Pierluigi Colli

Weak solutions to the Penrose-Fife phase field model for a class of admissible heat flux laws

Abstract This paper is concerned with a thermodynamically consistent model for diffusive phase transitions proposed by Penrose and Fife. The model has recently received a good deal of attention, and the related initial-boundary value problems have been investigated from the mathematical point of view. In the case where the order parameter is not conserved, the common working assumption for the heat flow was that such a flux is proportional to the gradient of the inverse absolute temperature. This position seems to be helpful for the analysis, due to the coupling term in the phase-field equation which depends right on the inverse temperature. Here we prove the existence of weak solutions in a wide setting of constitutive laws for the heat flux, different from the previous ones and considerably more significant owing to their behaviour for high temperatures. We also discuss the limiting situation of zero interfacial energy for the order parameter.

On a Penrose-Fife model with zero interfacial energy leading to a phase-field system of relaxed Stefan type

In this paper we study an initial-boundary value Stefan-type problem with phase relaxation where the heat flux is proportional to the gradient of the inverse absolute temperature. This problem arise naturally as limiting case of the Penrose-Fife model for diffusive phase transitions with nonconserved order parameter if the coefficient of the interfacial energy is taken as zero. It is shown that the relaxed Stefan problem admits a weak solution which is obtained as limit of solutions to the Penrose-Fife phase-field equations. For a special boundary condition involving the heat exchange with the surrounding medium, also uniqueness of the solution is proved.

Weak solution to hyperbolic Stefan problems with memory

Abstract. A model for Stefan problems in materials with memory is considered. This model is mainly characterized by a nonlinear Volterra integrodifferential equation of hyperbolic type. Colli and Grasselli proved the uniqueness of a weak solution under the natural assumptions on data and the existence of a strong solution for smoother data. Taking advantage of these two results and assuming just the hypotheses ensuring uniqueness, the existence of a weak solution is here shown.

Convergence of phase field to phase relaxation models with memory

This paper is concerned with phase field models accounting for memory effects and based on the linearized Gurtin-Pipkin constitutive assumption for the heat flux. After recalling and generalizing existence, uniqueness, and regularity results recently obtained by the authors, here the asymptotic behaviour of the solutions to related initial-boundary value problems is investigated as the coefficient of the interfacial energy tends to zero. It is found that the limits of suitable subsequences solve the induced hyperbolic phase relaxation problem, for which the question of uniqueness is still open.