Uniqueness of weak solutions to the Ginzburg-Landau model for superconductivity



We prove the uniqueness for weak solutions of the time-dependent 2-D Ginzburg-Landau model for superconductivity with L2 initial data in the case of Coulomb gauge. This question was left open in Tang and Wang (Physica D, 88:139–166, 1995). We also prove the uniqueness of the 3-D radially symmetric solution in bounded annular domain with the choice of Lorentz gauge and L2 initial data.

Mathematics Subject Classification (2000)



Uniqueness Ginzburg-Landau model Superconductivity Coulomb gauge Lorentz gauge 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Chen Z.M., Elliott C., Tang Q.: Justification of a two-dimensional evolutionary Ginzburg-Landau superconductivity model. RAIRO Model. Math. Anal. Numer. 32, 25–50 (1998)MathSciNetMATHGoogle Scholar
  2. 2.
    Chen Z.M., Hoffmann K.H.: Global classical solutions to a non-isothermal dynamical Ginzburg-Landau model in superconductivity. Numer. Funct. Anal. Optim. 18, 901–920 (1997)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Chen Z.M., Hoffmann K.H., Liang J.: On a nonstationary Ginzburg-Landau superconductivity model. Math. Methods Appl. Sci. 16, 855–875 (1993)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Danchin R.: Global existence in critical spaces for compressible Navier-Stokes equations. Invent. Math. 141, 579–614 (2000)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Du Q.: Global existence and uniqueness of solutions of the time-dependent Ginzburg-Landau model for superconductivity. Appl. Anal. 521, 1–17 (1994)CrossRefGoogle Scholar
  6. 6.
    Fan J.: The long-time behavior of the transient Ginzburg-Landau equations of superconductivity II. Appl. Math. Lett. 9(5), 107–109 (1996)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Fan J., Gao H.: Uniqueness of weak solutions in critical spaces of the 3-D time-dependent Ginzburg-Landau equations for superconductivity. Math. Nachr. 283(8), 1134–1143 (2010)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Fan J., Jiang S.: Global existence of weak solutions of a time-dependent 3-D Ginzburg-Landau model for superconductivity. Appl. Math. Lett. 16, 435–440 (2003)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Koch H., Tataru D.: Well-posedness for the Navier-Stokes equations. Adv. Math. 157(1), 22–35 (2001)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Liang J.: The regularity of solutions for the curl boundary problems and Ginzburg-Landau superconductivity model. Math. Model. Methods Appl. Sci. 5, 528–542 (1995)CrossRefGoogle Scholar
  11. 11.
    Phillips D., Shin E.: On the analysis of a non-isothermal model for superconductivity. Eur. J. Appl. Math. 15, 147–179 (2004)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Tang Q.: On an evolutionary system of Ginzburg-Landau equations with fixed total magnetic flux. Commun. PDE 20, 1–36 (1995)MATHGoogle Scholar
  13. 13.
    Tang Q., Wang S.: Time dependent Ginzburg-Landau equation of superconductivity. Physica D 88, 139–166 (1995)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Wang B.X.: Uniqueness of solutions for the Ginzburg-Landau model of superconductivity in three spatial dimensions. J. Math. Anal. Appl. 266, 1–20 (2002)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Zaouch F.: Time-periodic solutions of the time-dependent Ginzburg-Landau equations of surperconductivity. Z Angew Math. Phys. 54, 905–918 (2003)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Zhan M.: Well-posedness of phase-lock equations of superconductivity. Appl. Math. Lett. 18(11), 1210–1215 (2005)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Department of Applied MathematicsNanjing Forestry UniversityNanjingPeople’s Republic of China
  2. 2.Department of Applied PhysicsWaseda UniversityTokyoJapan

Personalised recommendations