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Nonlinear Partial Differential Equations pp 271–298Cite as

Nonlinear Diffusion with Fractional Laplacian Operators

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Part of the book series: Abel Symposia ((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.

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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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  1. Dpto. de Matemáticas, Universidad Autónoma de Madrid, 28049, Madrid, Spain

    Juan Luis Vázquez

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  1. Juan Luis Vázquez
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  1. of Science and Technology, Dept. of Mathematics, Norwegian University, Trondheim, 7491, Norway

    Helge Holden

  2. Centre of Mathematics for Applications, University of Oslo, Oslo, 0316, Norway

    Kenneth H. Karlsen

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Vázquez, J.L. (2012). Nonlinear Diffusion with Fractional Laplacian Operators. In: Holden, H., Karlsen, K. (eds) Nonlinear Partial Differential Equations. Abel Symposia, vol 7. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25361-4_15

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