A free boundary problem of type-I superconductivity

In this paper we consider a free boundary problem of superconductivity. Under isothermal conditions, a superconductor material of Type I will develop two phases separated by a sharp interface Γ(t). In the normal conducting phase the magnetic field is divergence free and satisfies the heat equation, whereas on the interface Γ(t), curl is the normal of Γ(t) andV _n is the velocity of Γ(t) in the direction of (constant) on Γ(t). Here our result consists of two parts: the first part is for the fixed boundary problem in 3-dimensional case with curl boundary condition, which has a unique global classical solution; the second part is for the free boundary problem in 2-dimensional case, a unique classical solution locally in time is established by Newton’s iteration method.