| Abstract: |
| The Muskat problem describes the evolution of the interface between oil and water in a tar sand. It is parabolic and regularizing for small initial data, but there is known to be blowup for some large data. This has lead to a lot of work and study of what kinds of conditions and bounds on the initial condition can guarantee global existence and wellposedness. We will show that whenever the slope of the interface is less than 5^{-1/2}, the equation is fundamentally parabolic with a comparison principle. We then combine this ellipticity with the nonlinear maximum principle argument, first introduced by Kiselev, Nazarov, and Volberg in the study of the surface quasi-geostraphic equation, in order to prove global wellposedness whenever the initial slope is less than 5^{-1/2}. |
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