Matioc, Anca-Voichita and Matioc, Bogdan-Vasile (2019) Well-posedness and stability results for a quasilinear periodic Muskat problem. JOURNAL OF DIFFERENTIAL EQUATIONS, 266 (9). pp. 5500-5531. ISSN 0022-0396, 1090-2732
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We study the Muskat problem describing the spatially periodic motion of two fluids with equal viscosities under the effect of gravity in a vertical unbounded two-dimensional geometry. We first prove that the classical formulation of the problem is equivalent to a nonlocal and nonlinear evolution equation expressed in terms of singular integrals and having only the interface between the fluids as unknown. Secondly, we show that this evolution equation has a quasilinear structure, which is at a formal level not obvious, and we also disclose the parabolic character of the equation. Exploiting these aspects, we establish the local well-posedness of the problem for arbitrary initial data in H-s (S), with s is an element of (3/2, 2), determine a new criterion for the global existence of solutions, and uncover a parabolic smoothing property. Besides, we prove that the zero steady-state solution is exponentially stable. (C) 2018 Elsevier Inc. All rights reserved.
| Item Type: | Article |
|---|---|
| Uncontrolled Keywords: | GLOBAL EXISTENCE; POROUS-MEDIA; TURNING WAVES; HELE-SHAW; INTERFACE; WATER; PARABOLICITY; REGULARITY; BREAKDOWN; EVOLUTION; Muskat problem; Singular integral; Well-posedness; Parabolic smoothing; Stability |
| Subjects: | 500 Science > 510 Mathematics |
| Divisions: | Mathematics |
| Depositing User: | Dr. Gernot Deinzer |
| Date Deposited: | 15 Apr 2020 08:51 |
| Last Modified: | 15 Apr 2020 08:51 |
| URI: | https://pred.uni-regensburg.de/id/eprint/27170 |
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