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 super‐conductivity with initial data ( ψ 0 , A 0 )∈ L 2 under the hypothesis that ( ψ , A ) ∈ L s (0, T ; L r ,∞ ) × $ L^{\bar s} $ (0, T ; $ L^{\bar r, \infty}) $ with Coulomb gauge for any ( r , s ) and $ (\bar r, \bar s) $ satisfying $ {2 \over {s}} $ + $ {3 \over {r}} $ = 1, $ {1 \over {\bar s}} $ + $ {3 \over {\bar r}} $ = 1, $ \bar s $ ≥ $ {{2s} \over {s-2}} $ , $ \bar r $ ≥ $ {{2r} \over {r-2}} $ and 3 < r ≤ 6, 3 < $ \bar r $ ≤ ∞. Here L r ,∞ ≡ $ L^r_w $ is the Lorentz space. As an application, we prove a uniqueness result with periodic boundary condition when ψ 0 ∈ $ L{{25} \over {7}} $ , A 0 ∈ L 3 (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)