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 ${theta_n}, {chi_n}$ (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 $nup+infty$ and that the time integrals of a suitable ``space\u27\u27 functional evaluated on $theta_n, chi_n$ are uniformly bounded with respect to $n$. Can we deduce that $theta_n$ and $nchi_n$ 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