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Mean field limits of the Gross-Pitaevskii and parabolic Ginzburg-Landau equations


Author: Sylvia Serfaty
Journal: J. Amer. Math. Soc. 30 (2017), 713-768
MSC (2010): Primary 35Q56, 35K55, 35Q55, 35Q31, 35Q35
DOI: https://doi.org/10.1090/jams/872
Published electronically: October 18, 2016
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Abstract: We prove that in a certain asymptotic regime, solutions of the Gross-Pitaevskii equation converge to solutions of the incompressible Euler equation, and solutions to the parabolic Ginzburg-Landau equations converge to solutions of a limiting equation which we identify.

We work in the setting of the whole plane (and possibly the whole three-dimensional space in the Gross-Pitaevskii case), in the asymptotic limit where $ \varepsilon $, the characteristic lengthscale of the vortices, tends to 0, and in a situation where the number of vortices $ N_\varepsilon $ blows up as $ \varepsilon \to 0$. The requirements are that $ N_\varepsilon $ should blow up faster than $ \vert\mathrm {log } \, \varepsilon \vert$ in the Gross-Pitaevskii case, and at most like $ \vert\mathrm {log } \, \varepsilon \vert$ in the parabolic case. Both results assume a well-prepared initial condition and regularity of the limiting initial data, and use the regularity of the solution to the limiting equations.

In the case of the parabolic Ginzburg-Landau equation, the limiting mean-field dynamical law that we identify coincides with the one proposed by Chapman-Rubinstein-Schatzman and E in the regime $ N_\varepsilon \ll \vert\mathrm {log } \, \varepsilon \vert$, but not if $ N_\varepsilon $ grows faster.


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  • [AGS] Luigi Ambrosio, Nicola Gigli, and Giuseppe Savaré, Gradient flows in metric spaces and in the space of probability measures, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel, 2005. MR 2129498
  • [AS] Luigi Ambrosio and Sylvia Serfaty, A gradient flow approach to an evolution problem arising in superconductivity, Comm. Pure Appl. Math. 61 (2008), no. 11, 1495-1539. MR 2444374, https://doi.org/10.1002/cpa.20223
  • [BBH] Fabrice Bethuel, Haïm Brezis, and Frédéric Hélein, Ginzburg-Landau vortices, Progress in Nonlinear Differential Equations and their Applications, 13, Birkhäuser Boston, Inc., Boston, MA, 1994. MR 1269538
  • [AK] Igor S. Aranson and Lorenz Kramer, The world of the complex Ginzburg-Landau equation, Rev. Modern Phys. 74 (2002), no. 1, 99-143. MR 1895097, https://doi.org/10.1103/RevModPhys.74.99
  • [BCD] Hajer Bahouri, Jean-Yves Chemin, and Raphaël Danchin, Fourier analysis and nonlinear partial differential equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 343, Springer, Heidelberg, 2011. MR 2768550
  • [BJS] Fabrice Bethuel, Robert L. Jerrard, and Didier Smets, On the NLS dynamics for infinite energy vortex configurations on the plane, Rev. Mat. Iberoam. 24 (2008), no. 2, 671-702. MR 2459209, https://doi.org/10.4171/RMI/552
  • [BOS1] F. Bethuel, G. Orlandi, and D. Smets, Collisions and phase-vortex interactions in dissipative Ginzburg-Landau dynamics, Duke Math. J. 130 (2005), no. 3, 523-614. MR 2184569, https://doi.org/10.1215/S0012-7094-05-13034-4
  • [BOS2] Didier Smets, Fabrice Bethuel, and Giandomenico Orlandi, Quantization and motion law for Ginzburg-Landau vortices, Arch. Ration. Mech. Anal. 183 (2007), no. 2, 315-370. MR 2278409, https://doi.org/10.1007/s00205-006-0018-4
  • [BOS3] F. Bethuel, G. Orlandi, and D. Smets, Dynamics of multiple degree Ginzburg-Landau vortices, Comm. Math. Phys. 272 (2007), no. 1, 229-261. MR 2291808, https://doi.org/10.1007/s00220-007-0206-6
  • [BR1] Fabrice Bethuel and Tristan Rivière, Vortices for a variational problem related to superconductivity, Ann. Inst. H. Poincaré Anal. Non Linéaire 12 (1995), no. 3, 243-303 (English, with English and French summaries). MR 1340265
  • [BDGSS] Fabrice Béthuel, Raphaël Danchin, Philippe Gravejat, Jean-Claude Saut, and Didier Smets, Les équations d'Euler, des ondes et de Korteweg-de Vries comme limites asymptotiques de l'équation de Gross-Pitaevskii, Séminaire: Équations aux Dérivées Partielles. 2008-2009, Sémin. Équ. Dériv. Partielles, École Polytech., Palaiseau, 2010, pp. Exp. No. I, 12 (French). MR 2668303
  • [BS] Fabrice Bethuel and Didier Smets, A remark on the Cauchy problem for the 2D Gross-Pitaevskii equation with nonzero degree at infinity, Differential Integral Equations 20 (2007), no. 3, 325-338. MR 2293989
  • [Br] Y. Brenier, Convergence of the Vlasov-Poisson system to the incompressible Euler equations, Comm. Partial Differential Equations 25 (2000), no. 3-4, 737-754. MR 1748352, https://doi.org/10.1080/03605300008821529
  • [CDS] Rémi Carles, Raphaël Danchin, and Jean-Claude Saut, Madelung, Gross-Pitaevskii and Korteweg, Nonlinearity 25 (2012), no. 10, 2843-2873. MR 2979973, https://doi.org/10.1088/0951-7715/25/10/2843
  • [CRS] S. J. Chapman, J. Rubinstein, and M. Schatzman, A mean-field model of superconducting vortices, European J. Appl. Math. 7 (1996), no. 2, 97-111. MR 1388106, https://doi.org/10.1017/S0956792500002242
  • [Ch1] Jean-Yves Chemin, Perfect incompressible fluids, Oxford Lecture Series in Mathematics and its Applications, vol. 14, The Clarendon Press, Oxford University Press, New York, 1998. Translated from the 1995 French original by Isabelle Gallagher and Dragos Iftimie. MR 1688875
  • [Ch2] Jean-Yves Chemin, Théorèmes d'unicité pour le système de Navier-Stokes tridimensionnel, J. Anal. Math. 77 (1999), 27-50 (French). MR 1753481, https://doi.org/10.1007/BF02791256
  • [CJ1] J. E. Colliander and R. L. Jerrard, Vortex dynamics for the Ginzburg-Landau-Schrödinger equation, Internat. Math. Res. Notices 7 (1998), 333-358. MR 1623410, https://doi.org/10.1155/S1073792898000221
  • [CJ2] J. E. Colliander and R. L. Jerrard, Ginzburg-Landau vortices: weak stability and Schrödinger equation dynamics, J. Anal. Math. 77 (1999), 129-205. MR 1753485, https://doi.org/10.1007/BF02791260
  • [De] Jean-Marc Delort, Existence de nappes de tourbillon en dimension deux, J. Amer. Math. Soc. 4 (1991), no. 3, 553-586 (French). MR 1102579, https://doi.org/10.2307/2939269
  • [Do] A. Dorsey, Vortex motion and the Hall effect in type II superconductors: a time-dependent Ginzburg-Landau approach, Phys. Rev. B 46 (1992), 8376-8392.
  • [DZ] Qiang Du and Ping Zhang, Existence of weak solutions to some vortex density models, SIAM J. Math. Anal. 34 (2003), no. 6, 1279-1299 (electronic). MR 2000970, https://doi.org/10.1137/S0036141002408009
  • [Du] M. Duerinckx, Well-posedness for mean-field evolutions arising in superconductivity, forthcoming.
  • [E1] Weinan E, Dynamics of vortices in Ginzburg-Landau theories with applications to superconductivity, Phys. D 77 (1994), no. 4, 383-404. MR 1297726, https://doi.org/10.1016/0167-2789(94)90298-4
  • [E2] W. E, Dynamics of vortex liquids in Ginzburg-Landau theories with applications to superconductivity, Phys. Rev. B 50 (1994), No. 2, 1126-1135.
  • [Ga] Clément Gallo, The Cauchy problem for defocusing nonlinear Schrödinger equations with non-vanishing initial data at infinity, Comm. Partial Differential Equations 33 (2008), no. 4-6, 729-771. MR 2424376, https://doi.org/10.1080/03605300802031614
  • [Ge] P. Gérard, The Cauchy problem for the Gross-Pitaevskii equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 23 (2006), no. 5, 765-779 (English, with English and French summaries). MR 2259616, https://doi.org/10.1016/j.anihpc.2005.09.004
  • [GE] L. P. Gor'kov and G. M. Eliashberg, Generalization of the Ginzburg-Landau equations for nonstationary problems in the case of alloys with paramagnetic impurities, Sov. Phys. JETP 27 (1968), 328-334.
  • [HL] Zheng-Chao Han and Yan Yan Li, Degenerate elliptic systems and applications to Ginzburg-Landau type equations. I, Calc. Var. Partial Differential Equations 4 (1996), no. 2, 171-202. MR 1379199, https://doi.org/10.1007/s005260050034
  • [J1] Robert L. Jerrard, Lower bounds for generalized Ginzburg-Landau functionals, SIAM J. Math. Anal. 30 (1999), no. 4, 721-746. MR 1684723, https://doi.org/10.1137/S0036141097300581
  • [J2] Robert L. Jerrard, Vortex filament dynamics for Gross-Pitaevsky type equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 1 (2002), no. 4, 733-768. MR 1991001
  • [JS1] Robert Leon Jerrard and Halil Mete Soner, Dynamics of Ginzburg-Landau vortices, Arch. Ration. Mech. Anal. 142 (1998), no. 2, 99-125. MR 1629646, https://doi.org/10.1007/s002050050085
  • [JS2] Robert L. Jerrard and Halil Mete Soner, The Jacobian and the Ginzburg-Landau energy, Calc. Var. Partial Differential Equations 14 (2002), no. 2, 151-191. MR 1890398, https://doi.org/10.1007/s005260100093
  • [JSp1] Robert Leon Jerrard and Daniel Spirn, Refined Jacobian estimates and Gross-Pitaevsky vortex dynamics, Arch. Ration. Mech. Anal. 190 (2008), no. 3, 425-475. MR 2448324, https://doi.org/10.1007/s00205-008-0167-8
  • [JSp2] Robert L. Jerrard and Daniel Spirn, Hydrodynamic limit of the Gross-Pitaevskii equation, Comm. Partial Differential Equations 40 (2015), no. 2, 135-190. MR 3277924, https://doi.org/10.1080/03605302.2014.963604
  • [KIK] N. B. Kopnin, B. I. Ivlev, and V. A. Kalatsky, The flux-flow Hall effect in type II superconductors. An explanation of the sign reversal, J. Low Temp. Phys. 90 (1993), 1-13.
  • [KS1] Matthias Kurzke and Daniel Spirn, $ \Gamma $-stability and vortex motion in type II superconductors, Comm. Partial Differential Equations 36 (2011), no. 2, 256-292. MR 2763341, https://doi.org/10.1080/03605302.2010.520182
  • [KS2] Matthias Kurzke and Daniel Spirn, Vortex liquids and the Ginzburg-Landau equation, Forum Math. Sigma 2 (2014), e11, 63. MR 3264248, https://doi.org/10.1017/fms.2014.6
  • [KMMS] Matthias Kurzke, Christof Melcher, Roger Moser, and Daniel Spirn, Dynamics for Ginzburg-Landau vortices under a mixed flow, Indiana Univ. Math. J. 58 (2009), no. 6, 2597-2621. MR 2603761, https://doi.org/10.1512/iumj.2009.58.3842
  • [Li1] Fang Hua Lin, Some dynamical properties of Ginzburg-Landau vortices, Comm. Pure Appl. Math. 49 (1996), no. 4, 323-359. MR 1376654, https://doi.org/10.1002/(SICI)1097-0312(199604)49:4$ \langle $323::AID-CPA1$ \rangle $3.0.CO;2-E
  • [Li2] Fang Hua Lin, Vortex dynamics for the nonlinear wave equation, Comm. Pure Appl. Math. 52 (1999), no. 6, 737-761. MR 1676761, https://doi.org/10.1002/(SICI)1097-0312(199906)52:6$ \langle $737::AID-CPA3$ \rangle $3.3.CO;2-P
  • [LX1] F.-H. Lin and J. X. Xin, On the dynamical law of the Ginzburg-Landau vortices on the plane, Comm. Pure Appl. Math. 52 (1999), no. 10, 1189-1212. MR 1699965, https://doi.org/10.1002/(SICI)1097-0312(199910)52:10$ \langle $1189::AID-CPA1$ \rangle $3.0.CO;2-T
  • [LX2] 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), no. 2, 249-274. MR 1674000, https://doi.org/10.1007/s002200050529
  • [LZ1] Fanghua Lin and Ping Zhang, On the hydrodynamic limit of Ginzburg-Landau vortices, Discrete Contin. Dyn. Syst. 6 (2000), no. 1, 121-142. MR 1739596, https://doi.org/10.3934/dcds.2000.6.121
  • [LZ2] Fanghua Lin and Ping Zhang, Semiclassical limit of the Gross-Pitaevskii equation in an exterior domain, Arch. Ration. Mech. Anal. 179 (2006), no. 1, 79-107. MR 2208290, https://doi.org/10.1007/s00205-005-0383-4
  • [MB] Andrew J. Majda and Andrea L. Bertozzi, Vorticity and incompressible flow, Cambridge Texts in Applied Mathematics, vol. 27, Cambridge University Press, Cambridge, 2002. MR 1867882
  • [MZ] Nader Masmoudi and Ping Zhang, Global solutions to vortex density equations arising from sup-conductivity, Ann. Inst. H. Poincaré Anal. Non Linéaire 22 (2005), no. 4, 441-458 (English, with English and French summaries). MR 2145721, https://doi.org/10.1016/j.anihpc.2004.07.002
  • [Mi] Evelyne Miot, Dynamics of vortices for the complex Ginzburg-Landau equation, Anal. PDE 2 (2009), no. 2, 159-186. MR 2547133, https://doi.org/10.2140/apde.2009.2.159
  • [O] Felix Otto, The geometry of dissipative evolution equations: the porous medium equation, Comm. Partial Differential Equations 26 (2001), no. 1-2, 101-174. MR 1842429, https://doi.org/10.1081/PDE-100002243
  • [RuSt] Jacob Rubinstein and Peter Sternberg, On the slow motion of vortices in the Ginzburg-Landau heat flow, SIAM J. Math. Anal. 26 (1995), no. 6, 1452-1466. MR 1356453, https://doi.org/10.1137/S0036141093259403
  • [PR] L. Peres and J. Rubinstein, Vortex dynamics in $ {\rm U}(1)$ Ginzburg-Landau models, Phys. D 64 (1993), no. 1-3, 299-309. MR 1214555, https://doi.org/10.1016/0167-2789(93)90261-X
  • [SR] Laure Saint-Raymond, Hydrodynamic limits of the Boltzmann equation, Lecture Notes in Mathematics, vol. 1971, Springer-Verlag, Berlin, 2009. MR 2683475
  • [Sa] Etienne Sandier, Lower bounds for the energy of unit vector fields and applications, J. Funct. Anal. 152 (1998), no. 2, 379-403. MR 1607928, https://doi.org/10.1006/jfan.1997.3170
  • [Sch] A. Schmid, A time dependent Ginzburg-Landau equation and its application to the problem of resistivity in the mixed state, Phys. Kondens. Materie 5 (1966), 302-317.
  • [SS1] Etienne Sandier and Sylvia Serfaty, Global minimizers for the Ginzburg-Landau functional below the first critical magnetic field, Ann. Inst. H. Poincaré Anal. Non Linéaire 17 (2000), no. 1, 119-145 (English, with English and French summaries). MR 1743433, https://doi.org/10.1016/S0294-1449(99)00106-7
  • [SS2] Etienne Sandier and Sylvia Serfaty, Limiting vorticities for the Ginzburg-Landau equations, Duke Math. J. 117 (2003), no. 3, 403-446. MR 1979050, https://doi.org/10.1215/S0012-7094-03-11732-9
  • [SS3] Etienne Sandier and Sylvia Serfaty, A product-estimate for Ginzburg-Landau and corollaries, J. Funct. Anal. 211 (2004), no. 1, 219-244. MR 2054623, https://doi.org/10.1016/S0022-1236(03)00199-X
  • [SS4] Etienne Sandier and Sylvia Serfaty, Gamma-convergence of gradient flows with applications to Ginzburg-Landau, Comm. Pure Appl. Math. 57 (2004), no. 12, 1627-1672. MR 2082242, https://doi.org/10.1002/cpa.20046
  • [SS5] Etienne Sandier and Sylvia Serfaty, Vortices in the magnetic Ginzburg-Landau model, Progress in Nonlinear Differential Equations and Their Applications, 70, Birkhäuser Boston, Inc., Boston, MA, 2007. MR 2279839
  • [Sch2] Steven Schochet, The point-vortex method for periodic weak solutions of the 2-D Euler equations, Comm. Pure Appl. Math. 49 (1996), no. 9, 911-965. MR 1399201, https://doi.org/10.1002/(SICI)1097-0312(199609)49:9$ \langle $911::AID-CPA2$ \rangle $3.0.CO;2-A
  • [Se1] Sylvia Serfaty, Vortex collisions and energy-dissipation rates in the Ginzburg-Landau heat flow. I. Study of the perturbed Ginzburg-Landau equation, J. Eur. Math. Soc. (JEMS) 9 (2007), no. 2, 177-217. MR 2293954, https://doi.org/10.4171/JEMS/77
  • [Se2] Sylvia Serfaty, Vortex collisions and energy-dissipation rates in the Ginzburg-Landau heat flow. II. The dynamics, J. Eur. Math. Soc. (JEMS) 9 (2007), no. 3, 383-426.
  • [Se3] Sylvia Serfaty, Gamma-convergence of gradient flows on Hilbert and metric spaces and applications, Discrete Contin. Dyn. Syst. 31 (2011), no. 4, 1427-1451. MR 2836361, https://doi.org/10.3934/dcds.2011.31.1427
  • [ST1] Sylvia Serfaty and Ian Tice, Lorentz space estimates for the Ginzburg-Landau energy, J. Funct. Anal. 254 (2008), no. 3, 773-825. MR 2381162, https://doi.org/10.1016/j.jfa.2007.11.010
  • [ST2] Sylvia Serfaty and Ian Tice, Ginzburg-Landau vortex dynamics with pinning and strong applied currents, Arch. Ration. Mech. Anal. 201 (2011), no. 2, 413-464. MR 2820354, https://doi.org/10.1007/s00205-011-0428-9
  • [SV] Sylvia Serfaty and Juan Luis Vázquez, A mean field equation as limit of nonlinear diffusions with fractional Laplacian operators, Calc. Var. Partial Differential Equations 49 (2014), no. 3-4, 1091-1120. MR 3168624, https://doi.org/10.1007/s00526-013-0613-9
  • [Sp1] Daniel Spirn, Vortex dynamics of the full time-dependent Ginzburg-Landau equations, Comm. Pure Appl. Math. 55 (2002), no. 5, 537-581. MR 1880643, https://doi.org/10.1002/cpa.3018
  • [Sp2] Daniel Spirn, Vortex motion law for the Schrödinger-Ginzburg-Landau equations, SIAM J. Math. Anal. 34 (2003), no. 6, 1435-1476 (electronic). MR 2000979, https://doi.org/10.1137/S0036141001396667
  • [Ti] Ian Tice, Ginzburg-Landau vortex dynamics driven by an applied boundary current, Comm. Pure Appl. Math. 63 (2010), no. 12, 1622-1676. MR 2742009, https://doi.org/10.1002/cpa.20328
  • [TT] J. Tilley and D. Tilley, Superfluidity and superconductivity, 2nd ed., Adam Hilger, Ltd., Bristol, 1986.
  • [T] M. Tinkham, Introduction to Superconductivity, 2nd edition, McGraw-Hill, (1996).
  • [Yau] Horng-Tzer Yau, Relative entropy and hydrodynamics of Ginzburg-Landau models, Lett. Math. Phys. 22 (1991), no. 1, 63-80. MR 1121850, https://doi.org/10.1007/BF00400379

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Additional Information

Sylvia Serfaty
Affiliation: Sorbonne Universités, UPMC Université Paris 06, CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, 4, place Jussieu 75005, Paris, France; and Institut Universitaire de France; and Courant Institute, New York University, 251 Mercer Street, New York, New York 10012
Email: serfaty@ann.jussieu.fr

DOI: https://doi.org/10.1090/jams/872
Keywords: Ginzburg-Landau, Gross-Pitaevskii, vortices, vortex liquids, mean-field limit, hydrodynamic limit, Euler equation
Received by editor(s): July 21, 2015
Received by editor(s) in revised form: June 26, 2016, and July 2, 2016
Published electronically: October 18, 2016
Article copyright: © Copyright 2016 American Mathematical Society

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