Rigidity of Acute Angled Corners for One Phase Muskat Interfaces

TL;DR

This paper proves local well-posedness for the one-phase Muskat problem with interfaces that can have singularities like corners or cusps, and shows that such corners are rigid and preserve their angles over time.

Contribution

It establishes the existence and uniqueness of solutions for interfaces with acute corners and cusps, demonstrating their rigidity and angle preservation.

Findings

Corners and cusps are preserved over time.

No rotation occurs at the corners.

Particle at the corner remains at the tip and moves downward.

Abstract

We consider the one-phase Muskat problem modeling the dynamics of the free boundary of a single fluid in porous media. We prove local well-posedness for fluid interfaces that are general curves and can have singularities. In particular, the free boundary can have acute angle corners or cusps. Moreover, we show that isolated corners/cusps on the interface must be rigid, meaning the angle of the corner is preserved for a finite time, there is no rotation at the tip, the particle at the tip remains at the tip and the velocity of that particle at the tip points vertically downward.