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

Matioc, Bogdan-Vasile

doi:10.2140/apde.2019.12.281

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN 1948-206X (online)

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 $H^{s}$ , $s \in (\frac{3}{2}, 2)$ , when neglecting surface tension, respectively in $H^{s}$ , $s \in (2, 3)$ , 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

Vol. 12, No. 2, 2019