Local-in-time regularity results for some flux-limited diffusion equations of porous media type

We prove some regularity results for solutions of some flux limited diffusion equations of porous media type for Lipschitz initial data (or assuming a uniform gradient bound on some power of the data), including the fast diffusion case in which the results are global in time. We also develop the existence and uniqueness theory for solutions of the fast diffusion case, which was not covered in the current literature.