Communications in Mathematical Physics

, Volume 330, Issue 3, pp 1021–1094 | Cite as

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

  • Yaniv AlmogEmail author
  • Bernard Helffer


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.


Weak Solution Global Existence Critical Current Global Stability Variational Solution 
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© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of MathematicsLouisiana State UniversityBaton RougeUSA
  2. 2.Laboratoire de MathématiquesUniversité Paris-Sud 11 et CNRSOrsay CedexFrance

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