Analysis of a two-phase field model for the solidification of an alloy

In this paper we present some theoretical results for a system of nonlinear partial differential equations that provide a phase field model for the solidification/melting of a metallic alloy. It is assumed that two different kinds of crystallization are possible. Consequently, the unknowns are the temperature τ and the phase field functions u and v. The time derivatives u t and v t appear in the equation for τ (the heat equation). On the other hand, the equations for u and v contain nonlinear terms where we find τ.