×

zbMATH — the first resource for mathematics

Gamma-convergence of gradient flows with applications to Ginzburg-Landau. (English) Zbl 1065.49011
This paper is devoted to the study of gradient flows of the Ginzburg-Landau energy functional \(E_\varepsilon (u)=\int_\Omega \left[\varepsilon | \nabla u| ^2+\varepsilon^{-1}(1-| u| ^2)^2\right]dx\), where \(\varepsilon>0\) and \(\Omega\subset\mathbb R^2\) is a smooth, bounded and simply connected domain. The authors establish several properties of gradient flows from the viewpoint of the Gamma-convergence theory. One of the first results of the paper provides lower-bound criteria to deduce an appropriate convergence property. Using this result the authors prove the limiting dynamical law of a finite number of vortices for the heat flow of the Ginzburg-Landau energy. The proofs combine powerful elliptic estimates and adequate minimization methods.

MSC:
49J45 Methods involving semicontinuity and convergence; relaxation
35J20 Variational methods for second-order elliptic equations
58E50 Applications of variational problems in infinite-dimensional spaces to the sciences
82D55 Statistical mechanical studies of superconductors
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Aftalion, J Math Pures Appl (9) 80 pp 339– (2001)
[2] ; ; Gradient flows in metric spaces and in the Wasserstein spaces of probability measures. Forthcoming.
[3] Ambrosio, Ann Scuola Norm Sup Pisa Cl Sci (4) 25 pp 27– (1997)
[4] ; ; Ginzburg-Landau vortices. Progress in Nonlinear Differential Equations and Their Applications, 13. Birkh?user, Boston, 1994. · doi:10.1007/978-1-4612-0287-5
[5] Bethuel, Ann of Math (2)
[6] Bethuel, Ann Inst H Poincar? Anal Non Lin?aire 12 pp 243– (1995)
[7] ?-convergence for beginners. Oxford Lecture Series in Mathematics and Its Applications, 22. Oxford University Press, Oxford, 2002. · doi:10.1093/acprof:oso/9780198507840.001.0001
[8] Op?rateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland Mathematics Studies, 5. · Zbl 0252.47055
[9] Notas de Matem?tica, 50. North-Holland, Amsterdam?London; American Elsevier, New York, 1973.
[10] Chapman, European J Appl Math 7 pp 97– (1996)
[11] Chen, J Differential Equations 96 pp 116– (1992)
[12] Colliander, Internat Math Res Notices 1998 pp 333–
[13] Colliander, J Anal Math 77 pp 129– (1999)
[14] de Mottoni, Proc Roy Soc Edinburgh Sect A 116 pp 207– (1990) · Zbl 0725.35009 · doi:10.1017/S0308210500031486
[15] Du, Appl Anal 53 pp 1– (1994)
[16] E, Phys D 77 pp 383– (1994)
[17] Evans, Comm Pure Appl Math 45 pp 1097– (1992)
[18] Ilmanen, J Differential Geom 38 pp 417– (1993)
[19] Jerrard, SIAM J Math Anal 30 pp 721– (1999)
[20] Jerrard, Calc Var Partial Differential Equations 9 pp 1– (1999)
[21] Jerrard, Arch Rational Mech Anal 142 pp 99– (1998)
[22] Jerrard, Calc Var Partial Differential Equations 14 pp 151– (2002)
[23] Lin, Comm Pure Appl Math 49 pp 323– (1996)
[24] Lin, Comm Pure Appl Math 52 pp 737– (1999)
[25] Lin, Comm Pure Appl Math 54 pp 206– (2001)
[26] Lin, Comm Math Phys 200 pp 249– (1999)
[27] Otto, Comm Partial Differential Equations 26 pp 101– (2001)
[28] Pismen, Phys D 47 pp 353– (1991)
[29] Sandier, J Funct Anal 152 pp 379– (1998)
[30] Sandier, Ann Inst H Poincar? Anal Non Lin?aire 17 pp 119– (2000)
[31] Sandier, Rev Math Phys 12 pp 1219– (2000)
[32] Sandier, Ann Sci ?cole Norm Sup (4) 33 pp 561– (2000)
[33] Sandier, Calc Var Partial Differential Equations 17 pp 17– (2003)
[34] Sandier, Duke Math J 117 pp 403– (2003)
[35] Sandier, J Funct Anal 211 pp 219– (2004)
[36] ; Vortices in the magnetic Ginzburg-Landau model. Monograph. In preparation.
[37] Serfaty, Commun Contemp Math 1 pp 213– (1999)
[38] Commun Contemp Math 1 pp 295– (1999)
[39] Serfaty, Arch Ration Mech Anal 149 pp 329– (1999)
[40] Serfaty, Indiana Math J
[41] Vortex collision and energy dissipation rates in the Ginzburg-Landau heat flow. In preparation.
[42] Spirn, Comm Pure Appl Math 55 pp 537– (2002)
[43] Introduction to superconductivity. 2nd ed. McGraw-Hill, New York, 1996.
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.