Matioc, Bogdan-Vasile (2020) Well-Posedness and Stability Results for Some Periodic Muskat Problems. JOURNAL OF MATHEMATICAL FLUID MECHANICS, 22 (3): 31. ISSN 1422-6928, 1422-6952
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We study the two-dimensional Muskat problem in a horizontally periodic setting and for fluids with arbitrary densities and viscosities. We show that in the presence of surface tension effects the Muskat problem is a quasilinear parabolic problem which is well-posed in the Sobolev space H-r(S) for each r is an element of(2,3). When neglecting surface tension effects, the Muskat problem is a fully nonlinear evolution equation and of parabolic type in the regime where the Rayleigh-Taylor condition is satisfied. We then establish the well-posedness of the Muskat problem in the open subset of H-2(S) defined by the Rayleigh-Taylor condition. Besides, we identify all equilibrium solutions and study the stability properties of trivial and of small finger-shaped equilibria. Also other qualitative properties of solutions such as parabolic smoothing, blow-up behavior, and criteria for global existence are outlined.
| Item Type: | Article |
|---|---|
| Uncontrolled Keywords: | HELE-SHAW; GLOBAL EXISTENCE; POROUS-MEDIA; TURNING WAVES; INTERFACE; EVOLUTION; WATER; PARABOLICITY; REGULARITY; EQUATIONS; Muskat problem; Singular integral; Well-posedness; Parabolic smoothing; Stability |
| Subjects: | 500 Science > 510 Mathematics |
| Divisions: | Mathematics |
| Depositing User: | Dr. Gernot Deinzer |
| Date Deposited: | 22 Mar 2021 07:18 |
| Last Modified: | 22 Mar 2021 07:18 |
| URI: | https://pred.uni-regensburg.de/id/eprint/44411 |
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