On the Thin Film Muskat and the Thin Film Stokes Equations
The present paper is concerned with the analysis of two strongly coupled systems of degenerate parabolic partial differential equations arising in multiphase thin film flows. In particular, we consider the two-phase thin film Muskat problem and the two-phase thin film approximation of the Stokes flow under the influence of both, capillary and gravitational forces. The existence of global weak solutions for medium size initial data in large function spaces is proved. Moreover, exponential decay results towards the equilibrium state are established, where the decay rate can be estimated by explicit constants depending on the physical parameters of the system. Eventually, it is shown that if the initial datum satisfies additional (low order) Sobolev regularity, we can propagate Sobolev regularity for the corresponding solution. The proofs are based on a priori energy estimates in Wiener and Sobolev spaces.
KeywordsMuskat problem moving interfaces two-phase thin film approximation free-boundary problems stokes flow
Mathematics Subject Classification35K25 35D30 35R35 35Q35 76B03
RGB was partially supported by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program “Investissements d’Avenir” (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR). GB recognizes the support of Grant No. 250070 of the Research Council of Norway. Part of the research leading to results presented here was conducted during a short stay of GB at Institut Camille Jordan under Projects DYFICOLTI and ANR-13-BS01-0003-01 support.
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Conflict of interest
The authors declares that there is no conflict of interest.
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