Weak Solutions of a Phase-Field Model for Phase Change of an Alloy with Thermal Properties

The phase-field method provides an alternative mathematical description for free-boundary problems corresponding to physical process with phase transitions. It postulates the existence of a function, called the phase-field, whose value identifies the phase at a particular point in space and time, and it is particularly suitable for cases with complex growth strutures occuring during phase transitions. The mathematical model studied in this work describes the solidification process occurring in a binary alloy with temperature dependent properties. It is based on a highly nonlinear degenerate parabolic system of partial differential equations with three independent variables: phase-field, solute concentration and temperature. Existence of weak solutions of such system is obtained via the introduction of a regularized problem, followed by the derivation of suitable estimates and the application of compactness arguments.