Gauss Curvature flows are nonlinear parabolic equations which have been used to describe geometric and topological properties of manifolds. W. Firey first considered the evolution of the Gauss curvature flow of compact surfaces to describe the erosion of rolling stones under water, and R. Hamilton showed that if a two-dimensional Gauss curvature flow initially contains a flat side, then there will be a smaller flat side a little later and it takes some time for the surface to become strictly convex. In this talk, I will discuss the near-the-interface behavior of compact convex curvature flows of general dimension with a flat side, moving by either scalar curvature or Gauss curvature. Under suitable initial conditions, I will present the all-time existence of a solution which is smooth up to the interface in its support for the scalar curvature flow, and a locally smooth solution with Hölder-continuous derivatives up to the free boundary in its support for the Gauss curvature flow. To this end, I will explain how to control the shrinking speed of level sets and get the optimal derivative estimates of order one and two of the pressure, along with Hölder continuity of curvatures divided by optimal decay rates.