Ginzburg - Landau vortex dynamics driven by an applied boundary current

In this paper we study the time-dependent Ginzburg-Landau equations on a smooth, bounded domain Ω ⊂ R2, subject to both an applied magnetic field and an applied boundary current. We model the boundary current by a gauge invariant inhomogeneous Neumann boundary condition. After proving well-posedness of the equations with this boundary condition, we study the evolution of the energy of the solutions, deriving an upper bound for the energy growth. We then turn to the study of the dynamics of the vortices of the solutions in the limit ε → 0. We first consider the original time scale, in which the vortices do not move and the solutions undergo a “phase relaxation.” Then we study an accelerated time scale in which the vortices move according to a derived dynamical law. In the dynamical law, we identify a novel Lorentz force term induced by the applied boundary current.