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

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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.

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