Uniqueness of weak solutions in critical space of the 3-D time-dependent Ginzburg-Landau equations for superconductivity

We prove the uniqueness of weak solutions of the 3-D time-dependent Ginzburg-Landau equations for superconductivity with initial data (psi(0), A(0)) is an element of L-2 under the hypothesis that (psi, A) is an element of L-s (0,T; L-r,L-infinity) x L-(s) over bar(0,T; L-(r) over bar,L-infinity) with Coulomb gauge for any (r, s) and ((r) over bar,(s) over bar) satisfying 2/s + 3/r = 1, 2/(s) over bar + 3/(r) over bar = 1, (s) over bar >= 2s/s-2, (r) over bar >= 2r/r-2 and 3 < r <= 6, 3 < (r) over bar <= infinity. Here L-r,L-infinity equivalent to L-w(r) is the Lorentz space. As an application, we prove a uniqueness result with periodic boundary condition when psi(0) is an element of L-25/7, A(0) is an element of L-3. (C) 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim