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Sergio Conti; Michael Goldman; Felix Otto; Sylvia Serfaty
A branched transport limit of the Ginzburg-Landau functional
(Dérivation d’une fonctionnelle de type transport branché à partir du modèle de Ginzburg-Landau)
Journal de l'École polytechnique — Mathématiques, 5 (2018), p. 317-375, doi: 10.5802/jep.72
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Class. Math.: 35Q56, 49S05, 82D55, 49J10, 49S05
Keywords: Gamma convergence, Ginzburg-Landau, branched transportation, pattern formation, type-I superconductors

Résumé - Abstract

We study the Ginzburg-Landau model of type-I superconductors in the regime of small external magnetic fields. We show that, in an appropriate asymptotic regime, flux patterns are described by a simplified branched transportation functional. We derive the simplified functional from the full Ginzburg-Landau model rigorously via $\Gamma $-convergence. The detailed analysis of the limiting procedure and the study of the limiting functional lead to a precise understanding of the multiple scales contained in the model.


[1] G. Alberti, R. Choksi & F. Otto, “Uniform energy distribution for an isoperimetric problem with long-range interactions”, J. Amer. Math. Soc. 22 (2009) no. 2, p. 569-605
[2] L. Ambrosio, N. Fusco & D. Pallara, Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs, Oxford University Press, New York, 2000
[3] L. Ambrosio, N. Gigli & G. Savaré, Gradient flows in metric spaces and in the space of probability measures, Lectures in Math. ETH Zürich, Birkhäuser Verlag, Basel, 2005
[4] P. Bella & M. Goldman, “Nucleation barriers at corners for a cubic-to-tetragonal phase transformation”, Proc. Roy. Soc. Edinburgh Sect. A 145 (2015) no. 4, p. 715-724
[5] H. Ben Belgacem, S. Conti, A. DeSimone & S. Müller, “Rigorous bounds for the Föppl-von Kármán theory of isotropically compressed plates”, J. Nonlinear Sci. 10 (2000), p. 661-683
[6] H. Ben Belgacem, S. Conti, A. DeSimone & S. Müller, “Energy scaling of compressed elastic films”, Arch. Rational Mech. Anal. 164 (2002), p. 1-37
[7] M. Bernot, V. Caselles & J.-M. Morel, Optimal transportation networks. Models and theory, Lect. Notes in Math. 1955, Springer-Verlag, Berlin, 2009
[8] F. Bethuel, “A counterexample to the weak density of smooth maps between manifolds in Sobolev spaces”, arXiv:1401.1649, 2014
[9] A. Braides, $\Gamma $-convergence for beginners, Oxford Lecture Series in Mathematics and its Applications 22, Oxford University Press, Oxford, 2002
[10] A. Brancolini, C. Rossmanith & B. Wirth, “Optimal micropatterns in 2D transport networks and their relation to image inpainting”, Arch. Rational Mech. Anal. 228 (2018) no. 1, p. 279-308
[11] A. Brancolini & B. Wirth, “Optimal micropatterns in transport networks”, arXiv:1511.08467, 2015
[12] A. Chan & S. Conti, “Energy scaling and branched microstructures in a model for shape-memory alloys with $SO(2)$ invariance”, Math. Models Methods Appl. Sci. 25 (2015), p. 1091-1124
[13] R. Choksi, S. Conti, R. V. Kohn & F. Otto, “Ground state energy scaling laws during the onset and destruction of the intermediate state in a type-I superconductor”, Comm. Pure Appl. Math. 61 (2008), p. 595-626
[14] R. Choksi, R. V. Kohn & F. Otto, “Domain branching in uniaxial ferromagnets: a scaling law for the minimum energy”, Comm. Math. Phys. 201 (1999), p. 61-79
[15] R. Choksi, R. V. Kohn & F. Otto, “Energy minimization and flux domain structure in the intermediate state of a type-I superconductor”, J. Nonlinear Sci. 14 (2004), p. 119-171
[16] E. Cinti & F. Otto, “Interpolation inequalities in pattern formation”, J. Funct. Anal. 271 (2016) no. 11, p. 3348-3392
[17] S. Conti, “Branched microstructures: scaling and asymptotic self-similarity”, Comm. Pure Appl. Math. 53 (2000), p. 1448-1474
[18] S. Conti, J. Diermeier & B. Zwicknagl, “Deformation concentration for martensitic microstructures in the limit of low volume fraction”, Calc. Var. Partial Differential Equations 56 (2017) no. 1
[19] S. Conti, F. Otto & S. Serfaty, “Branched microstructures in the Ginzburg-Landau model of type-I superconductors”, SIAM J. Math. Anal. 48 (2016) no. 4, p. 2994-3034
[20] S. Conti & B. Zwicknagl, “Low volume-fraction microstructures in martensites and crystal plasticity”, Math. Models Methods Appl. Sci. 26 (2016) no. 7, p. 1319-1355
[21] G. Dal Maso, An introduction to $\Gamma $-convergence, Progress in Nonlinear Differential Equations and their Applications 8, Birkhäuser Boston Inc., Boston, MA, 1993
[22] R. L. Frank, C. Hainzl, R. Seiringer & J P. Solovej, “Microscopic derivation of Ginzburg-Landau theory”, J. Amer. Math. Soc. 25 (2012) no. 3, p. 667-713
[23] M. Goldman, “Self-similar minimizers of a branched transport functional”, arXiv:1704.05342, 2017
[24] M. Goldman & B. Merlet, “Phase segregation for binary mixtures of Bose-Einstein Condensates”, SIAM J. Math. Anal. 49 (2017) no. 3, p. 1947-1981
[25] A. Jaffe & C. Taubes, Vortices and monopoles. Structure of static gauge theories, Progress in Physics 2, Birkhäuser, Boston MA, 1980
[26] W. Jin & P. Sternberg, “Energy estimates of the von Kármán model of thin-film blistering”, J. Math. Phys. 42 (2001), p. 192-199
[27] H. Knüpfer, R. V. Kohn & F. Otto, “Nucleation barriers for the cubic-to-tetragonal phase transformation”, Comm. Pure Appl. Math. 66 (2013) no. 6, p. 867-904
[28] H. Knüpfer & C. B. Muratov, “Domain structure of bulk ferromagnetic crystals in applied fields near saturation”, J. Nonlinear Sci. 21 (2011) no. 6, p. 921-962
[29] R. V. Kohn & S. Müller, “Branching of twins near an austenite-twinned-martensite interface”, Phil. Mag. A 66 (1992), p. 697-715
[30] R. V. Kohn & S. Müller, “Surface energy and microstructure in coherent phase transitions”, Comm. Pure Appl. Math. 47 (1994), p. 405-435
[31] L. Modica, “The gradient theory of phase transitions and the minimal interface criterion”, Arch. Rational Mech. Anal. 98 (1987), p. 123-142
[32] F. Otto & T. Viehmann, “Domain branching in uniaxial ferromagnets: asymptotic behavior of the energy”, Calc. Var. Partial Differential Equations 38 (2010) no. 1-2, p. 135-181
[33] E. Oudet & F. Santambrogio, “A Modica-Mortola approximation for branched transport and applications”, Arch. Rational Mech. Anal. 201 (2011) no. 1, p. 115-142
[34] R. Prozorov, “Equilibrium topology of the intermediate state in type-I superconductors of different shapes”, Phys. Rev. Lett. 98 (2007)
[35] R. Prozorov, R. W. Giannetta, A. A. Polyanskii & G. K. Perkins, “Topological hysteresis in the intermediate state of type I superconductors”, Phys. Rev. B 72 (2005)
[36] R. Prozorov & J. Hoberg, Dynamic formation of metastable intermediate state patterns in type-I superconductors, 25th international conference on low temperature physics, Journal of Physics: Conference Series 150, IOP Publishing, 2009
[37] E. Sandier & S. Serfaty, Vortices in the magnetic Ginzburg-Landau model, Progress in Nonlinear Differential Equations and their Applications 70, Birkhäuser Boston, Inc., Boston, MA, 2007
[38] S. Serfaty, Coulomb gases and Ginzburg-Landau vortices, Zürich Lectures in Advanced Mathematics, European Mathematical Society, Zürich, 2015
[39] M. Tinkham, Introduction to superconductivity, Int. series in Pure and Appl. physics, McGraw Hill, New York, 1996
[40] T. Viehmann, Uniaxial ferromagnets, Ph. D. Thesis, Universität Bonn, 2009
[41] C. Villani, Topics in optimal transportation, Graduate Studies in Mathematics 58, American Mathematical Society, Providence, RI, 2003
[42] Q. Xia, “Interior regularity of optimal transport paths”, Calc. Var. Partial Differential Equations 20 (2004) no. 3, p. 283-299
[43] B. Zwicknagl, “Microstructures in low-hysteresis shape memory alloys: scaling regimes and optimal needle shapes”, Arch. Rational Mech. Anal. 213 (2014), p. 355-421