The $N$-dimensional gravity driven Muskat problem

TL;DR

This paper analyzes the $N$-dimensional gravity-driven Muskat problem, reformulating it as a nonlinear nonlocal evolution equation, and establishes well-posedness and smoothing properties in Sobolev spaces for the interface dynamics.

Contribution

It extends the analysis of the Muskat problem to higher dimensions, proving well-posedness and regularity results using harmonic analysis and abstract parabolic theory.

Findings

The evolution is of parabolic type under the Rayleigh-Taylor condition.

The problem defines a semiflow in subcritical Sobolev spaces $H^s( abla)$.

Solutions exhibit parabolic smoothing up to ${ m C}^ abla$.

Abstract

We study the Muskat problem, which describes the motion of two immiscible, incompressible fluids in a homogeneous porous medium occupying the full space ${\mathbb{R}^{N+1}}$, $N \geq 2$, driven by gravity. The interface between the fluids is given as graph of a function over $\mathbb{R}^N$. The problem is reformulated as a nonlinear, nonlocal evolution problem for this function, involving singular integrals arising from potential representations of the velocity and pressure fields. Using results from harmonic analysis, we demonstrate that the evolution is of parabolic type in the open set identified by the Rayleigh-Taylor condition. We use the abstract theory of such problems to establish that the Muskat problem defines a semiflow on this set in all subcritical Sobolev spaces $H^s(\mathbb{R}^N)$, $s>s_c$, where ${s_c=1+N/2}$ is the critical exponent. We additionally obtain parabolic smoothing up to ${\rm C}^\infty$.