## 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

Bogdan-Vasile Matioc

#### 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 $H s$, $s ∈ ( 3 2 , 2 )$, when neglecting surface tension, respectively in $H s$, $s ∈ ( 2 , 3 )$, when surface-tension effects are included. Besides, we provide new criteria for the global existence of solutions.

#### Article information

Source
Anal. PDE, Volume 12, Number 2 (2019), 281-332.

Dates
Revised: 17 January 2018
Accepted: 7 May 2018
First available in Project Euclid: 9 October 2018

https://projecteuclid.org/euclid.apde/1539050439

Digital Object Identifier
doi:10.2140/apde.2019.12.281

Mathematical Reviews number (MathSciNet)
MR3861893

Zentralblatt MATH identifier
06974515

#### Citation

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

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