Global solution to the Penrose-Fife phase-field model with zero interfacial energy and Fourier Law
- Colli, Pierluigi
- Sprekels, Jürgen
2010 Mathematics Subject Classification
- 35K50 47H20 80A22 80A20
- Penrose-Fife model, Fourier law, phase transitions, phase-field models, phase relaxation systems, initial-boundary value problems, maximum principle, existence results
In this paper we study a system of field equations of Penrose-Fife type governing the dynamics of phase transitions with a nonconserved order parameter. In many recent contributions on this subject, the heat flux law has been assumed in the form q = ∇ (1/θ). In contrast to that, here we consider the (more realistic) case of the Fourier law when q is proportional to the negative gradient -∇θ of the (absolute) temperature θ. The assumption of Fourier heat conduction presents particular difficulties in the framework of the Penrose-Fife model, since then the field equation representing the balance of internal energy does not seem to have a maximum principle from which the positivity of θ could be derived. In this connection, we recall that the main difficulty in proving existence for phase-field systems of Penrose-Fife type is the proof of the positivity of θ. It is shown in this paper that in the case without interfacial energy, that is, when the free energy does not contain a quadratic gradient term of the order parameter, there exists a comparatively easy way to conclude the positivity of θ under rather weak and quite natural conditions on the data of the system. Having established this result, the existence of a weak solution is readily obtained using known results on general phase-field systems.
- Adv. Math. Sci. Appl., 9 (1999), no. 1, pp. 383-391