#### Vol. 12, No. 4, 2019

 Recent Issues
 The Journal About the Journal Editorial Board Subscriptions Editors’ Interests Scientific Advantages Submission Guidelines Submission Form Editorial Login Ethics Statement ISSN: 1948-206X (e-only) ISSN: 2157-5045 (print) Author Index To Appear Other MSP Journals
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