On the well-posedness of two-dimensional Muskat problem with an elastic interface

TL;DR

This paper proves local well-posedness of the 2D Muskat problem with an elastic interface in Sobolev spaces and establishes global well-posedness for small initial data in certain stable cases.

Contribution

It extends the well-posedness theory of the Muskat problem to include elastic interfaces and provides global results for small initial data in stable configurations.

Findings

Local well-posedness in $H^s$ for $s \\geq 2$

Global well-posedness for small data in $H^s$, $s > 3/2$, in stable cases

Applicable to both one-phase and two-phase scenarios

Abstract

We investigate the two-dimensional Muskat problem with a nonlinear elastic interface, for both one-phase and two-phase scenarios. Following the framework developed by Nguyen [35,36], we demonstrate that the problem is locally well-posed in $H^s$ for $s\geq 2$ for arbitrary initial data. Furthermore, for the one-phase case and the stable two-phase case $(\rho^+ \leq \rho^-)$, we establish global well-posedness for small initial data in $H^s$ when $s> \frac{3}{2}$.