This work is concerned with the study of an initial boundary value problem for a non-conserved phase field system arising from the Penrose-Fife approach to the kinetics of phase transitions. The system couples a nonlinear parabolic equation for the absolute temperature with a nonlinear hyperbolic equation for the phase variable χ, which is characterized by the presence of an inertial term multiplied by a small positive coefficient µ. This feature is the main consequence of supposing that the response of χ to the generalized force (which is the functional derivative of a free energy potential and arises as a consequence of the tendency of the free energy to decay towards a minimum) is subject to delay. We first obtain well-posedness for the resulting initial-boundary value problem in which the heat flux law contains a special function of the absolute temperature ϑ, i.e. α(ϑ) ∼ ϑ − 1/ϑ. Then we prove convergence of any family of weak solutions of the parabolic-hyperbolic model to a weak solution of the standard Penrose-Fife model as µ ↘ 0. However, the main novelty of this paper consists in proving some regularity results on solutions of the parabolic-hyperbolic system (including also estimates of Moser type) that could be useful for the study of the longterm dynamics.
|Titolo:||Well-posedness and regularity for a parabolic-hyperbolic Penrose-Fife phase field system|
ROCCA, ELISABETTA (Primo)
|Parole Chiave:||Penrose-Fife model - hyperbolic equation - continuous dependence - regularity|
|Settore Scientifico Disciplinare:||Settore MAT/05 - Analisi Matematica|
|Data di pubblicazione:||2005|
|Digital Object Identifier (DOI):||10.1007/s10492-005-0031-1|
|Appare nelle tipologie:||01 - Articolo su periodico|