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.