Viscosity solutions of a degenerate parabolic-elliptic system arising in the mean-field theory of superconductivity

In a Type-II superconductor the magnetic field penetrates the superconducting body through the formation of vortices. In an extreme Type-II superconductor these vortices reduce to line singularities. Because the number of vortices is large it seems feasible to model their evolution by an averaged problem, known as the mean-held model of superconductivity. We assume that the evolution law of an individual vortex, which underlies the averaging process, involves the current of the generated magnetic field as well as the curvature vector. In the present paper we study a two-dimensional reduction, assuming all vortices to be perpendicular to a given direction. Since both the magnetic field H and the averaged vorticity omega are curl-free, we may represent them via a scalar magnetic potential q and a scalar stream function psi, respectively. We study existence, uniqueness and asymptotic behaviour of solutions (psi, q) of the resulting degenerate elliptic-parabolic system (with curvature taken into account or not) by means of viscosity and weak solutions. In addition we relate (psi, q) to solutions (omega, H) of the mean-field equations without curvature. Finally we construct special solutions of the corresponding stationary equations with two or more superconducting phases.