Global wellposedness for the 3D Muskat problem with medium size slope

We prove the existence and uniqueness of global, classical solutions to the 3D Muskat problem in the stable regime whenever the initial interface has sublinear growth and slope $\mathrm{\mid }\mathrm{\mid }{\mathrm{\nabla }}_x{f}_0\mathrm{\mid }{\mathrm{\mid }}_{L^{\mathrm{\infty }}}<5^{-1\mathrm{/}2}$. We show under these assumptions that the equation is fundamentally parabolic, satisfying a comparison principle. Applying the modulus of continuity technique, we show that rough initial data instantly becomes $C^{1,1}$ with the curvature decaying like $O(t^{-1})$.