Free Boundary Regularity of the Porous Medium Equation with nonlocal drifts in Dimension One

We study the free boundary of the porous medium equation with nonlocal drifts in dimension one. Under the assumption that the initial data has super-quadratic growth at the free boundary, we show that the solution is smooth in space and $C_{\text{loc}}^{2,1}$ in time, and then the free boundary is $C_{\text{loc}}^{2,1}$. Moreover if the drift is local, both the solution and the free boundary are smooth.