Minimizing movement for a fractional porous medium equation in a periodic setting

Abstract

We consider a fractional porous medium equation that extends the classical porous medium and fractional heat equations. The flow is studied in the space of periodic probability measures endowed with a non-local transportation distance constructed in the spirit of the Benamou-Brenier formula. For initial periodic probability measures, we show the existence of absolutely continuous curves that are generalized minimizing movements associated to Rényi entropy. For that, we need to obtain entropy and distance properties and to develop a subdifferential calculus in our setting.

Degenerate Cahn-Hilliard equation: From nonlocal to local

“…To conclude, let us mention that optimal transport was used to study several PDEs on the torus via JKO scheme, in particular fractional porous medium equation [36], continuity equation with nonlocal velocity in 1D [37], systems of continuity equations with nonlinear diffusion and nonlocal drifts [13] and certain fourth-order equation in one dimension [26].…”