The Stability of the Normal State of Superconductors in the Presence of Electric Currents

The stability of the normal state of superconductors in the presence of electric currents is studied in the large domain limit. The model being used is the time-dependent Ginzburg-Landau model, in the absence of an applied magnetic field, and with the effect of the induced magnetic field being neglected. We find that if the current is nowhere perpendicular to the boundary, or if the minimal current on the boundary, at points where it is perpendicular to it, is greater than the critical current in the one-dimensional case, then the normal state is stable. We also prove some short-time instability when the current is both perpendicular to the boundary and smaller then the one-dimensional critical current.