Well-posedness and decay to equilibrium for the Muskat problem with discontinuous permeability

TL;DR

This paper establishes local well-posedness and decay to equilibrium for the Muskat problem with discontinuous permeability, introducing a novel PDE analysis approach that handles unbounded curvature and proves the first global existence result in this setting.

Contribution

It develops a new methodology for analyzing the Muskat problem with permeability jumps, allowing for unbounded curvature and proving global existence and decay for small initial data.

Findings

Proved local-in-time well-posedness for the inhomogeneous Muskat problem.

Established global existence and decay to equilibrium for small initial data.

Developed a new energy-dissipation inequality coupling velocity derivatives and interface curvature.

Abstract

We first prove local-in-time well-posedness for the Muskat problem, modeling fluid flow in a two-dimensional inhomogeneous porous media. The permeability of the porous medium is described by a step function, with a jump discontinuity across the fixed-in-time curve $(x_1,-1+f(x_1))$, while the interface separating the fluid from the vacuum region is given by the time-dependent curve $(x_1,h(x_1,t))$. Our estimates are based on a new methodology that relies upon a careful study of the PDE system, coupling Darcy's law and incompressibility of the fluid, rather than the analysis of the singular integral contour equation for the interface function $h$. We are able to develop an existence theory for any initial interface given by $h_0 \in H^2$ and any permeability curve-of-discontinuity that is given by $f \in H^{2.5}$. In particular, our method allows for both curves to have (pointwise) unbounded curvature. In the case that the permeability discontinuity is the set $f=0$, we prove global existence and decay to equilibrium for small initial data. This decay is obtained using a new energy-energy dissipation inequality that couples tangential derivatives of the velocity in the bulk of the fluid with the curvature of the interface. To the best of our knowledge, this is the first global existence result for the Muskat problem with discontinuous permeability.