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

Global solutions for the Muskat problem in the scaling invariant Besov space $\dot B^1_{\infty, 1}$