The Ginzburg-Landau equations of superconductivity and the one-phase Stefan problem

We study the time-dependent Ginzburg-Landau model for type I superconductivity in a cylindrically symmetric setting. We show that under appropriate monotonicity properties for the initial data, the singular limit (as the penetration depth tends to zero and the Ginzburg-Landau parameter is kept fixed) is a classical one-phase Stefan problem for the magnetic field H. We combine energy methods with monotonicity properties obtained via maximum principles.