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 (representing, e.g., suitable approximations of the temperature and the phase variable) we know that their sum converges in some Lp -space and that they satisfy a suitable tigtness condition. Can we deduce that the sequences 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.