Thin film approximations to the Muskat problem. (English) Zbl 1328.35330

Dogbe, Christian (ed.), Actes du colloque “EDP-Normandie”, Le Havre, France, Octobre 23–24, 2012. [s.l.]: Fédération Normandie-Mathématiques (ISBN 978-2-9541221-1-3/pbk). Normandie-Mathématiques, 83-91 (2013).
Summary: Existence of nonnegative weak solutions is shown for thin film approximations of the Muskat problem which includes either gravity forces or capillary forces or both. The model describes the space-time evolution of the heights of the two fluid layers and is a fully coupled system of two fourth-order degenerate parabolic equations in either \(\mathbb{R}^d\) or a bounded domain of \(\mathbb{R}^d\), \(d= 1,2\).
In the latter case, it is supplemented with no-flux boundary conditions. Two approaches are used to show the existence of solutions: either the original system is approximated by a non-degenerate parabolic system and a solution is constructed by a compactness method. Or, observing that the original system can be viewed as a gradient flow for the 2-Wasserstein distance in the space of probability measures with finite second moment, existence of a solution follows from a variational approach.
For the entire collection see [Zbl 1296.35006].


35R35 Free boundary problems for PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence