Nonlinear Porous Medium Flow with Fractional Potential Pressure

Archive for Rational Mechanics and Analysis volume 202, pages 537–565 (2011)Cite this article

Abstract

We study a porous medium equation with nonlocal diffusion effects given by an inverse fractional Laplacian operator:

$$\partial_t u-\nabla\cdot(u\nabla p)=0, \quad p=(-\Delta)^{-s}u,\quad 0 < s < 1.$$

We pose the problem for \({x\in {\mathbb{R}^n}}\) and t > 0 with bounded and compactly supported initial data, and prove the existence of weak and bounded solutions that propagate with finite speed, a property that is not shared by other fractional diffusion models.

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Author information

Affiliations

  1. School of Mathematics, University of Texas at Austin, 1 University Station, C1200, Austin, TX, 78712-1082, USA

    Luis Caffarelli

  2. Institute for Computational Engineering and Sciences, Austin, USA

    Luis Caffarelli

  3. Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049, Madrid, Spain

    Juan Luis Vazquez

  4. Institute ICMAT, Madrid, Spain

    Juan Luis Vazquez

Authors
  1. Luis Caffarelli
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  2. Juan Luis Vazquez
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Corresponding author

Correspondence to Luis Caffarelli.

Additional information

Communicated by F. Lin

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Caffarelli, L., Vazquez, J.L. Nonlinear Porous Medium Flow with Fractional Potential Pressure. Arch Rational Mech Anal 202, 537–565 (2011). https://doi.org/10.1007/s00205-011-0420-4

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Keywords

  • Porous Medium
  • Weak Solution
  • Contact Point
  • Obstacle Problem
  • Energy Inequality