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Limiting Motion for the Parabolic Ginzburg–Landau Equation with Infinite Energy Data

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Abstract

We study a class of solutions to the parabolic Ginzburg–Landau equation in dimension 2 or higher, with ill-prepared infinite energy initial data. We show that, asymptotically, the vorticity evolves according to motion by mean curvature in Brakke’s weak formulation. Then, we prove that in the plane, point vortices do not move in the original time scale. These results extend the works of Bethuel, Orlandi and Smets (Ann Math (2) 163(1):37–163, 2006; Duke Math J 130(3):523–614, 2005) to infinite energy data; they allow us to consider point vortices on a lattice (in dimension 2), or filament vortices of infinite length (in dimension 3).

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References

  1. Ambrosio L., Soner M.: A measure theoretic approach to higher codimension mean curvature flow. Ann. Sc. Norm. Sup. Pisa Cl. Sci. 25, 27–49 (1997)

    MathSciNet  MATH  Google Scholar 

  2. Baldo S., Orlandi G., Weitkamp S.: Convergence of minimizers with local energy bounds for the Ginzburg–Landau functionals. Indiana Math. J. 58(5), 2369–2408 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bethuel F., Bourgain J., Brezis H., Orlandi G.: W 1,p estimates for solutions to the Ginzburg–Landau functional with boundary data in H 1/2. C. R. Acad. Sci. Paris I 333, 1–8 (2001)

    Article  ADS  MATH  Google Scholar 

  4. Bethuel F., Brezis H., Hélein F.: Ginzburg–Landau Vortices. Progr. Nonlinear Differential Equations Appl., vol. 13. Birkhäuser, Boston (1994)

    MATH  Google Scholar 

  5. Bethuel, F., Brezis, H., Orlandi, G.: Asymptotics for the Ginzburg–Landau equation in arbitrary dimensions. J. Funct. Anal. 186, 432–520 (2001). [Erratum: 188, 548–549 (2002)]

  6. Bethuel F., Brezis H., Orlandi G.: Improved estimates for the Ginzburg–Landau equation: the elliptic case. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 4(2), 319–355 (2005)

    MathSciNet  MATH  Google Scholar 

  7. Bethuel F., Orlandi G.: Uniform estimates for the parabolic Ginzburg–Landau equation. ESAIM Control Optim. Calc. Var. 8, 219–238 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bethuel F., Orlandi G., Smets D.: Convergence of the parabolic Ginzburg–Landau equation to motion by mean curvature. Ann. Math. (2) 163(1), 37–163 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bethuel F., Orlandi G., Smets D.: Collisions and phase-vortex interactions in dissipative Ginzburg–Landau dynamics. Duke Math. J. 130(3), 523–614 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Bethuel F., Orlandi G., Smets D.: Quantization and motion law for Ginzburg–Landau vortices. Arch. Ration. Mech. Anal. 183(2), 315–370 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bethuel F., Orlandi G., Smets D.: Dynamics of multiple degree Ginzburg–Landau vortices. Commun. Math. Phys. 272(1), 229–261 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. Bethuel, F., Orlandi, G., Smets, D.: Dynamique des tourbillons de vorticité pour l’équation de Ginzburg–Landau parabolique. Séminaire: É. Sémin. Équ. Dériv. Partielles. Exp. No. 18. École Polytechnique, Palaiseau (2007)

  13. Brakke K.: The Motion of a Surface by its Mean Curvature. Princeton University Press, Princeton (1978)

    MATH  Google Scholar 

  14. Chen Y., Struwe M.: Existence and partial regularity results for the heat flow for harmonic maps. Math. Z. 201, 83–103 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  15. Delphine C.: Unbounded solutions to defocusing parabolic systems. Differ. Integral Equ. 28(9–10), 899–940 (2015)

    MathSciNet  MATH  Google Scholar 

  16. Federer H.: Geometric Measure Theory. Springer, Berlin (1969)

    MATH  Google Scholar 

  17. Ginibre J., Velo G.: The Cauchy problem in local spaces for the complex Ginzburg–Landau equation I. Contraction methods. Phys. D 95, 191–228 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ginibre J., Velo G.: The Cauchy problem in local spaces for the complex Ginzburg–Landau equation II. Contraction methods. Commun. Math. Phys. 187, 45–79 (1997)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. Ilmanen T.: Convergence of the Allen–Cahn equation to Brakke’s motion by mean curvature. J. Differ. Geom. 38, 417–461 (1993)

    MathSciNet  MATH  Google Scholar 

  20. Ilmanen, T.: Elliptic regularization and partial regularity for motion by mean curvature. Mem. Am. Math. Soc. 108(520) (1994)

  21. Jerrard R.L., Soner H.M.: Dynamics of Ginzburg–Landau vortices. Arch. Rational Mech. Anal. 142, 99–125 (1998)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. Jerrard R.L., Soner H.M.: Scaling limits and regularity results for a class of Ginzburg–Landau systems. Ann. Inst. H. Poincaré 16, 423–466 (1999)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  23. Jerrard R.L., Soner H.M.: The Jacobian and the Ginzburg–Landau energy. Calc. Var. PDE 14, 151–191 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  24. Lin F.H.: Some dynamical properties of Ginzburg–Landau vortices. Commun. Pure Appl. Math. 49, 323–359 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lin F.H.: Complex Ginzburg–Landau equations and dynamics of vortices, filaments, and codimension-2 submanifolds. Commun. Pure Appl. Math. 51, 385–441 (1998)

    Article  MathSciNet  Google Scholar 

  26. Lin, F.H., Riviè, T.: Complex Ginzburg–Landau equation in high dimension and codimension two area minimizing currents. J. Eur. Math. Soc. 1, 237–311 (1999) [Erratum: Ibid. 2, 87–91 (2000)]

  27. Lin F.H., Rivière T.: A quantization property for moving line vortices. Commun. Pure Appl. Math. 54, 825–850 (2001)

    MathSciNet  MATH  Google Scholar 

  28. Preiss D.: Geometry of measures in \({\mathbb{R}^n}\): distribution, rectifiability, and densities. Ann. Math. 125, 537–643 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  29. Reshetnyak Y.: Weak convergence of completely additive functions on a set. Siberian Math. J. 9, 487–498 (1968)

    Article  MATH  Google Scholar 

  30. Sandier E., Serfaty S.: Gamma-convergence of gradient flow with applications to Ginzburg–Landau. Commun. Pure Appl. Math. 57, 1627–1672 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  31. Serfaty S.: Vortex collision and energy dissipation rates in the Ginzburg–Landau heat flow, part I: study of the perturbed Ginzburg–Landau equation. J. Eur. Math. Soc. 9(2), 177–217 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  32. Serfaty S.: Vortex collision and energy dissipation rates in the Ginzburg–Landau heat flow, part II: the dynamics. J. Eur. Math. Soc. 9(3), 383–426 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  33. Soner, H.M.: Ginzburg–Landau equation and motion by mean curvature. I. Convergence, and II. Development of the initial interface. J. Geom. Anal. 7, 437–475, 477–491 (1997)

  34. Spirn D.: Vortex dynamics of the full time-dependent Ginzburg–Landau equations. Commun. Pure Appl. Math. 55, 537–581 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  35. Spirn D.: Vortex motion law for the Schrödinger–Ginzburg–Landau equations. SIAM J. Math. Anal. 34(6), 1435–1476 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  36. Struwe M.: On the evolution of harmonic maps in higher dimensions. J. Differ. Geom. 28, 485–502 (1988)

    MathSciNet  MATH  Google Scholar 

  37. Struwe, M.: On the asymptotic behavior of the Ginzburg–Landau model in 2 dimensions. Differ. Integral Equ. 7, 1613–1624 (1994) [Erratum: 8, 224 (1995)]

  38. Wang C.: On moving Ginzburg–Landau filament vortices. Commun. Anal. Geom. 12, 1185–1199 (2004)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, 4 Place Jussieu, 75005, Paris, France

    Delphine Côte

  2. Centre de Mathématiques Laurent Schwartz, CNRS et École polytechnique, 91128, Palaiseau Cedex, France

    Raphaël Côte

Authors
  1. Delphine Côte
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  2. Raphaël Côte
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Correspondence to Raphaël Côte.

Additional information

Communicated by W. Schlag

R.C. gratefully acknowledges support from the ANR contract MAToS ANR-14-CE25-0009-01, and from the ERC Advanced Grant 291214 BLOWDISOL.

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Côte, D., Côte, R. Limiting Motion for the Parabolic Ginzburg–Landau Equation with Infinite Energy Data. Commun. Math. Phys. 350, 507–568 (2017). https://doi.org/10.1007/s00220-016-2736-2

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  • DOI: https://doi.org/10.1007/s00220-016-2736-2

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