Well-posedness and regularity for a parabolic-hyperbolic Penrose-Fife phase field system

Elisabetta Rocca

Applications of Mathematics (2005)

  • Volume: 50, Issue: 5, page 415-450
  • ISSN: 0862-7940

Abstract

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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.

How to cite

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Rocca, Elisabetta. "Well-posedness and regularity for a parabolic-hyperbolic Penrose-Fife phase field system." Applications of Mathematics 50.5 (2005): 415-450. <http://eudml.org/doc/33231>.

@article{Rocca2005,
abstract = {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 $\chi $, which is characterized by the presence of an inertial term multiplied by a small positive coefficient $\mu $. This feature is the main consequence of supposing that the response of $\chi $ 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 $\vartheta $, i.e. $\alpha (\vartheta )\sim \vartheta -1/\vartheta $. 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 $\mu \searrow 0$. 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.},
author = {Rocca, Elisabetta},
journal = {Applications of Mathematics},
keywords = {Penrose-Fife model; hyperbolic equation; continuous dependence; regularity; Penrose-Fife model; hyperbolic equation; continuous dependence; regularity; weak solutions; initial boundary value problem; well-posedness},
language = {eng},
number = {5},
pages = {415-450},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Well-posedness and regularity for a parabolic-hyperbolic Penrose-Fife phase field system},
url = {http://eudml.org/doc/33231},
volume = {50},
year = {2005},
}

TY - JOUR
AU - Rocca, Elisabetta
TI - Well-posedness and regularity for a parabolic-hyperbolic Penrose-Fife phase field system
JO - Applications of Mathematics
PY - 2005
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 50
IS - 5
SP - 415
EP - 450
AB - 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 $\chi $, which is characterized by the presence of an inertial term multiplied by a small positive coefficient $\mu $. This feature is the main consequence of supposing that the response of $\chi $ 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 $\vartheta $, i.e. $\alpha (\vartheta )\sim \vartheta -1/\vartheta $. 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 $\mu \searrow 0$. 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.
LA - eng
KW - Penrose-Fife model; hyperbolic equation; continuous dependence; regularity; Penrose-Fife model; hyperbolic equation; continuous dependence; regularity; weak solutions; initial boundary value problem; well-posedness
UR - http://eudml.org/doc/33231
ER -

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