  The one-phase and two-phase Muskat problems with arbitrary viscosity contrast
are studied in all dimensions. They are quasilinear parabolic equations for the
graph free boundary. We prove that small data in the scaling invariant
homogeneous Besov space $\dot B^1_{\infty, 1}$ lead to unique global solutions.
The proof exploits a new structure of the Dirichlet-Neumann operator which
allows us to implement a robust fixed-point argument. As a consequence of this
method, the initial data is only assumed to be in $\dot B^1_{\infty, 1}$ and
the solution map is Lipschitz continuous in the same topology. For the general
Muskat problem, the only known scaling invariant result was obtained in the
Wiener algebra (plus an $L^2$ assumption) which is strictly contained in $\dot
B^1_{\infty, 1}$.

Analysis of PDEs