Eventual regularization for the 3D Muskat problem: Lipschitz for finite time implies global existence

We show that for any fixed Lipschitz constant $L$, there is a time $T^*<\infty$ depending only on $L$ such that if $f:[0,T^*]\times \mathbb{R}^{2}\to [0,1]$ is a classical solution of the stable Muskat problem with $||\nabla_x f||_{L^\infty}\leq L$, then $f$ extends uniquely to a classical solution defined for all time. Our proof is short, quantitative, and explicit.