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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The following typical problem occurs in passing to the limit in some phase field models: for two sequences of space-time dependent functions (representing, e.g., suitable approximations of the temperature and the phase variable) we know that the sum converges in some -space as and that the time integrals of a suitable ``space'' functional evaluated on are uniformly bounded with respect to . Can we deduce that and 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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Additional Information

**Giuseppe Savaré**

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

Email:
savare@ian.pv.cnr.it

DOI:
http://dx.doi.org/10.1090/S0002-9947-02-03035-0

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