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Global well-posedness
for the two-dimensional Muskat problem with slope less than
1
Stephen Cameron |
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Vol. 12 (2019), No. 4, 997–1022
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DOI: 10.2140/apde.2019.12.997
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Abstract |
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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 goes to infinity, and they are unique when the initial data is . 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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Keywords
Muskat problem, porous media, fluid interface, global
well-posedness
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Mathematical Subject Classification 2010
Primary: 35K55, 35Q35, 35R09
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Milestones
Received: 9 May 2017
Revised: 14 January 2018
Accepted: 30 July 2018
Published: 20 October 2018
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Authors |
| Stephen Cameron | |
| Department of Mathematics University of Chicago Chicago, IL United States |
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