## Analysis & PDE

• Anal. PDE
• Volume 12, Number 4 (2019), 997-1022.

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

Stephen Cameron

#### 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.

#### Article information

Source
Anal. PDE, Volume 12, Number 4 (2019), 997-1022.

Dates
Revised: 14 January 2018
Accepted: 30 July 2018
First available in Project Euclid: 30 October 2018

https://projecteuclid.org/euclid.apde/1540864858

Digital Object Identifier
doi:10.2140/apde.2019.12.997

Mathematical Reviews number (MathSciNet)
MR3869383

Zentralblatt MATH identifier
06991224

#### Citation

Cameron, Stephen. Global well-posedness for the two-dimensional Muskat problem with slope less than 1. Anal. PDE 12 (2019), no. 4, 997--1022. doi:10.2140/apde.2019.12.997. https://projecteuclid.org/euclid.apde/1540864858

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