Non-splat singularity for the one-phase Muskat problem

TL;DR

This paper proves that splat singularities, where interfaces self-intersect, do not occur in the one-phase Muskat problem, contrasting with water wave equations where such singularities can form.

Contribution

It establishes the absence of splat singularities in the one-phase Muskat problem, providing new insights into interface behavior in porous media flows.

Findings

No splat singularities form in the one-phase Muskat problem.

Contrast with water wave equations where splat singularities can occur.

Advances understanding of interface regularity in porous media flows.

Abstract

For the water waves equations, the existence of splat singularities has been shown in [3], i.e., the interface self-intersects along an arc in finite time. The aim of this paper is to show the absence of splat singularities for the incompressible fluid dynamics in porous media.