Existence and stability of superconducting solutions for the   Ginzburg-Landau equations in the presence of weak electric currents
  For a reduced Ginzburg-Landau model in which the magnetic field is neglected, we prove, for weak electric currents, the existence of a steady-state solution in a vicinity of the purely superconducting state. We further show that this solution is linearly stable. 
Yaniv Almog, Leonid Berlyand, Dmitry Golovaty, Itai Shafrir
math-ph math.MP