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Existence of solutions to a phase transition model with microscopic movements
Author(s) -
Feireisl Eduard,
Petzeltová Hana,
Rocca Elisabetta
Publication year - 2008
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.1089
Subject(s) - mathematics , a priori and a posteriori , limit (mathematics) , phase transition , entropy production , stability (learning theory) , entropy (arrow of time) , property (philosophy) , mathematical analysis , statistical physics , thermodynamics , physics , philosophy , epistemology , machine learning , computer science
We prove the existence of weak solutions for a 3D phase change model introduced by Michel Frémond in ( Non‐smooth Thermomechanics . Springer: Berlin, 2002) showing, via a priori estimates, the weak sequential stability property in the sense already used by the first author in ( Comput. Math. Appl. 2007; 53 :461–490). The result follows by passing to the limit in an approximate problem obtained adding a superlinear part (in terms of the gradient of the temperature) in the heat flux law. We first prove well posedness for this last problem and then—using proper a priori estimates—we pass to the limit showing that the total energy is conserved during the evolution process and proving the non‐negativity of the entropy production rate in a suitable sense. Finally, these weak solutions turn out to be the classical solution to the original Frémond's model provided all quantities in question are smooth enough. Copyright © 2008 John Wiley & Sons, Ltd.