Mixed normal-superconducting states in the presence of strong electric currents

We study the Ginzburg-Landau equations in the presence of large electric currents, that are smaller than the critical current where the normal state losses its stability. For steady-state solutions in the large $\kappa$ limit, we prove that the superconductivity order parameter is exponentially small in a significant part of the domain, and small in the rest of it. Similar results are obtained for the time-dependent problem, in continuation of the paper by the two first authors [3]. We conclude by obtaining some weaker results, albeit similar, for steady-state solutions in the large domain limit.