Analysis & PDE
- Anal. PDE
- Volume 12, Number 2 (2019), 281-332.
The Muskat problem in two dimensions: equivalence of formulations, well-posedness, and regularity results
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 , , when neglecting surface tension, respectively in , , when surface-tension effects are included. Besides, we provide new criteria for the global existence of solutions.
Anal. PDE, Volume 12, Number 2 (2019), 281-332.
Received: 18 October 2016
Revised: 17 January 2018
Accepted: 7 May 2018
First available in Project Euclid: 9 October 2018
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 35R37: Moving boundary problems 35K59: Quasilinear parabolic equations 35K93: Quasilinear parabolic equations with mean curvature operator 35Q35: PDEs in connection with fluid mechanics 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)
Matioc, Bogdan-Vasile. The Muskat problem in two dimensions: equivalence of formulations, well-posedness, and regularity results. Anal. PDE 12 (2019), no. 2, 281--332. doi:10.2140/apde.2019.12.281. https://projecteuclid.org/euclid.apde/1539050439