Global regularity for 2D Muskat equations with finite slope

TL;DR

This paper proves that the 2D Muskat equation maintains regularity over time if the interface's slope remains bounded, and establishes global regularity for small initial slopes using nonlinear bounds and local existence results.

Contribution

It introduces new nonlinear lower bounds for nonlocal operators to prove regularity and global existence for the 2D Muskat equation under bounded slope conditions.

Findings

Solution remains regular if interface slope stays bounded.

Global regularity achieved for small initial slopes.

Local existence holds for large initial data in $W^{2,p}$.

Abstract

We consider the 2D Muskat equation for the interface between two constant density fluids in an incompressible porous medium, with velocity given by Darcy's law. We establish that as long as the slope of the interface between the two fluids remains bounded and uniformly continuous, the solution remains regular. The proofs exploit the nonlocal nonlinear parabolic nature of the equations through a series of nonlinear lower bounds for nonlocal operators. These are used to deduce that as long as the slope of the interface remains uniformly bounded, the curvature remains bounded. The nonlinear bounds then allow us to obtain local existence for arbitrarily large initial data in the class $W^{2,p}$, $1<p\leq \infty$. We provide furthermore a global regularity result for small initial data: if the initial slope of the interface is sufficiently small, there exists a unique solution for all time.