Minimizing Movement for a Fractional Porous Medium Equation in a Periodic Setting

TL;DR

This paper studies a fractional porous medium equation in a periodic setting using a non-local transportation framework, establishing existence of generalized minimizing movements related to Rényi entropy.

Contribution

It introduces a novel approach combining fractional porous medium equations with a non-local transportation distance in periodic spaces, extending classical methods.

Findings

Existence of absolutely continuous curves as generalized minimizing movements.

Development of entropy and distance properties in the non-local setting.

Subdifferential calculus adapted to the fractional porous medium context.

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 R\'enyi entropy. For that, we need to obtain entropy and distance properties and to develop a subdifferential calculus in our setting.