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

✍ Scribed by Jishan Fan; Hongjun Gao

• Publisher: John Wiley and Sons
• Year: 2010
• Tongue: English
• Weight: 144 KB
• Volume: 283
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200710083

✦ Synopsis

Abstract

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)