Abstract
Taking account of a microscopical model for dynamical supercooling and superheating effects, the usual equilibrium condition prescribing a fixed temperature at the interface between two phases is replaced by relaxation dynamics for the phase variable x, representing the concentration of one of the two phases. At first the approach of “non-equilibrium thermodynamics” is followed and a parabolic system is formulated; existence, uniqueness and regularity properties of the L2-solution are obtained by means of the theory of non-linear semigroups of contractions. These developments are also generalized to phase transitions in heterogeneous systems. Then the heat diffusion equation is coupled with the relaxation dynamics for x and the well-posedness of an initial- and boundary-value problem is proved. The standard Stefan problem is obtained as a limit case. Also, a model for glass formation by very fast cooling is proposed. Finally, Fourier's conduction law is replaced by relaxation dynamics for the heat flux; this corresponds to assuming wave propagation for the heat. An existence result is proved for the corresponding problem and the limit behaviour as the relaxation time vanishes is studied.
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