Paralinearization of the Muskat Equation and Application to the Cauchy Problem

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 H˙ ¹(R) ∩ H˙ ^s(R) with s> 3 / 2. This paper is essentially self-contained and does not rely on general results from paradifferential calculus.