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VORTEX LIQUIDS AND THE GINZBURG–LANDAU EQUATION

Part of: Parabolic equations and systems Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  27 May 2014

MATTHIAS KURZKE  and
DANIEL SPIRN
MATTHIAS KURZKE
Affiliation:
School of Mathematical Sciences, University of Nottingham, Nottingham NG7 2RD, UK; matthias.kurzke@nottingham.ac.uk
DANIEL SPIRN
Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA; spirn@math.umn.edu

Abstract

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We establish vortex dynamics for the time-dependent Ginzburg–Landau equation for asymptotically large numbers of vortices for the problem without a gauge field and either Dirichlet or Neumann boundary conditions. As our main tool, we establish quantitative bounds on several fundamental quantities, including the kinetic energy, that lead to explicit convergence rates. For dilute vortex liquids, we prove that sequences of solutions converge to the hydrodynamic limit.

MSC classification

Secondary: 35Q56: Ginzburg-Landau equations 35K51: Initial-boundary value problems for second-order parabolic systems

Information

Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 2 , 01 July 2014 , e11
Creative Commons
Creative Common License - CCCreative Common License - BY
The online version of this article is published within an Open Access environment subject to the conditions of the Creative Commons Attribution licence .
Copyright
© The Author(s) 2014

References

Ambrosio, L. and Serfaty, S., ‘A gradient flow approach to an evolution problem arising in superconductivity’, Comm. Pure Appl. Math. 61 11 (2008), 1495–1539.Google Scholar
Ambrosio, L., Mainini, E. and Serfaty, S., ‘Gradient flow of the Chapman-Rubinstein-Schatzman model for signed vortices’, Ann. IHP, Analyse nonlinéaire. 28 2 (2011), 217–246.Google Scholar
Bethuel, F., Brezis, H. and Hélein, F., ‘Ginzburg–Landau vortices’, in Progress in Nonlinear Differential Equations and their Applications, vol. 13, (Birkhäuser, Boston, 1994).Google Scholar
Bethuel, F., Orlandi, G. and Smets, D., ‘Collisions and phase-vortex interactions in dissipative Ginzburg–Landau dynamics’, Duke Math. J. 130 (2005), 523–614.CrossRefGoogle Scholar
Brezis, H., Coron, J.-M. and Lieb, E., ‘Harmonic maps with defects’, Comm. Math. Phys. 107 (1986), 649–705.Google Scholar
Chapman, S. J., ‘A hierarchy of models for type-II superconductors’, SIAM Rev. 42 (2000), 555–598.Google Scholar
Chapman, S. J., Rubinstein, J. and Schatzman, M., ‘A mean-field model of superconducting vortices’, Eur. J. Appl. Math. 7 (1996), 97–111.Google Scholar
Colliander, J. E. and Jerrard, R. L., ‘Ginzburg–Landau vortices: weak stability and Schrödinger equation dynamics’, J. Anal. Math. 77 (1999), 129–205.Google Scholar
Delort, J.-M., ‘Existence de nappes de tourbillon en dimension deux’, J. Amer. Math. Soc. 4 (1991), 553–586.Google Scholar
DiPerna, R. J. and Majda, A. J., ‘Concentrations in regularizations for 2-D incompressible flow’, Comm. Pure Appl. Math. 40 (1987), 301–345.CrossRefGoogle Scholar
Dorsey, A. T., ‘Vortex motion and the Hall effect in type-II superconductors: A time-dependent Ginzburg–Landau theory approach’, Phys. Rev. B 46 13 (Oct 1992), 8376–8392.Google Scholar
E, W., ‘Dynamics of vortices in Ginzburg–Landau theories with applications to superconductivity’, Phys. D 77 (1994), 383–404.Google Scholar
E, W., ‘Dynamics of vortex liquids in Ginzburg–Landau theories with applications to superconductivity’, Phys. Rev. B 50 (1994), 1126–1135.Google Scholar
Essmann, U. and Träuble, H., ‘The flux-line arrangement in the “intermediate state” of type II superconductors’, Phys. Lett. 27A (1968), 156–157.Google Scholar
Evans, L. C., ‘Partial differential equations’, in Graduate Studies in Mathematics, vol. 19, (American Mathematical Society, Providence, RI, 1998.Google Scholar
Evans, L. C. and Müller, S., ‘Hardy spaces and the two-dimensional Euler equations with nonnegative vorticity’, J. Amer. Math. Soc. 7 (1994), 199–219.Google Scholar
Lopes Filho, M. C., Nussenzveig Lopes, H. J. and Xin, Z., ‘Existence of vortex sheets with reflection symmetry in two space dimensions’, Arch. Ration. Mech. Anal. 158 (2001), 235–257.CrossRefGoogle Scholar
Jerrard, R. L., ‘Vortex dynamics for the Ginzburg–Landau wave equation’, Calc. Var. Partial Differential Equations 9 (1999), 683–688.CrossRefGoogle Scholar
Jerrard, R. L., ‘Lower bounds for generalized Ginzburg–Landau functionals’, SIAM J. Math. Anal. 30 (1999), 721–746.Google Scholar
Jerrard, R. L. and Soner, H. M., ‘The Jacobian and the Ginzburg–Landau energy’, Calc. Var. Partial Differential Equations 14 (2002), 151–191.Google Scholar
Jerrard, R. L. and Soner, H. M., ‘Dynamics of Ginzburg–Landau vortices’, Arch. Rational Mech. Anal. 142 (1998), 99–125.Google Scholar
Jerrard, R. L. and Spirn, D., ‘Refined Jacobian estimates for Ginzburg–Landau functionals’, Indiana Univ. Math. J. 56 (2007), 135–186.Google Scholar
Jerrard, R. L. and Spirn, D., ‘Refined Jacobian estimates and Gross–Pitaevsky vortex dynamics’, Arch. Ration. Mech. Anal. (2008).Google Scholar
Kopnin, N. B., Ivlev, B. I. and Kalatsky, V. A., ‘The flux-flow hall effect in type ii superconductors: an explanation of the sign reversal’, J. Low Temp. Phys. 90 1 (01 1993), 1–13.Google Scholar
Kurzke, M., Melcher, C. and Moser, R., ‘Vortex motion for the Landau–Lifshitz–Gilbert equation with spin transfer torque’, SIAM J. Math. Anal. 43 1099–1121.CrossRefGoogle Scholar
Kurzke, M., Melcher, C., Moser, R. and Spirn, D., ‘Dynamics for Ginzburg–Landau vortices under a mixed flow’, Indiana Univ. Math. J. 58 6 (2009), 2597–2621.CrossRefGoogle Scholar
Kurzke, M. and Spirn, D., ‘ $\Gamma $ -stability and vortex motion in type II superconductors’, Commun. Partial Differential Equations 36 2 (2011), 256–292.Google Scholar
Kurzke, M. and Spirn, D., ‘Quantitative equipartition of the Ginzburg–Landau energy with applications’, Indiana 59 6 (2010), 2077–2092.Google Scholar
Lin, F.-H., ‘Some dynamical properties of Ginzburg–Landau vortices’, Comm. Pure Appl. Math. 49 (1996), 323–359.Google Scholar
Lin, F.-H., ‘Vortex dynamics for the nonlinear wave equation’, Comm. Pure Appl. Math. 52 (1999), 737–761.Google Scholar
Lin, F. H. and Lin, T. C., ‘Minimax solutions of the Ginzburg–Landau equations’, Selecta Math. 3 (1997), 99–113.Google Scholar
Lin, F. H. and Zhang, P., ‘On the hydrodynamic limit of Ginzburg–Landau vortices’, Discrete Contin. Dyn. Syst. 6 (1) (2000), 121–142.Google Scholar
Lin, F. and Zhang, P., ‘On the hydrodynamic limit of Ginzburg–Landau wave vortices’, Comm. Pure Appl. Math. 55 7 (2002), 831–856.Google Scholar
Liu, J. G. and Xin, Z., ‘Convergence of the point vortex method for 2-D vortex sheet’, Math. Comp. 70 (2001), 595–606.Google Scholar
Liu, J. G. and Xin, Z., ‘Convergence of vortex methods for weak solutions to the 2-D Euler equations with vortex sheet data’, Comm. Pure Appl. Math. 48 (1995), 611–628.Google Scholar
Majda, A. J., ‘Remarks on weak solutions for vortex sheets with a distinguished sign’, Indiana Univ. Math. J. 42 (1993), 921–939.Google Scholar
Miot, E., ‘Dynamics of vortices for the complex Ginzburg–Landau equation’, Anal. PDE 2 2 (2009), 159–186.Google Scholar
Peres, L. and Rubinstein, J., ‘Vortex dynamics in $U(1)$ Ginzburg–Landau models’, Phys. D 64 (1993), 299–309.Google Scholar
Roitberg, Y., ‘Elliptic boundary value problems in the space of distributions’, in Mathematics and its Applications, vol. 498, (Kluwer Academic Publications, Dordrecht, 1999.Google Scholar
Sandier, E., ‘Lower bounds for the energy of unit vector fields and applications’, J. Funct. Anal. 152 (1998), 379–403.Google Scholar
Sandier, E. and Serfaty, S., ‘Gamma-convergence of gradient flows with applications to Ginzburg–Landau’, Comm. Pure Appl. Math. 57 (2004), 1627–1672.Google Scholar
Sandier, E. and Serfaty, S., ‘Product Estimates for Ginzburg–Landau and corollaries’, J. Funct. Anal. 211 (2004), 219–244.Google Scholar
Sandier, E. and Soret, M., ‘ $\mathbb{S}^1$ -valued harmonic maps with high topological degree: Asymptotic behavior of the singular set’, Potential Anal. 13 (2000), 169–184.CrossRefGoogle Scholar
Schochet, S., ‘The point vortex method for periodic weak solutions of the 2D Euler equations’, Comm. Pure Appl. Math. 49 (1996), 911–965.Google Scholar
Serfaty, S., ‘Vortex collisions and energy-dissipation rates in the Ginzburg–Landau heat flow, part I: Study of the perturbed Ginzburg–Landau equation’, J. Eur. Math. Soc 9 (2007), 177–217.Google Scholar
Serfaty, S., ‘Vortex collisions and energy-dissipation rates in the Ginzburg–Landau heat flow. II. The dynamics’, J. Eur. Math. Soc. (JEMS) 9 (2007), 383–426.CrossRefGoogle Scholar
Serfaty, S., ‘Gamma-convergence of gradient flows on Hilbert and metric spaces and applications’, Discrete Contin. Dyn. Syst. Ser. A 31 4 (2011), 1427–1451.CrossRefGoogle Scholar
Serfaty, S. and Tice, I., ‘Ginzburg–Landau vortex dynamics with pinning and strong applied currents’, Arch. Ration. Mech. Anal. 201 (2011), 413–464.Google Scholar
Spirn, D., ‘Vortex dynamics of the full time-dependent Ginzburg–Landau model’, Comm. Pure Appl. Math. 55 (2002), 537–581.Google Scholar