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We consider a family of fractional porous media equations, recently studiedby Caffarelli and V\'azquez. We show the construction of a weak solution asWasserstein gradient flow of a square fractional Sobolev norm. Energydissipation inequality, regularizing effect and decay estimates for the $L^p$norms are established. Moreover, we show that a classical porous mediumequation can be obtained as a limit case.

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