Global wellposedness for the 3D Muskat problem with medium size slope

Abstract

We prove the existence and uniqueness of global, classical solutions to the 3D Muskat problem in the stable regime whenever the initial interface has sublinear growth and slope $||{\nabla }_x{f}_0|{|}_{L^{\infty }}<5^{-1/2}$. We show under these assumptions that the equation is fundamentally parabolic, satisfying a comparison principle. Applying the modulus of continuity technique, we show that rough initial data instantly becomes $C^{1,1}$ with the curvature decaying like $O(t^{-1})$.

36 pages

References in corpus (1)

• Global well-posedness for the critical 2D dissipative quasi-geostrophic equation

Cited by in corpus (1)

• Eventual regularization for the 3D Muskat problem: Lipschitz for finite time implies global existence