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 $\{\theta_n\}, \{{\raise.3ex\hbox{$\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.