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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Compactness properties for families of quasistationary solutions of some evolution equations

Author: Giuseppe Savaré
Journal: Trans. Amer. Math. Soc. 354 (2002), 3703-3722
MSC (2000): Primary 47J25, 80A22; Secondary 37L05, 47J35, 49J45
Published electronically: May 7, 2002
MathSciNet review: 1911517
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Abstract: The following typical problem occurs in passing to the limit in some phase field models: for two sequences of space-time dependent functions $\{\theta_n\}, \{{\raise.3ex\hbox{$\chi$ }}_n\}$ (representing, e.g., suitable approximations of the temperature and the phase variable) we know that the sum $\theta_n + {\raise.3ex\hbox{$\chi$ }}_n$ converges in some $L^p$-space as $n\uparrow+\infty$ and that the time integrals of a suitable ``space'' functional evaluated on $\theta_n, {\raise.3ex\hbox{$\chi$ }}_n$ are uniformly bounded with respect to $n$. Can we deduce that $\theta_n$ and ${\raise.3ex\hbox{$\chi$ }}_n$ converge separately? LUCKHAUS (1990) gave a positive answer to this question in the framework of the two-phase Stefan problem with Gibbs-Thompson law for the melting temperature. PLOTNIKOV (1993) proposed an abstract result employing the original idea of Luckhaus and arguments of compactness and reflexivity type. We present a general setting for this and other related problems, providing necessary and sufficient conditions for their solvability: these conditions rely on general topological and coercivity properties of the functionals and the norms involved, and do not require reflexivity.

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Giuseppe Savaré
Affiliation: Dipartimento di Matematica “F. Casorati”, Università di Pavia. Via Ferrata, 1. I-27100 Pavia, Italy

Keywords: Quasistationary solutions of evolution equations, phase-field models, Stefan problem with Gibbs-Thomson law, compactness methods
Received by editor(s): October 3, 2001
Published electronically: May 7, 2002
Additional Notes: This work was partially supported by the M.U.R.S.T. (Italy) through national research project funds, and by the Institute of Numerical Analysis of the C.N.R., Pavia, Italy
Article copyright: © Copyright 2002 American Mathematical Society