Time compactness tools for discretized evolution equations and applications to degenerate parabolic PDEs
Résumé
We discuss several techniques for proving compactness of sequences of approximate solutions to discretized evolution PDEs, with applications to convergence of finite volume discretizations of degenerate parabolic equations. While the well-known Aubin-Simon kind functional-analytic techniques were recently generalized to the discrete setting by Gallouët and Latché [12], here we discuss direct techniques for estimating the time translates of approximate solutions in the space $L^1$. One important result is the Kruzhkov time compactness lemma. Further, we describe a specific technique that benefits from the order-preservation for the underlying PDE, and recall the well-known methods based on nonlinear weak-* convergence and on the subsequent reduction of Young measures.
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