In this talk, we will first derive anisotropic, nonlocal porous media equations as the scaling limit of (moderately) interacting particle systems. This derivation gives rise to a novel class of anisotropic, nonlocal, degenerate partial differential equations. In the particular scenario of isotropic interactions, these PDEs reduce to nonlocal porous medium equations.
Subsequently, we introduce an approach to establish optimal regularity for their solutions in Sobolev spaces with respect to both temporal and spatial variables. The proof leverages the kinetic formulation of solutions, velocity-averaging techniques, and a careful micro-local analysis of pseudodifferential operators. This work extends to the nonlocal case the previous results [Tadmor, Tao; CPAM, 2007], [Gess, JEMS; 2021], [Gess, Sauer, Tadmor, Anal. PDE; 2020].