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

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.