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

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 ${\dot{H}}^2({\mathbb{R}}^2)\cap W^{1,\mathrm{\infty }}({\mathbb{R}}^2)$. Moreover, we prove the existence of solutions for initial data which are not Lipschitz.