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Archive for Rational Mechanics and Analysis

, Volume 231, Issue 3, pp 1531–1614 | Cite as

Three Dimensional Vortex Approximation Construction and \({\varepsilon}\)-Level Estimates for the Ginzburg–Landau Functional

  • Carlos RománEmail author
Article

Abstract

We provide a quantitative three dimensional vortex approximation construction for the Ginzburg–Landau functional. This construction gives an approximation of vortex lines coupled to a lower bound for the energy, optimal to leading order, analogous to the two dimensional ones, and valid for the first time at the \({\varepsilon}\)-level. These tools allow for a new approach to analyzing the behavior of global minimizers for the Ginzburg–Landau functional below and near the first critical field in three dimensions, followed in Román (On the first critical field in the three dimensional Ginzburg–Landau model of superconductivity, 2018). In addition, they allow one to obtain an \({\varepsilon}\)-quantitative product estimate for the study of Ginzburg–Landau dynamics.

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Notes

Acknowledgments

I would like to warmly thank my Ph.D. advisors Etienne Sandier and Sylvia Serfaty for suggesting the problem, for their helpful advice, careful reading, and useful comments. I also thank the anonymous referee for helpful comments that led to improvements in the presentation of the appendices of the paper. This work was supported by a public grant overseen by the French National Research Agency (ANR) as part of the “Investissements d’Avenir” program (reference: ANR-10-LABX-0098, LabEx SMP).

Conflict of interest

The author declares that they have no conflict of interest.

References

  1. 1.
    Alama S., Bronsard L., Montero J.A.: On the Ginzburg–Landau model of a superconducting ball in a uniform field. Ann. Inst. H. Poincaré Anal. Non Linéaire. 23(2), 237–267 (2006)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alberti G., Baldo S., Orlandi G.: Variational convergence for functionals of Ginzburg–Landau type. Indiana Univ. Math. J. 54(5), 1411–1472 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bethuel F., Brezis H., Hélein F.: Ginzburg–Landau vortices, Progress in Nonlinear Differential Equations and their Applications, vol. 13. Birkhäuser Boston, Inc., Boston, MA (1994)zbMATHGoogle Scholar
  4. 4.
    Bourgain J., Brezis H., Mironescu P.: H 1/2 maps with values into the circle: minimal connections, lifting, and the Ginzburg–Landau equation. Publ. Math. Inst. Hautes Études Sci. 99, 1–115 (2004)CrossRefzbMATHGoogle Scholar
  5. 5.
    Bethuel F., Brezis H., Orlandi G.: Asymptotics for the Ginzburg–Landau equation in arbitrary dimensions. J. Funct. Anal. 186(2), 432–520 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Brezis H., Coron J.-M., Lie E. H.: Harmonic maps with defects. Commun. Math. Phys. 107(4), 649–705 (1986)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Baldo S., Jerrard R. L., Orlandi G., Soner H. M.: Convergence of Ginzburg–Landau functionals in three-dimensional superconductivity. Arch. Ration. Mech. Anal. 205(3), 699–752 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Baldo S., Jerrard R. L., Orlandi G., Sone H. M.: Vortex density models for superconductivity and superfluidity. Commun. Math. Phys. 318(1), 131–171 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bethuel F., Riviére T.: Vortices for a variational problem related to superconductivity. Ann. Inst. H. Poincaré Anal. Non Linéaire 12(3), 243–303 (1995)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gruber P. M.: Asymptotic estimates for best and stepwise approximation of convex bodies. I.. Forum Math 5(3), 281–297 (1993)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Jerrard R. L.: Lower bounds for generalized Ginzburg–Landau functionals. SIAM J. Math. Anal. 30(4), 721–746 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Jerrard R., Montero A., Sternberg P.: Local minimizers of the Ginzburg–Landau energy with magnetic field in three dimensions. Commun. Math. Phys. 249(3), 549–577 (2004)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Jerrard R. L., Soner H. M.: The Jacobian and the Ginzburg–Landau energy. Calc. Var. Partial Differ. Equ. 14(2), 151–191 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lang S.: Real and Functional Analysis, Third, Graduate Texts in Mathematics, vol. 142. Springer, New York (1993)CrossRefGoogle Scholar
  15. 15.
    Lin F.-H., Riviére T.: A quantization property for static Ginzburg–Landau vortices. Commun. Pure Appl. Math. 54(2), 206–228 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Lin F., Riviére T.: Complex Ginzburg–Landau equations in high dimensions and codimension two area minimizing currents. J. Eur. Math. Soc. (JEMS) 1(3), 237–311 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Riviére, T.: Line vortices in the U(1)-Higgs model. ESAIM Contrôle Optim. Calc. Var. 1, 77–167, 1995/1996Google Scholar
  18. 18.
    Román, C.: On the first critical field in the three dimensional Ginzburg–Landau model of superconductivity. ArXiv e-prints 2018, available at arXiv:1802.09390. Under review in Commun. Math. Phys
  19. 19.
    Sandier E.: Ginzburg–Landau minimizers from \({\mathbb R^{n+1}}\) to \({\mathbb R^n}\) and minimal connections. Indiana Univ. Math. J. 50(4), 1807–1844 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Sandier E.: Lower bounds for the energy of unit vector fields and applications. J. Funct. Anal. 152(2), 379–403 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Serfaty S.: Mean field limits of the Gross–Pitaevskii and parabolic Ginzburg–Landau equations. J. Amer. Math. Soc. 30(3), 713–768 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Serfaty S.: Local minimizers for the Ginzburg–Landau energy near critical magnetic field. I. Commun. Contemp. Math. 1(2), 213–254 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Sandier E., Serfaty S.: Global minimizers for the Ginzburg–Landau functional below the first critical magnetic field. Ann. Inst. H. Poincaré Anal. Non Linéaire 17(1), 119–145 (2000)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Sandier E., Serfaty S.: On the energy of type-II superconductors in the mixed phase. Rev. Math. Phys. 12(9), 1219–1257 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Sandier É., Serfaty S.: A rigorous derivation of a free-boundary problem arising in superconductivity. Ann. Sci. École Norm. Sup. (4) 33(4), 561–592 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Sandier E., Serfaty S.: Ginzburg–Landau minimizers near the first critical field have bounded vorticity. Calc. Var. Partial Differ. Equ. 17(1), 17–28 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Sandier E., Serfaty S.: A product-estimate for Ginzburg–Landau and corollaries. J. Funct. Anal. 211(1), 219–244 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Sandier E., Serfaty S.: Vortices in the Magnetic Ginzburg–Landau Model, Progress in Nonlinear Differential Equations and Their Applications, vol. 70. Birkhäuser, Boston, Inc., Boston, MA (2007)zbMATHGoogle Scholar
  29. 29.
    Sandier E., Shafrir I.: Small Energy Ginzburg–Landau Minimizers in \({\mathbb{R}^3}\). J. Funct. Anal. 272(9), 3946–3964 (2017)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Sorbonne Universités, UPMC Univ Paris 06, CNRS, UMR 7598, Laboratoire Jacques-Louis LionsParisFrance
  2. 2.Mathematisches InstitutUniversität LeipzigLeipzigGermany

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