In this talk, we will present a free boundary problem that models the fluid flow through a partially saturated porous medium. In the saturated region, this phenomenon is governed by an elliptic equation, while in the unsaturated region, it is governed by a parabolic equation. The free boundary arises as the interface that separates these two regions, which are a priori unknown. The general theory of existence, uniqueness, and regularity of solutions was developed in the 1980s. Some of the most influential works are van Duyn--Peletier '82, Alt--Luckhaus '83, Hulshof--Peletier '86, DiBenedetto--Gariepy '87, and Hulshof--Wolanski '88.
One of the most interesting questions is to understand the behavior of the free boundary, which is strongly influenced by the regularity of the boundary data due to the nonlocal nature of the problem. When the data is sufficiently regular, it is known that the free boundary is continuous, or even differentiable, under certain initial conditions. However, local regularization results were unknown. For the 1+1 dimensional model, we show that when the data is merely bounded, the free boundary is of class $C^{1/2}$, and this regularity is optimal. This is a joint work with Dennis Kriventsov (Rutgers University).