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Convergence of phase field to phase relaxation models governed by an entropy equation with memory
Author(s) -
Gilardi Gianni,
Rocca Elisabetta
Publication year - 2006
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.765
Subject(s) - mathematics , preprint , logarithm , quadratic equation , relaxation (psychology) , convergence (economics) , limit (mathematics) , entropy (arrow of time) , mathematical analysis , geometry , quantum mechanics , physics , psychology , social psychology , economics , economic growth
The subject of the present paper consists in proving the convergence of a phase‐field model, based on the entropy equation with memory, to phase relaxation. The well‐posedness and the long‐time behaviour of solutions for the non‐linear and singular phase‐field system have been recently shown by Bonetti et al. (Preprint IMATI‐CNR, 2005; Discrete Contin. Dyn. Syst. Ser. B , in press). Here, we study the asymptotic behaviour of such solutions as the interfacial energy coefficient tends to zero. The limit problem is a phase relaxation problem with memory, which is new. We prove well‐posedness results through convergence under rather general assumptions. However, the case of a quadratic non‐linearity for the latent heat is excluded. Such a situation is dealt for the problem without memory in a generalized setting by introducing an ad hoc logarithm. Copyright © 2006 John Wiley & Sons, Ltd.