#### Vol. 12, No. 2, 2019

 Recent Issues
 The Journal About the Journal Editorial Board Editors’ Interests Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN: 1948-206X (e-only) ISSN: 2157-5045 (print) Author Index To Appear Other MSP Journals
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 ${H}^{s}$, $s\in \left(\frac{3}{2},2\right)$, when neglecting surface tension, respectively in ${H}^{s}$, $s\in \left(2,3\right)$, 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