Time dependent Ginzburg-Landau equations of superconductivity

Abstract

We study in this article the existence, uniqueness and long time behavior of the solutions of a nonstationary Ginzburg-Landau superconductivity model. We first prove the existence and uniqueness of solutions with H1 initial data, which are crucial for the study of the global attractor. We also obtain, for the first time, the existence of global weak solutions of the model with L2 initial data. It is then proved that the Ginzburg-Landau system admits a global attractor, which represents exactly all the long time dynamics of the system. The global attractor obtained consists of exactly the set of steady state solutions and its unstable manifold. Its Hausdorff and fractal dimensions are estimated in terms of the physically relevant Ginzburg-Landau parameter, diffusion parameter and applied magnetic field. We construct explicitly absorbing sets for some abstract semigroups having a Lyapunov functional and consequently prove the existence of global attractors. This abstract result is applied to the Ginzburg-Landau system, for which the existence of global attractor does not seem to be the direct consequence of some a priori estimates of solutions.