Quasilinearization of the 3D Muskat equation, and applications to the critical Cauchy problem

TL;DR

This paper introduces a new approach to analyze the 3D Muskat equation by decomposing its nonlinearity, enabling the study of solutions with critical regularity and establishing well-posedness for large initial data.

Contribution

It presents a novel decomposition of the nonlinearity in the Muskat equation and proves well-posedness for large data in critical spaces, including non-Lipschitz initial conditions.

Findings

First well-posedness result for large data in critical space

Existence of solutions with non-Lipschitz initial data

New decomposition technique for the Muskat equation

Abstract

We exhibit a new decomposition of the nonlinearity for the Muskat equation and use it to commute Fourier multipliers with the equation. This allows to study solutions with critical regularity. As a corollary, we obtain the first well-posedness result for arbitrary large data in the critical space $\dot{H}^2(\mathbb{R}^2)\cap W^{1,\infty}(\mathbb{R}^2)$. Moreover, we prove the existence of solutions for initial data which are not Lipschitz.