We study the time-dependent Ginzburg-Landau equations in the presence of strong currents, but weaker than the critical current where the normal state losses its stability. 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. Some results in the large domain limit will be presented as well. Joint work with Bernard Helffer and Xingbin Pan.
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