On a free boundary problem describing the phase transition in an incompressible viscous fluid

Abstract

Ice melts at $0^{◦}\mathrm{C}$ under a pressure of $1$ atm, and increasing the pressure decreases the melting temperature. In the present paper, a new problem is posed that describes the process of phase transition in an incompressible viscous fluid, taking into account the above-described pressure effect. This problem is described as a free boundary problem in terms of the Navier–Stokes equations coupled with the heat equation, where the equilibrium temperature is assumed to be related to the pressure by the Clapeyron–Clausius equation. We prove the existence of a global-in-time solution.