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A Gradient Flow Approach to the Porous Medium Equation with Fractional Pressure

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Abstract

We consider a family of porous media equations with fractional pressure, recently studied by Caffarelli and Vázquez. We show the construction of a weak solution as the Wasserstein gradient flow of a square fractional Sobolev norm. The 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.

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Authors and Affiliations

  1. Dipartimento di Matematica ”F. Casorati”, Università degli Studi di Pavia, via Ferrata 1, 27100, Pavia, Italy

    Stefano Lisini & Antonio Segatti

  2. Dipartimento di Ingegneria Meccanica, Energetica, Gestionale e dei Trasporti (DIME), Università degli Studi di Genova, via all’Opera Pia 15, 16145, Genova, Italy

    Edoardo Mainini

Authors
  1. Stefano Lisini
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  2. Edoardo Mainini
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  3. Antonio Segatti
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Correspondence to Stefano Lisini.

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Communicated by A. Figalli

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Lisini, S., Mainini, E. & Segatti, A. A Gradient Flow Approach to the Porous Medium Equation with Fractional Pressure. Arch Rational Mech Anal 227, 567–606 (2018). https://doi.org/10.1007/s00205-017-1168-2

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  • DOI: https://doi.org/10.1007/s00205-017-1168-2

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