Motion of concentration sets in Ginzburg-Landau equations
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 13 (2004) no. 1, p. 3-43
@article{AFST_2004_6_13_1_3_0,
     author = {B\'ethuel, Fabrice and Orlandi, Giandomenico and Smets, Didier},
     title = {Motion of concentration sets in Ginzburg-Landau equations},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques},
     address = {Toulouse},
     volume = {Ser. 6, 13},
     number = {1},
     year = {2004},
     pages = {3-43},
     zbl = {1063.35075},
     mrnumber = {2060028},
     language = {en},
     url = {http://www.numdam.org/item/AFST_2004_6_13_1_3_0}
}
Bethuel, Fabrice; Orlandi, Giandomenico; Smets, Didier. Motion of concentration sets in Ginzburg-Landau equations. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 13 (2004) no. 1, pp. 3-43. http://www.numdam.org/item/AFST_2004_6_13_1_3_0/

[1] Alberti (G. ) , Baldo ( S.) and Orlandi (G.), Variational convergence for functionals of Ginzburg-Landau type, preprint 2002.

[2] Alweida (L. ) and Bethuel (F.), Topological methods for the Ginzburg-Landau equation, J. Math Pures Appl. 11, p. 1-49 (1998). | MR 1617594 | Zbl 0904.35023

[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). | Numdam | MR 1655508 | Zbl 1043.35136

[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). | MR 1881236 | Zbl 01719347

[5] Bethuel (F. ) , Brezis ( H.) and Hélein (F.), Ginzburg-Landau vortices, Birkhäuser, Boston, 1994. | MR 1269538 | Zbl 0802.35142

[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. | MR 1864830 | Zbl 01703083

Erratum 188, p. 548-549 (2002). | MR 1883416

[7] Bethuel (F. ) and Orlandi (G.), Uniform estimates for the parabolic Ginzburg-Landau equation, ESAIM, C.O.C.V 8, p. 219-238 (2002). | MR 1932951 | Zbl 01967374

[8] Bethuel (F. ), Orlandi (G.) and Smets (D.), Vortex rings for the Gross-Pitaevskii equation, J. Eur. Math. Soc., to appear. | Zbl 02075133

[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). | MR 1988309 | Zbl 1031.35026

[10] Bethuel ( F.) and Rivière (T.), A minimization problem related to superconductivity, Annales IHP, Analyse Non Linéaire 12, p. 243-303 (1995). | Numdam

[11] Bethuel (F. ) and Saut ( J.C.), Travelling waves for the Gross-Pitaevskii equation, Annales IHP, Phys. Théor. 70, (1999). | Numdam | MR 1669387 | Zbl 0933.35177

[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. | MR 1781527 | Zbl 0970.35069

[13] Brakke (K.) , The motion of a surface by its mean curvature, Princeton University Press (1978). | MR 485012 | Zbl 0386.53047

[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). | MR 1288397 | Zbl 0805.49004

[15] Bronsard ( L.) and Kohn (R.V.), Motion by mean curvature as the singular limit of Ginzburg-Landau dynamics,. J. Differential Equations 90, p. 211-237 (1991). | MR 1101239 | Zbl 0735.35072

[16] Carr (J.) , Pego ( R. ), Metastable patterns in solutions of u t = ε2uxx - f(u, Comm. Pure Appl. Math. 42, p. 523-576 (1989). | MR 997567 | Zbl 0685.35054

[17] Chen (X.) , Generation and propagation of interfaces for reaction-diffusion equations, J. Differential Equations 96, p. 116-141 (1992). | MR 1153311 | Zbl 0765.35024

[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). | MR 990191 | Zbl 0652.58024

[19] De Giorgi ( E. ), Some conjectures on flow by mean curvature , Proc. of the Capri Workshop ( 1990), Benevento-Bruno-Sbordone (eds.) (1990).

[20] De Mottoni ( P.) and Schatzman (M.), Development of interfaces in RN, Proc. Roy. Soc. Edinburgh Sect. A 116, p. 207-220 (1990). | MR 1084732 | Zbl 0725.35009

[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 ). | MR 1177477 | Zbl 0801.35045

[22] Federer (H. ) , Geometric Measure Theory, Springer , Berlin, (1969). | MR 257325 | Zbl 0176.00801

[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). | MR 784476 | Zbl 0585.35051

[24] Gurtin (M. ), On a theory of phase transitions with interfacial energy, Arch. Rat. Mech. Anal. 87, p. 187-212 (1985). | MR 768066

[25] Huisken ( G.), Asymptotic behavior for singularities of the mean curvature flow, J. Diff. Geom. 31, p. 285-299 (1990). | MR 1030675 | Zbl 0694.53005

[26] Ilmanen ( T.), Convergence of the Allen-Cahn equation to Brakke's motion by mean curvature, J. Diff. Geom. 38, p. 417-461 (1993). | MR 1237490 | Zbl 0784.53035

[27] Ilmanen ( T.), Elliptic regularization and partial regularity for motion by mean curvature, Mem. Amer. Math. Soc. 108, no. 520 (1994). | MR 1196160 | Zbl 0798.35066

[28] Jerrard ( R.L.) and Soner (H.M.), Dynamics of Ginzburg-Landau vortices, Arch. Rational Mech. Anal. 142, p. 99-125 (1998). | MR 1629646 | Zbl 0923.35167

[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 Linéaire 16, p. 423-466 (1999). | Numdam | MR 1697561 | Zbl 0944.35006

[30] Jerrard ( R.L.) and Soner (H.M.), The Jacobian and the Ginzburg-Landau energy, Calc. Var. PDE 14, p. 151-191 (2002). | MR 1890398 | Zbl 1034.35025

[31] Lin (F.H. ), Some dynamical properties of Ginzburg-Landau vortices, Comm. Pure Appl. Math. 49, p. 323-359 (1996). | MR 1376654 | Zbl 0853.35058

[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). | MR 1491752 | Zbl 0932.35121

[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). | MR 1714735 | Zbl 0939.35056

Erratum, Ibid.

[34] Lin (F.H.) and Rivière ( T.), A quantization property for static Ginzburg-Landau vortices, Comm. Pure Appl. Math. 54, p. 206-228 (2001). | MR 1794353 | Zbl 1033.58013

[35] Lin (F.H. ) and Rivière (T.), A quantization property for moving line vortices, Comm. Pure Appl. Math. 54, p. 826-850 (2001). | MR 1823421 | Zbl 1029.35127

[36] Mermin (N.D. ), The topological theory of defects in ordered media, Rev. Modern Phys. 51, p. 591-648 (1979). | MR 541885

[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). | MR 1411048 | Zbl 0858.35038

[38] Modica (L. ), The gradient theory of phase transitions and the minimal interface criterion, Arch. Rat. Mech. Anal. 98, p. 123-142 (1987). | MR 866718 | Zbl 0616.76004

[39] Modica (L. ) and Mortola (S.), Un esempio di Γ-convergenza , Boll. Un. Mat. Ital. B 14, p. 285-299 (1977). | MR 445362 | Zbl 0356.49008

[40] Pacard (F.) and Rivière ( T.), Linear and non linear aspects of vortices , Birkhäuser, (2001). | Zbl 0948.35003

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

[42] Preiss (D. ), Geometry of measures in Rn: distribution, rectifiability, and densities, Ann. of Math. 125, p. 537-643 (1987). | MR 890162 | Zbl 0627.28008

[43] Rivière ( T.), Line vortices in the U(1) Higgs model, ESAIM, C.O.C.V. 1, p. 77-167 (1996). | Numdam | MR 1394302

[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). | MR 1696100 | Zbl 0944.49007

[45] Simon (L. ), Lectures on Geometric Measure Theory, Proc. of the Centre for Math. Anal., Austr. Nat. Univ., 1983 . | MR 756417 | Zbl 0546.49019

[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). | MR 1674799 | Zbl 0935.35061

[47] Sternberg ( P.), The effect of a singular perturbation on nonconvex variational problems, Arch. Rat. Mech. Anal. 101, p. 209-260 (1988). | MR 930124 | Zbl 0647.49021

[48] Struwe (M. ), On the evolution of harmonic maps in higher dimensions, J. Diff. Geom. 28, p. 485-502 (1988). | MR 965226 | Zbl 0631.58004

[49] Struwe (M. ), On the asymptotic behavior of the Ginzburg-Landau model in 2 dimensions, J. Diff. Equ. 7, p. 1613-1624 (1994), | MR 1269674 | Zbl 0809.35031

Erratum 8, p. 224 (1995).

[50] Wang (C.) , On moving Ginzburg-Landau filament vortices, Max-Planck-Institut Leipzig, preprint.

[51] Zhou (F.) and Zhou ( Q.), A remark on the multiplicity of solutions for the Ginzburg-Landau equation, Ann. IHP, Analyse Nonlinéaire 16, p. 255-267 (1999). | Numdam | MR 1674771 | Zbl 0928.35034