Supercooling and superheating effects in heterogeneous systems
In the model for phase transitions in binary systems based on Fourier's and Fick's laws, the interface equilibrium condition 6 = w, relating the temperature 8 and the chemical activity w, is here replaced by a relaxation dynamics for the liquid concentration -37 + ,w) (H 1: inverse of the Heaviside graph); here fi e C°(R2) and sign fi(6,w) = sign(# — w). In the case of a single dimension of space, with an interface jc = s(t), a different dynamics can be considered: s'(t) = fi(6{s(t),t),w(s(t),t)). Each of these laws is coupled with the energy conservation equation. Existence of at least one solution and maximum and minimum principles are proved for both problems. These results can be generalized in many ways, in particular, for temperature, concentration and phase-dependent coefficients. For the problem in a single space dimension, existence of a solution is proved also for the case of vanishing mass diffusivity in the solid. Uniqueness of the solution is shown for the problem in several dimensions of space.