The $N$-dimensional gravity driven Muskat problem

TL;DR

The paper addresses the gravity-driven Muskat problem in ${\bf R}^{N+1}$, where two immiscible fluids are separated by a graphically described interface $y=f(t,x)$. It develops a fully nonlinear, nonlocal evolution framework for $f$ using boundary potentials, establishes parabolicity under a Rayleigh–Taylor condition, and proves local well-posedness and parabolic smoothing in subcritical Sobolev spaces $H^s({\bf R}^N)$ with $s>s_c=1+N/2$. Central to the analysis is the resolvent theory for the double-layer potential $\mathbb{D}(f)$ and its extensions to $L_2$ and $H^s$ spaces via generalized Riesz transforms, along with localization by Fourier multipliers. The results yield that the Muskat problem defines a semiflow on an open RT-set ${\mathcal O}$, with solutions becoming infinitely smooth for $t>0$, significantly advancing understanding of high-dimensional gravity-driven interfacial flows in porous media. This framework provides a robust foundation for further study of stability, long-time behavior, and potential generalizations to related moving-boundary problems.

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 ${\rm C}^\infty$.

Theorems & Definitions (93)

• Theorem 1.1
• Proposition 2.1
• proof
• Lemma 3.1
• Corollary 3.2
• proof
• Theorem 3.3
• Lemma 3.4
• proof
• proof : Proof of Theorem \ref{['T:1']}