Fouzi Zaouch
Global Existence and Boundedness of Solutions of the Time-Dependent Ginzburg-Landau Equations with a Time-Dependent Magnetic Field
The paper is published: Rostocker Mathematisches Kolloquium, Rostock. Math. Kolloq. 57, 53-70(2003)
MSC:
35K55 Nonlinear PDE of parabolic type
35B40 Asymptotic behavior of solutions
35B65 Smoothness/regularity of solutions of PDE
Abstract: This paper is concerned with existence, uniqueness and long-time
asymptotic behavior of the solutions of the time-dependent
Ginzburg-Landau equations of superconductivity, in the case where
the applied magnetic field $\H$ is time-dependent. We first prove
existence and uniqueness of solutions with $H^1$-initial data.
This result is obtained under the $\phi=-\omega(\nabla\cdot\A)$''
gauge with $\omega>0$. These solutions become then uniformly
bounded in time for the $H^1$-norm, by assuming time-uniform
boundedness on $\H$ and its time derivative.

Keywords: Superconductivity, Ginzburg-Landau equation, gauge, initial boundary value problems, global existence and uniqueness