# Paralinearization of the Muskat equation and application to the Cauchy problem

## TL;DR

This paper introduces a paralinearization technique for the Muskat equation, leading to a simplified proof of local well-posedness for rough initial data in certain Sobolev spaces.

## Contribution

It provides a novel paralinearization of the Muskat equation and applies it to establish well-posedness without relying on general paradifferential calculus results.

## Key findings

- Explicit parabolic evolution equation derived
- Proves local well-posedness for rough initial data
- Self-contained approach avoiding general paradifferential calculus

## Abstract

We paralinearize the Muskat equation to extract an explicit parabolic evolution equation having a compact form. This result is applied to give a simple proof of the local well-posedness of the Cauchy problem for rough initial data, in homogeneous Sobolev spaces $\dot{H}^1(\mathbb{R})\cap \dot{H}^s(\mathbb{R})$ with $s>3/2$. This paper is essentially self-contained and does not rely on general results from paradifferential calculus.