# A paradifferential approach for well-posedness of the Muskat problem

## TL;DR

This paper proves local well-posedness for the Muskat problem in arbitrary dimensions using a paradifferential approach, accommodating large initial data and rough boundaries, and clarifies stability conditions.

## Contribution

It introduces a paradifferential method to establish well-posedness of the Muskat problem for large data and rough boundaries in any dimension, extending previous results.

## Key findings

- Proved local well-posedness in subcritical Sobolev spaces for large data.
- Allowed Lipschitz boundaries with large variations.
- Established automatic Rayleigh-Taylor stability for single-fluid cases.

## Abstract

We study the Muskat problem for one fluid or two fluids, with or without viscosity jump, with or without rigid boundaries, and in arbitrary space dimension $d$ of the interface. The Muskat problem is scaling invariant in the Sobolev space $H^{s_c}(\mathbb{R}^d)$ where $s_c=1+\frac{d}{2}$. Employing a paradifferential approach, we prove local well-posedness for large data in any subcritical Sobolev spaces $H^s(\mathbb{R}^d)$, $s>s_c$. Moreover, the rigid boundaries are only required to be Lipschitz and can have arbitrarily large variation. The Rayleigh-Taylor stability condition is assumed for the case of two fluids with viscosity jump but is proved to be automatically satisfied for the case of one fluid. The starting point of this work is a reformulation solely in terms of the Drichlet-Neumann operator. The key elements of proofs are new paralinearization and contraction results for the Drichlet-Neumann operator in rough domains.