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 (4)

• Global well-posedness for the critical 2D dissipative quasi-geostrophic equation
• On the global existence for the Muskat problem
• Global well-posedness of slightly supercritical active scalar equations
• On The Muskat Problem With Viscosity Jump: Global In Time Results

Cited by in corpus (9)

• The Peskin Problem with Viscosity Contrast
• Global well-posedness for the three dimensional Muskat problem in the critical Sobolev space
• Global solutions for the Muskat problem in the scaling invariant Besov space ${\dot{B}}_{\infty ,1}^1$
• The Muskat problem with $C^1$ data
• Global existence and decay of the inhomogeneous Muskat problem with Lipschitz initial data
• Global well-posedness for the one-phase Muskat problem
• Eventual regularization for the 3D Muskat problem: Lipschitz for finite time implies global existence
• Global Results for the Inhomogeneous Muskat Problem
• The Second Iterate of the Muskat Equation in Supercritical Spaces