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2019 Global well-posedness for the two-dimensional Muskat problem with slope less than 1
Stephen Cameron
Anal. PDE 12(4): 997-1022 (2019). 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.

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Stephen Cameron. "Global well-posedness for the two-dimensional Muskat problem with slope less than 1." Anal. PDE 12 (4) 997 - 1022, 2019. https://doi.org/10.2140/apde.2019.12.997

Information

Received: 9 May 2017; Revised: 14 January 2018; Accepted: 30 July 2018; Published: 2019
First available in Project Euclid: 30 October 2018

zbMATH: 06991224
MathSciNet: MR3869383
Digital Object Identifier: 10.2140/apde.2019.12.997

Subjects:
Primary: 35K55, 35Q35, 35R09

Rights: Copyright © 2019 Mathematical Sciences Publishers

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Vol.12 • No. 4 • 2019
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