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Compactness properties for families of quasistationary solutions of some evolution equations

The following typical problem occurs in passing to the limit in some phase field models: for two sequences of space-time dependent functions {θ n }, {X n } (representing, e.g., suitable approximations of the temperature and the phase variable) we know that the sum θ n + X n converges in some L P -space as n ↑ +∞ and that the time integrals of a suitable space functional evaluated on θ n ,X n are uniformly bounded with respect to n. Can we deduce that θ n and X 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.