On The Muskat Problem With Viscosity Jump: Global In Time Results

TL;DR

This paper establishes global existence and regularity results for the Muskat problem with viscosity jump in 2D and 3D, overcoming significant mathematical challenges due to non-locality and lack of maximum principles.

Contribution

It proves global in time existence for stable initial data, demonstrates instant analyticity, and improves decay rates and constants in 3D, advancing understanding of the problem's well-posedness.

Findings

Global existence for medium size stable data

Instant analyticity of solutions

Sharp decay rates of analytic norms

Abstract

The Muskat problem models the filtration of two incompressible immiscible fluids of different characteristics in porous media. In this paper, we consider both the 2D and 3D setting of two fluids of different constant densities and different constant viscosities. In this situation, the related contour equations are non-local, not only in the evolution system, but also in the implicit relation between the amplitude of the vorticity and the free interface. Among other extra difficulties, no maximum principles are available for the amplitude and the slopes of the interface in $L^\infty$. We prove global in time existence results for medium size initial stable data in critical spaces. We also enhance previous methods by showing smoothing (instant analyticity), improving the medium size constant in 3D, together with sharp decay rates of analytic norms. The found technique is twofold, giving ill-posedness in unstable situations for very low regular solutions.