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

We exhibit a new decomposition of the nonlinearity for Muskat equation and use it to commute Fourier multipliers with equation. This allows study solutions critical regularity. As corollary, we obtain first well-posedness result arbitrary large data in space H?2(R2)?W1,?(R2). Moreover, prove existence initial which are not Lipschitz.