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

Applications of Mathematics (2005)

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

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topRocca, 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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