Boldrini, José Luiz; Dias Vaz, Cristina Lúcia Existence and regularity of solutions of a phase field model for solidification with convection of pure materials in two dimensions. (English) Zbl 1034.76057 Electron. J. Differ. Equ. 2003, Paper No. 109, 25 p. (2003). Summary: We study existence and regularity of weak solutions of a phase field type model for pure material solidification in the presence of natural convection. We assume that a non-stationary solidification occurs in a two-dimensional bounded domain. The governing equations are the phase field equation coupled with a nonlinear heat equation and modified Navier-Stokes equations. These equations include buoyancy forces modelled by Boussinesq approximation, and a Carman-Koseny term to model the flow in mushy regions. Since these modified Navier-Stokes equations hold only in the non-solid regions, which are not known a priori, we have a free boundary problem. Cited in 7 Documents MSC: 76T99 Multiphase and multicomponent flows 80A22 Stefan problems, phase changes, etc. 76R10 Free convection 76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids 35Q30 Navier-Stokes equations 35R35 Free boundary problems for PDEs Keywords:weak solutions; phase transition; natural convection; non-stationary solidification; nonlinear heat equation; modified Navier-Stokes equations; buoyancy; Boussinesq approximation; Carman-Koseny term; mushy regions; free boundary problem PDFBibTeX XMLCite \textit{J. L. Boldrini} and \textit{C. L. Dias Vaz}, Electron. J. Differ. Equ. 2003, Paper No. 109, 25 p. (2003; Zbl 1034.76057) Full Text: EuDML EMIS