Skip to main content
SpringerLink
Log in

Global Stability of the Normal State of Superconductors in the Presence of a Strong Electric Current

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We consider the time-dependent Ginzburg–Landau model of superconductivity in the presence of an electric current flowing through a two-dimensional wire. We show that when the current is sufficiently strong the solution converges in the long-time limit to the normal state. We provide two types of upper bounds for the critical current where such global stability is achieved: by using the principal eigenvalue of the magnetic Laplacian associated with the normal magnetic field, and through the norm of the resolvent of the linearized steady-state operator. In the latter case we estimate the resolvent norm in large domains by the norms of approximate operators defined on the plane and the half-plane. We also obtain a lower bound, in large domains, for the above critical current by obtaining the current for which the normal state looses its local stability.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Almog Y.: The stability of the normal state of superconductors in the presence of electric currents. SIAM J. Math. Anal. 40, 824–850 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  2. Almog Y., Helffer B., Pan X.-B.: Superconductivity near the normal state under the action of electric currents and induced magnetic fields in \({\mathbb R^2}\) . Commun. Math. Phys. 300, 147–184 (2010)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  3. Almog Y., Helffer B., Pan X.-B.: Superconductivity near the normal state in a half-plane under the action of a perpendicular electric current and an induced magnetic field, Part II: The large conductivity limit. SIAM J. Math. Anal. 44, 3671–3733 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  4. Almog Y., Helffer B., Pan X.-B.: Superconductivity near the normal state in a half-plane under the action of a perpendicular electric current and an induced magnetic field. Trans. Am. Math. Soc. 365, 1183–1217 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bender, C.M., Jones, H.F.: WKB analysis of PT-symmetric Sturm–Liouville problems. J. Phys. A-Math. Theor. 45, 444004 (2012)

    Google Scholar 

  6. Bonnaillie-Noël V., Dauge M.: Asymptotics for the low-lying eigenstates of the Schrödinger operator with magnetic field near corners. Ann. Henri Poincaré 7, 899–931 (2006)

    Article  ADS  MATH  Google Scholar 

  7. Chen Z.M., Hoffmann K.-H., Liang J.: On a nonstationary Ginzburg-Landau superconductivity model. Math. Methods Appl. Sci. 16, 855–875 (1993)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  8. Davies, E.B.: Linear operators and their spectra. In: Cambridge studies in advanced mathematics, Vol. 106, Cambridge: Cambridge University Press, 2007

  9. Du Q.: Global existence and uniqueness of solutions of the time-dependent Ginzburg-Landau model for superconductivity. Appl. Anal. 53, 1–17 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  10. Engel, K.J., Nagel, R.: One-parameter semigroups for linear evolution equations. In: Graduate texts in Mathematics, Vol. 194, New York: Springer-Verlag, 2000

  11. Evans, L.C.: Partial differential equations. Graduate Studies in Mathematics, vol. 19, Providence: AMS, 1998

  12. Feireisl E., Takáč P.: Long-time stabilization of solutions to the Ginzburg-Landau equations of superconductivity. Monatsh. Math. 133, 197–221 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  13. Fleckinger-Pellé J., Kaper H.G., Takáč P.: Dynamics of the Ginzburg–Landau equations of superconductivity. Nonlinear Anal. {\bf 32caron;, 647–665 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  14. Fournais, S., Helffer, B.: Spectral methods in surface superconductivity. Progress in Nonlinear Differential Equations and their Applications, vol. 77, Boston: Birkhäuser, 2010

  15. Giorgi T., Philips D.: The breakdown of superconductivity due to strong fields for the Ginzburg–Landau model. SIAM J. Math. Anal. 30, 341–359 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  16. Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. In: Classics in Mathematics, Berlin: Springer, 2001. Reprint of the 1998 edition

  17. Girault, V., Raviart, P.-A.: Finite element approximation of the Navier–Stokes equations. In: Lecture Notes in Mathematics, Vol. 749, Berlin: Springer, 1986. Extended version

  18. Grisvard, P.: Elliptic problems in nonsmooth domains. In: Monographs and Studies in Mathematics, Vol. 24. Boston: Pitman (Advanced Publishing Program), 1985

  19. Grisvard, P.: Singularities in boundary value problems. Berlin: Springer, 1992

  20. Hartman, P.: Ordinary differential equations. Classics in Applied Mathematics, vol. 38, Philadelphia: SIAM, 2002

  21. Helffer, B., Sjöstrand, J.: From resolvent bounds to semigroup bounds. (2010). Preprint : arXiv:1001.4171v1

  22. Henry, D.: Geometric theory of semilinear parabolic equations, Vol. 840. In: Lecture notes in mathematics. Berlin: Springer, 1981

  23. Ivlev B.I., Kopnin N.B.: Electric currents and resistive states in thin superconductors. Adv. Phys. 33, 47–114 (1984)

    Article  ADS  Google Scholar 

  24. Kato, T.: Perturbation theory for linear operators, 3rd ed. Berlin: Springer, 1980

  25. Kondratiev, V.A.: Boundary Value Problems for elliptic equations in domain with conical or angular points. Trans. Moscow Math. Soc. 1967, pp. 227–313

  26. Lu K., Pan X.-B.: Estimates of the upper critical field for the Ginzburg-Landau equations of superconductivity. Phys. D 127, 73–104 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  27. Mazya, V.G., Plamenevskii, V.A.: L p estimates of solutions of elliptic boundary problems in domains with edges. In: Transactions of the Moscow Mathematical Society, Issue 1, pp. 49–97, 1980

  28. Montgomery R.: Hearing the zero locus of a magnetic field. Commun. Math. Phys. 168, 651–675 (1995)

    Article  ADS  MATH  Google Scholar 

  29. Pan, X.-B., Kwek, K.-H.: Schrödinger operators with non-degenerately vanishing magnetic fields in bounded domains. Trans. Am. Math. Soc. 354, 4201–4227 (electronic). (2002)

    Google Scholar 

  30. Peres-Hari, L., Rubinstein, J., Sternberg, P.: Kinematic and dynamic vortices in a thin film driven by an applied current and magnetic field. Accepted for publication in Physica D

  31. Protter, M.H., Weinberger, H.F.: Maximum principles in differential equations. Englewood Cliffs: Prentice-Hall, 1967

  32. Rubinstein, J., Sternberg, P., Ma, Q.: Bifurcation diagram and pattern formation of phase slip centers in superconducting wires driven with electric currents. Phys. Rev. Lett. 99, 167003 (2007)

    Google Scholar 

  33. Rubinstein J., Sternberg P., Kim J.: On the behavior of a superconducting wire subjected to a constant voltage difference. SIAM J. Appl. Math. 70, 1739–1760 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  34. Rubinstein J., Sternberg P., Zumbrun K.: The resistive state in a superconducting wire: bifurcation from the normal state. Arch. Ration. Mech. Anal. 195, 117–158 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  35. Serfaty S., Tice I.: Ginzburg–Landau vortex dynamics with pinning and strong applied currents. Arch. Ration. Mech. Anal. 201, 413–464 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  36. Tice I.: Ginzburg–Landau vortex dynamics driven by an applied boundary current. Commun. Pure Appl. Math. 63, 1622–1676 (2010)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Louisiana State University, Baton Rouge, LA, 70803, USA

    Yaniv Almog

  2. Laboratoire de Mathématiques, Université Paris-Sud 11 et CNRS, Bât 425, 91405, Orsay Cedex, France

    Bernard Helffer

Authors
  1. Yaniv Almog
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bernard Helffer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yaniv Almog.

Additional information

Communicated by W. Schlag

Rights and permissions

Reprints and permissions

About this article

Cite this article

Almog, Y., Helffer, B. Global Stability of the Normal State of Superconductors in the Presence of a Strong Electric Current. Commun. Math. Phys. 330, 1021–1094 (2014). https://doi.org/10.1007/s00220-014-1970-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-014-1970-8

Keywords

Search

Navigation