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We present a result on existence of solutions for a system of highly nonlinear and singular partial differential equations obtained by coupling the two-crystallization Allen-Cahn equations to a singular Navier-Stokes system and a nonlinear heat equation. Such a system constitutes a phase field model for non-isothermal solidification/melting processes of certain metallic alloys for which two different kinds of crystallization are possible. In this model, the liquid phase and each one of the possible crystallization states are described by their own phase fields. The possibility of occurrence of fluid flow in a a priori unknown non-solid region is also considered, turning the model into a free-boundary value problem.

On a system coupling two-crystallization Allen-Cahn equations and a singular Navier-Stokes system