Global well-posedness for the 2D Muskat problem with slope less than 1

Abstract

We prove the existence of global, smooth solutions to the 2D Muskat problem in the stable regime whenever the product of the maximal and minimal slopes is strictly less than 1. The curvature of these solutions solutions decays to 0 as $t$ goes to infinity, and they are unique when the initial data is $C^{1,\epsilon }$. We do this by constructing a modulus of continuity generated by the equation, just as Kiselev, Nazarov, and Volberg did in their proof of the global well-posedness for the quasi-geostraphic equation.

22 pages

References in corpus (3)

• Global well-posedness for the critical 2D dissipative quasi-geostrophic equation
• A maximum principle for the Muskat problem for fluids with different densities
• Large Time Decay Estimates for the Muskat Equation

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• The Peskin Problem with Viscosity Contrast
• The vanishing surface tension limit of the Muskat problem
• Regularity of solutions to the Muskat equation
• Global existence and decay of the inhomogeneous Muskat problem with Lipschitz initial data
• Entropy solutions to macroscopic IPM
• The Second Iterate of the Muskat Equation in Supercritical Spaces