Weak solutions to a multi-phase field system of parabolic equations related to alloy solidification

Existence of weak solutions to a phase field model for solidification of alloys is studied. The model consists of balance equations for the energy and the concentrations of the alloy components which are coupled to a system of Allen-Cahn equations describing the motion of phase and grain boundaries. The system is stated in terms of thermodynamic potentials corresponding to (inverse) temperature and chemical potentials divided by the temperature and phase field variables describing the presence of the possible phases. The fields of the conserved quantities are functions of these variables, and difficulties arise from the growth properties. The existence proof is based on a perturbation method. The differential equations are solved for functions with nicer growth properties. After, appropriate estimates are derived in order to let the perturbation vanish