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Nonlinear Diffusion with Fractional Laplacian Operators

  • Juan Luis Vázquez
Part of the Abel Symposia book series (ABEL, volume 7)

Abstract

We describe two models of flow in porous media including nonlocal (long-range) diffusion effects. The first model is based on Darcy’s law and the pressure is related to the density by an inverse fractional Laplacian operator. We prove existence of solutions that propagate with finite speed. The model has the very interesting property that mass preserving self-similar solutions can be found by solving an elliptic obstacle problem with fractional Laplacian for the pair pressure-density. We use entropy methods to show that these special solutions describe the asymptotic behavior of a wide class of solutions.

The second model is more in the spirit of fractional Laplacian flows, but nonlinear. Contrary to usual Porous Medium flows (PME in the sequel), it has infinite speed of propagation. Similarly to them, an L 1-contraction semigroup is constructed and it depends continuously on the exponent of fractional derivation and the exponent of the nonlinearity.

Keywords

Porous Medium Obstacle Problem Fractional Diffusion Porous Medium Equation Schwartz Class 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

Author partially supported by Spanish Project MTM2008-06326-C02. The author is grateful to the referee and N. Guillén for carefully reading the manuscript.

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Authors and Affiliations

  1. 1.Dpto. de MatemáticasUniversidad Autónoma de MadridMadridSpain

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