Advertisement

Communications in Mathematical Physics

, Volume 350, Issue 2, pp 507–568 | Cite as

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

  • Delphine Côte
  • Raphaël CôteEmail author
Article

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).

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 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)MathSciNetzbMATHGoogle Scholar
  2. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 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)ADSCrossRefzbMATHGoogle Scholar
  4. 4.
    Bethuel F., Brezis H., Hélein F.: Ginzburg–Landau Vortices. Progr. Nonlinear Differential Equations Appl., vol. 13. Birkhäuser, Boston (1994)zbMATHGoogle Scholar
  5. 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)]Google Scholar
  6. 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)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Bethuel F., Orlandi G.: Uniform estimates for the parabolic Ginzburg–Landau equation. ESAIM Control Optim. Calc. Var. 8, 219–238 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bethuel F., Orlandi G., Smets D.: Quantization and motion law for Ginzburg–Landau vortices. Arch. Ration. Mech. Anal. 183(2), 315–370 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bethuel F., Orlandi G., Smets D.: Dynamics of multiple degree Ginzburg–Landau vortices. Commun. Math. Phys. 272(1), 229–261 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 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)Google Scholar
  13. 13.
    Brakke K.: The Motion of a Surface by its Mean Curvature. Princeton University Press, Princeton (1978)zbMATHGoogle Scholar
  14. 14.
    Chen Y., Struwe M.: Existence and partial regularity results for the heat flow for harmonic maps. Math. Z. 201, 83–103 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Delphine C.: Unbounded solutions to defocusing parabolic systems. Differ. Integral Equ. 28(9–10), 899–940 (2015)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Federer H.: Geometric Measure Theory. Springer, Berlin (1969)zbMATHGoogle Scholar
  17. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 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)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Ilmanen T.: Convergence of the Allen–Cahn equation to Brakke’s motion by mean curvature. J. Differ. Geom. 38, 417–461 (1993)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Ilmanen, T.: Elliptic regularization and partial regularity for motion by mean curvature. Mem. Am. Math. Soc. 108(520) (1994)Google Scholar
  21. 21.
    Jerrard R.L., Soner H.M.: Dynamics of Ginzburg–Landau vortices. Arch. Rational Mech. Anal. 142, 99–125 (1998)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. 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)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Jerrard R.L., Soner H.M.: The Jacobian and the Ginzburg–Landau energy. Calc. Var. PDE 14, 151–191 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Lin F.H.: Some dynamical properties of Ginzburg–Landau vortices. Commun. Pure Appl. Math. 49, 323–359 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 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)MathSciNetCrossRefGoogle Scholar
  26. 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)]Google Scholar
  27. 27.
    Lin F.H., Rivière T.: A quantization property for moving line vortices. Commun. Pure Appl. Math. 54, 825–850 (2001)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Preiss D.: Geometry of measures in \({\mathbb{R}^n}\): distribution, rectifiability, and densities. Ann. Math. 125, 537–643 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Reshetnyak Y.: Weak convergence of completely additive functions on a set. Siberian Math. J. 9, 487–498 (1968)CrossRefzbMATHGoogle Scholar
  30. 30.
    Sandier E., Serfaty S.: Gamma-convergence of gradient flow with applications to Ginzburg–Landau. Commun. Pure Appl. Math. 57, 1627–1672 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 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)Google Scholar
  34. 34.
    Spirn D.: Vortex dynamics of the full time-dependent Ginzburg–Landau equations. Commun. Pure Appl. Math. 55, 537–581 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Spirn D.: Vortex motion law for the Schrödinger–Ginzburg–Landau equations. SIAM J. Math. Anal. 34(6), 1435–1476 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Struwe M.: On the evolution of harmonic maps in higher dimensions. J. Differ. Geom. 28, 485–502 (1988)MathSciNetzbMATHGoogle Scholar
  37. 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)]Google Scholar
  38. 38.
    Wang C.: On moving Ginzburg–Landau filament vortices. Commun. Anal. Geom. 12, 1185–1199 (2004)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Laboratoire Jacques-Louis LionsUniversité Pierre et Marie CurieParisFrance
  2. 2.Centre de Mathématiques Laurent SchwartzCNRS et École polytechniquePalaiseau CedexFrance

Personalised recommendations