A gradient flow approach to the porous medium equation with fractional pressure

We consider a family of porous media equations with fractional pressure, recently studied by Caffarelli and Vazquez. We show the construction of a weak solution as Wasserstein gradient flow of a square fractional Sobolev norm. Energy dissipation inequality, regularizing effect and decay estimates for the L^p norms are established. Moreover, we show that a classical porous medium equation can be obtained as a limit case.