Vol. 12, No. 4, 2019

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Global well-posedness for the two-dimensional Muskat problem with slope less than 1

Stephen Cameron

Vol. 12 (2019), No. 4, 997–1022
DOI: 10.2140/apde.2019.12.997
Abstract

We prove the existence of global, smooth solutions to the two-dimensional Muskat problem in the stable regime whenever the product of the maximal and minimal slope is less than 1. The curvature of these solutions decays to 0 as t goes to infinity, and they are unique when the initial data is C1,ϵ . We do this by getting a priori estimates using a nonlinear maximum principle first introduced in a paper by Kiselev, Nazarov, and Volberg (2007), where the authors proved global well-posedness for the surface quasigeostraphic equation.

Keywords
Muskat problem, porous media, fluid interface, global well-posedness
Mathematical Subject Classification 2010
Primary: 35K55, 35Q35, 35R09
Milestones
Received: 9 May 2017
Revised: 14 January 2018
Accepted: 30 July 2018
Published: 20 October 2018
Authors
Stephen Cameron
Department of Mathematics
University of Chicago
Chicago, IL
United States