Ginzburg–Landau Equations and Stable Solutions in a Rotational Domain

Abstract

The Ginzburg–Landau (GL) equations, with or without magnetic effect, are studied in the case of a rotational domain in $\mathbb{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.