Global weak solutions to a phase-field model for seawater solidification with melt convection

This paper is concerned with the study of mathematical analysis of a phase-field model of solidification with the possibility of flow occurring in non-solidified regions of the pure seawater. The governing equations of the model are a convective phase-field equation coupled with a nonlinear heat equation and a modified incompressible Navier–Stokes system. The phase-field equation describes phase transitions in solid–liquid phases, where an order parameter is used to distinguish the phases. The Navier–Stokes system is modelled by the Boussinesq approximation and a Carman–Koseny term. Due to the presence of the Carman–Koseny term, the Navier–Stokes system only holds in the non-solid regions, which are a priori unknown. Thus, it becomes a moving boundary-value problem. First, we regularize our problem by introducing a parameter \(\delta >0\) $\delta >0$ , and obtain a sequence of approximate solutions for this regularized problem by using the semi-Galerkin method. Then, we use compactness arguments to pass to the limit in the approximate solutions, and obtain a weak solution to the original problem in the two-dimensional case.