Fisher information and continuity estimates for nonlinear but 1-homogeneous diffusive PDEs (via the JKO scheme)

In this short paper we prove, using the JKO scheme, that quantities such as the Fisher information stay bounded or decrease across time for a family of 1-homogeneous diffusive PDEs. As a corollary, we prove that moduli of continuity are conserved across time for the solutions of those PDEs.