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

2016/06/30 by Stefano Lisini, Edoardo Mainini, Antonio Segatti · 1 citation
Mathematics · #Balanced flow #Dissipation #Flow (mathematics) #Geometric Analysis and Curvature Flows #Limit (mathematics) #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations #Porous medium #Pressure gradient #Sobolev space #math.AP #msc:35A01 #msc:49K20

paper · pdf · doi:10.1007/s00205-017-1168-2

published as ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS (2017) · 34 pages

openalex created_date 2016/07/22 · openalex publication_date 2017/09/12 · arxiv created 2017/10/10 · arxiv updated 2017/10/11 · openalex updated_date 2026/08/06

Abstract

We consider a family of fractional porous media equations, recently studied by Caffarelli and Vázquez. 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.

Citations

• Convergence of the Mass-Transport Steepest Descent Scheme for the Subcritical Patlak–Keller–Segel Model
• A mean field equation as limit of nonlinear diffusions with fractional Laplacian operators
• Nonlinear Porous Medium Flow with Fractional Potential Pressure

Cited by

• Fractional high order thin film equation: gradient flow approach

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