Paralinearization of the Muskat equation and application to the Cauchy problem

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.