Global well-posedness for the two-dimensional Muskat problem with slope less than 1

Cameron, Stephen

doi:10.2140/apde.2019.12.997

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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 $C^{1, ϵ}$ . 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

Vol. 12, No. 4, 2019