Weak solutions to the Ginzburg-Landau model in superconductivity with the temporal gauge

We first prove the uniqueness of weak solutions (ψ, A) to the 3-D Ginzburg-Landau system in superconductivity with the temporal gauge if(ψ, A)W :={(ψ, Aψ∈ L²(0, T; L∞), which is a critical space for some positive constant T. We also prove the global existence of solutions for the Ginzburg-Landau system with magnetic diffusivity μ>f 0 or μ=0. Finally, we prove the uniform bounds with respect to μ of strong solutions in space dimensions d=2. Consequently, the existence of the limit as μ→0 can be established.