Vol. 12, No. 2, 2019

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The Muskat problem in two dimensions: equivalence of formulations, well-posedness, and regularity results

Bogdan-Vasile Matioc

Vol. 12 (2019), No. 2, 281–332
DOI: 10.2140/apde.2019.12.281
Abstract

We consider the Muskat problem describing the motion of two unbounded immiscible fluid layers with equal viscosities in vertical or horizontal two-dimensional geometries. We first prove that the mathematical model can be formulated as an evolution problem for the sharp interface separating the two fluids, which turns out to be, in a suitable functional-analytic setting, quasilinear and of parabolic type. Based upon these properties, we then establish the local well-posedness of the problem for arbitrary large initial data and show that the solutions become instantly real-analytic in time and space. Our method allows us to choose the initial data in the class Hs , s (3 2,2), when neglecting surface tension, respectively in Hs , s (2,3), when surface-tension effects are included. Besides, we provide new criteria for the global existence of solutions.

Keywords
Muskat problem, surface tension, singular integral
Mathematical Subject Classification 2010
Primary: 35R37, 35K59, 35K93, 35Q35, 42B20
Milestones
Received: 18 October 2016
Revised: 17 January 2018
Accepted: 7 May 2018
Published: 14 August 2018
Authors
Bogdan-Vasile Matioc
Fakultät für Mathematik
Universität Regensburg
Regensburg
Germany