Istituto di Matematica Applicata e Tecnologie Informatiche     
Savare' G. Compactness Properties for Families of Quasistationary Solutions of some Evolution Equations. Preprint ercim.cnr.ian//1999-1143, 1999.
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'' 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.
Subject Quasistationary solutions of evolution equations
Phase field models
Compactness methods

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