Ginzburg-Landau equations and stable solutions in a rotational domain

The Ginzburg-Landau (GL) equations, with or without magnetic effect, are studied in the case of a rotational domain in R(3). It can be shown that there exist rotational solutions which describe the physical state of permanent current of electrons in a ring-shaped superconductor. Moreover, if a physical parameter-called the GL parameter-is sufficiently large, then these solutions are stable, that is, they are local minimizers of an energy functional (GL energy). This is proved by the spectral analysis on the linearized equation.