Global solution to a phase field model with irreversible and constrained phase evolution

Abstract

Abstract.This note deals with a nonlinear system of PDEs describing some irreversible phase change phenomena that account for a bounded limit velocity of the phase transition process. An existence result is established by using time discretization, compactness arguments, and techniques of subdifferential operators. Introduction.The present analysis is concerned with a nonlinear system of partial differential equations describing irreversible phase change phenomena with bounded velocity of phase transition.Such a system comes from a recent model, proposed by Fremond, which relies on the consideration that the movements of the microscopic particles may affect the macroscopic phase transition process as well. In particular, our system is to be regarded in the more general framework of phase field models. The literature concerning this kind of model is rather wide and the reader is referred, e.g., to [6],

Well-posedness results and asymptotic behaviour for a phase transition model taking into account microscopic accelerations

“…The paper [15] proves the existence of a solution to the system (1.1)-(1.2) in the special case where a finite maximum speed λ > 0 is imposed to the phase transition process. As for the analytical device, (1.3) is replaced by…”

“…Finally, the result of [15] is used to show that the full problem turns out to have a global strong solution in the one-dimensional setting [12] and a local (in time) one in the threedimensional case [16]. Now, it must be noted that (1.2) is derived by means of the virtual power principle, neglecting the power of the acceleration forces.…”

Well-posedness results for a model of damage in thermoviscoelastic materials

“…where, in particular, the monotonicity of β yields for a.a. t (the notation is formal) (see [15,Lemma 4.1], for a rigorous justification). Then, we estimate the right-hand side of (3.16) as follows …”

Well-posedness and long-time behavior for a class of doubly nonlinear equations

“…Equations of the type of (1.1) stem in connection with phase change phenomena [14,17,23,32,46,47], gas flow through porous media [68], damage [15,16,30,31,54], and, in the specific case α(λr) = α(r) for all λ > 0 and r ∈ R (which is however not included in the present analysis), elastoplasticity [25,50,51,52], brittle fractures [26], ferroelectricity [56], and general rate-independent systems [29,48,49,53,55].…”