Long-time stabilization of solutions to the Ginzburg-Landau equations of superconductivity

The long-time dynamical properties of solutions (phi ,A) to the time-dependent Ginzburg-Landau (TDGL) equations of superconductivity are investigated. The applied magnetic field varies with time, but it is assumed to approach a long-time asymptotic limit. Sufficient conditions (in terms of the time rate of change of the applied magnetic field) are given which guarantee that the dynamical process defined by the TDGL equations is asymptotically autonomous, i.e., it approaches a dynamical system as time goes to infinity. Analyticity of an energy functional is used to show that every solution of the TDGL equations asymptotically approaches a (single) stationary solution of the (time-independent) Ginzburg-Landau equations. The standard "phi = -del . A" gauge is chosen.