The 2D Muskat Problem II: Stable Regime Small Data Singularity on the Half-plane

TL;DR

This paper demonstrates that finite-time interface singularities can occur in the Muskat problem on a half-plane, even from small smooth initial data, by establishing maximum principles and local well-posedness.

Contribution

It shows the existence of small-data singularities in the Muskat problem on the half-plane, advancing understanding of interface instability in porous media.

Findings

Finite-time interface singularities occur on the half-plane.

Maximum principles for potential energy and slope are established.

Small initial data can lead to singularities.

Abstract

We study the Muskat problem on the half-plane, which models motion of an interface between two fluids of distinct densities (e.g., oil and water) in a porous medium (e.g., an aquifer) that sits atop an impermeable layer (e.g., bedrock). Existence of finite time stable regime interface curve singularities is still open on the whole plane, but we show that they do arise on the half-plane, including from arbitrarily small smooth initial data. To obtain this result, we establish maximum principles for both the potential energy and the slope of solutions in this model, as well as develop a general local well-posedness theory in the companion paper [25].