Rigidity of Acute Angled Corners for One Phase Muskat Interfaces

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.

Rigorous thin film approximations of the one-phase unstable Muskat problem

“…These facts have been used to give well-posedness for arbitrary slope [20]. The solutions have been shown to do not smooth instantly [1]. For small slope there is instant smoothing in the stable regime and ill-posedness in the unstable regime [30].…”