Nonlinear Porous Medium Flow with Fractional Potential Pressure

2010/01/31 by Luis A. Caffarelli, Luis Caffarelli, Juan Luis Vazquez +1 · 2 citations
Mathematics · #Bounded function #Complex system #Flow (mathematics) #Fractional Differential Equations Solutions #Inverse #Inverse problem #Mathematical Biology Tumor Growth #Nonlinear Partial Differential Equations #Nonlinear system #Porous medium #math.AP #msc:35K55 #msc:35K65 #msc:76S05.

paper · pdf · doi:10.1007/s00205-011-0420-4

32 pages, Latex

arxiv created 2010/02/01 · openalex publication_date 2011/04/20 · arxiv updated 2015/05/14 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

Abstract

We study a porous medium equation, with nonlocal diffusion effects given by an inverse fractional Laplacian operator. We pose the problem in n-dimensional space for all t>0 with bounded and compactly supported initial data, and prove existence of a weak and bounded solution that propagates with finite speed, a property that is nor shared by other fractional diffusion models.

Citations

• On explosions of solutions to a system of partial differential equations modelling chemotaxis
• Holder estimates for solutions of integro differential equations like the fractional laplace
• Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian

Cited by

• A Gradient Flow Approach to the Porous Medium Equation with Fractional Pressure
• Stochastic models associated to a Nonlocal Porous Medium Equation