15] Bronsard ( L.) and Kohn ( R.V.), Motion by mean curvature as the singular limit of Ginzburg-Landau dynamics,. J. Differential Equations90, p. 211-237 (1991). Zbl0735.35072MR1101239
17. [16] Carr ( J.) , Pego ( R. ), Metastable patterns in solutions of u t = ε2uxx - f(u, Comm. Pure Appl. Math.42, p. 523-576 (1989). Zbl0685.35054MR997567
18. [17] Chen ( X.) , Generation and propagation of interfaces for reaction-diffusion equations, J. Differential Equations96, p. 116-141 (1992). Zbl0765.35024MR1153311
19. [18] Chen ( Y.) and Struwe ( M.), Existence and partial regularity results for the heat flow for harmonic maps, Math. Z.201, p. 83-103 (1989). Zbl0652.58024MR990191
20. [19] De Giorgi ( E. ), Some conjectures on flow by mean curvature , Proc. of the Capri Workshop ( 1990), Benevento-Bruno-Sbordone (eds.) (1990). 
21. [20] De Mottoni ( P.) and Schatzman ( M.), Development of interfaces in RN, Proc. Roy. Soc. Edinburgh Sect.A116, p. 207-220 (1990). Zbl0725.35009MR1084732
22. [21] Evans ( L.C. ), Soner ( H.M.) and Souganidis ( P.E.), Phase transitions and generalized motion by mean curvature, Comm. Pure Appl. Math.45, p. 1097-1123 (1992 ). Zbl0801.35045MR1177477
23. [22] Federer ( H. ) , Geometric Measure Theory, Springer , Berlin, (1969). Zbl0176.00801MR257325
24. [23] Giga ( Y.) and Kohn ( R.V.), Asymptotically self-similar blow-up of semilinear heat equations, Comm. Pure Appl. Math.38, p. 297-319 (1985). Zbl0585.35051MR784476
25. [24] Gurtin ( M. ), On a theory of phase transitions with interfacial energy, Arch. Rat. Mech. Anal.87, p. 187-212 (1985). MR768066
26. [25] Huisken ( G.), Asymptotic behavior for singularities of the mean curvature flow, J. Diff. Geom.31, p. 285-299 (1990). Zbl0694.53005MR1030675
27. [26] Ilmanen ( T.), Convergence of the Allen-Cahn equation to Brakke's motion by mean curvature, J. Diff. Geom.38, p. 417-461 (1993). Zbl0784.53035MR1237490
28. [27] Ilmanen ( T.), Elliptic regularization and partial regularity for motion by mean curvature, Mem. Amer. Math. Soc.108, no. 520 (1994). Zbl0798.35066MR1196160
29. [28] Jerrard ( R.L.) and Soner ( H.M.), Dynamics of Ginzburg-Landau vortices, Arch. Rational Mech. Anal.142, p. 99-125 (1998). Zbl0923.35167MR1629646
30. [29] Jerrard ( R.L.) and Soner ( H.M.), Scaling limits and regularity results for a class of Ginzburg-Landau systems, Ann. Inst. H. Poincaré Anal. Non Linéaire16, p. 423-466 (1999). Zbl0944.35006MR1697561
31. [30] Jerrard ( R.L.) and Soner ( H.M.), The Jacobian and the Ginzburg-Landau energy, Calc. Var. PDE14, p. 151-191 (2002). Zbl1034.35025MR1890398
32. [31] Lin ( F.H. ), Some dynamical properties of Ginzburg-Landau vortices, Comm. Pure Appl. Math.49, p. 323-359 (1996). Zbl0853.35058MR1376654
33. [32] Lin ( F.H. ), Complex Ginzburg-Landau equations and dynamics of vortices, filaments, and codimension-2 submanifolds, Comm. Pure Appl. Math.51, p. 385-441 (1998). Zbl0932.35121MR1491752
34. [33] Lin ( F.H.) and Rivière ( T.), Complex Ginzburg-Landau equation in high dimension and codimension 2 area minimizing currents, J. Eur. Math. Soc.1, p. 237-311 (1999). Zbl0939.35056MR1714735
35. Erratum, Ibid. 
36. [34] Lin ( F.H.) and Rivière ( T.), A quantization property for static Ginzburg-Landau vortices, Comm. Pure Appl. Math.54, p. 206-228 (2001). Zbl1033.58013MR1794353
37. [35] Lin ( F.H. ) and Rivière ( T.), A quantization property for moving line vortices, Comm. Pure Appl. Math.54, p. 826-850 (2001). Zbl1029.35127MR1823421
38. [36] Mermin ( N.D. ), The topological theory of defects in ordered media, Rev. Modern Phys.51, p. 591-648 (1979). MR541885
39. [37] Mironescu ( P.), Les minimiseurs locaux pour l'équation de Ginzburg-Landau sont à symétrie radiale, C.R. Acad. Sci. Paris (I) 323, p. 593-598 (1996). Zbl0858.35038MR1411048
40. [38] Modica ( L. ), The gradient theory of phase transitions and the minimal interface criterion, Arch. Rat. Mech. Anal.98, p. 123-142 (1987). Zbl0616.76004MR866718
41. [39] Modica ( L. ) and Mortola ( S.), Un esempio di Γ-convergenza , Boll. Un. Mat. Ital.B14, p. 285-299 (1977). Zbl0356.49008MR445362
42. [40] Pacard ( F.) and Rivière ( T.), Linear and non linear aspects of vortices , Birkhäuser, (2001). Zbl0948.35003
43. [41] Pismen ( L.M. ) and Rubinstein ( J.), Motion of vortex lines in the Ginzburg-Landau model, Phys.D47, p. 353-360 (1991). Zbl0728.35090MR1098255
44. [42] Preiss ( D. ), Geometry of measures in Rn: distribution, rectifiability, and densities, Ann. of Math.125, p. 537-643 (1987). Zbl0627.28008MR890162
45. [43] Rivière ( T.), Line vortices in the U(1) Higgs model, ESAIM, C.O.C.V.1, p. 77-167 (1996). Zbl0874.53019MR1394302
46. [44] Serfaty ( S.), Local minimizers for the Ginzburg-Landau energy near critical magnetic fields, I and II, Comm. Contemp. Math.1, p. 213-254 and p. 295-333 (1999). Zbl0944.49007MR1696100
47. [45] Simon ( L. ), Lectures on Geometric Measure Theory, Proc. of the Centre for Math. Anal., Austr. Nat. Univ., 1983 . Zbl0546.49019MR756417
48. [46] Soner ( H.M. ), Ginzburg-Landau equation and motion by mean curvature. I. Convergence, and II. Development of the initial interface, J. Geom. Anal.7, no. 3, p. 437-475 and p. 477-491 (1997). Zbl0935.35061MR1674799
49. [47] Sternberg ( P.), The effect of a singular perturbation on nonconvex variational problems, Arch. Rat. Mech. Anal.101, p. 209-260 (1988). Zbl0647.49021MR930124
50. [48] Struwe ( M. ), On the evolution of harmonic maps in higher dimensions, J. Diff. Geom.28, p. 485-502 (1988). Zbl0631.58004MR965226
51. [49] Struwe ( M. ), On the asymptotic behavior of the Ginzburg-Landau model in 2 dimensions, J. Diff. Equ.7, p. 1613-1624 (1994), Zbl0809.35031MR1269674
52. Erratum8, p. 224 (1995). 
53. [50] Wang ( C.) , On moving Ginzburg-Landau filament vortices, Max-Planck-InstitutLeipzig, preprint. Zbl1063.35081
54. [51] Zhou ( F.) and Zhou ( Q.), A remark on the multiplicity of solutions for the Ginzburg-Landau equation, Ann. IHP, Analyse Nonlinéaire16, p. 255-267 (1999). Zbl0928.35034MR1674771

1. Fabrice Bethuel, Giandomenico Orlandi, Didier Smets, Improved estimates for the Ginzburg-Landau equation : the elliptic case