Abstract Approach to Evolution Equations of Phase-Field Type and Applications

A general system of abstract nonlinear parabolic equations deriving from phase-field models of heat transfer in anisotropic fluids with phase transitions is studied through a backward finite differences approximation scheme and existence and uniqueness theorems are proved. Then, applications are presented to two physical situations: first, a standard heat diffusion problem is briefly discussed mainly as a justification of the choices of the abstract setting. Second, a more interesting transmission problem between two different adjoining fluids is studied under very general compatibility hypotheses and the abstract results are adapted to provide an existence and uniqueness theorem also in this case. Moreover, through convex analysis techniques, a mathematical study is performed of the subdifferential-type relations linking the phase fields and the temperatures to the internal energies of the fluids.