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

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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

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

Received: 9 May 2017
Revised: 14 January 2018
Accepted: 30 July 2018
First available in Project Euclid: 30 October 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35K55: Nonlinear parabolic equations 35Q35: PDEs in connection with fluid mechanics 35R09: Integro-partial differential equations [See also 45Kxx]

Muskat problem porous media fluid interface global well-posedness


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.

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