Modulus of Continuity Method for the Muskat Problem and Fractional Mean Curvature Flow

In this work, we consider two nonlinear, nonlocal, parabolic equations and show that under appropriate growth assumptions, solutions regularize. The first is the 2d Muskat problem, which models the evolution of the interface between oil and water in a tar sand. In the stable regime, the equation was known to be well-posed for smooth, flat initial data, but the slope can blow up in finite time for some large initial data [CCF+12]. It was conjectured that slope 1 was the cutoff for global existence of graphical solutions. We resolve half of this conjecture, proving that the equation is globally well-posed whenever the initial data has slope less than 1. This work was originally published in [Cam19]. The second equation we consider is fractional mean curvature flow, which is a nonlocal fractional order analogue of the usual mean curvature flow. As with the local case, the regularity of general solutions of fractional mean curvature flow is very difficult to study. Without some form of star convexity or graphical assumption, solutions are known to pinch and develop singularities [CSV18, CDNV18]. Despite this, we prove that if our initial set $E_0$ is bounded between two Lipschitz subgraphs, then the minimal viscosity solution becomes a Lipschitz subgraph itself in finite time. This is a purely nonlocal phenomena, as the corresponding theorem is false for classical mean curvature flow. For both equations we prove our results by showing that under our assumptions, evolving a solution over time quantitatively improves its modulus of continuity. This general method of proof was developed independently by Ishii-Lions in [IL90] for fully nonlinear elliptic equations and Kiselev, Nazarov, and Volberg in [AKV07] for active scalar equations.