Compactness properties for families of quasistationary solutions of some evolution equations

Savaré, Giuseppe

doi:10.1090/S0002-9947-02-03035-0

Compactness properties for families of quasistationary solutions of some evolution equations
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by Giuseppe Savaré

Trans. Amer. Math. Soc. 354 (2002), 3703-3722

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

Published electronically: May 7, 2002

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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 $\{\theta _n\}$, $\{\chi _n\}$ (representing, e.g., suitable approximations of the temperature and the phase variable) we know that the sum $\theta _n + \chi _n$ converges in some $L^p$-space as $n\uparrow +\infty$ and that the time integrals of a suitable “space” functional evaluated on $\theta _n$, $\chi _n$ are uniformly bounded with respect to $n$. Can we deduce that $\theta _n$ and $\chi _n$ 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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