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The Mathematical Theories of Diffusion: Nonlinear and Fractional Diffusion

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Nonlocal and Nonlinear Diffusions and Interactions: New Methods and Directions

Part of the book series: Lecture Notes in Mathematics ((LNMCIME,volume 2186))

Abstract

We describe the mathematical theory of diffusion and heat transport with a view to including some of the main directions of recent research. The linear heat equation is the basic mathematical model that has been thoroughly studied in the last two centuries. It was followed by the theory of parabolic equations of different types. In a parallel development, the theory of stochastic partial differential equations gives a foundation to the probabilistic study of diffusion.

Nonlinear diffusion equations have played an important role not only in theory but also in physics and engineering, and we focus on a relevant aspect, the existence and propagation of free boundaries. Due to our research, we use the porous medium and fast diffusion equations as case examples.

A large part of the paper is devoted to diffusion driven by fractional Laplacian operators and other nonlocal integro-differential operators representing nonlocal, long-range diffusion effects. Three main models are examined (one linear, two nonlinear), and we report on recent progress in which the author is involved.

To appear in Springer Lecture Notes in Mathematics, C.I.M.E. Subseries, 2017.

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Notes

  1. 1.

    Strictly speaking, priority goes to the latter, but the methods were different.

  2. 2.

    Nash and Nirenberg shared the Abel Prize for 2015.

  3. 3.

    Also called interface in the literature.

  4. 4.

    In [253] they were called Type I and Type II in reverse order.

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Acknowledgements

This work was partially supported by Spanish Project MTM2014-52240-P. The text is based on series of lectures given at the CIME Summer School held in Cetraro, Italy, in July 2016. The author is grateful to the CIME foundation for the excellent organization. The author is also very grateful to his collaborators mentioned in the text for an effort of many years. Special thanks are due to F. del Teso, N. Simonov and D. Stan for a careful reading and comments on the text.

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

    Juan Luis Vázquez

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Correspondence to Juan Luis Vázquez .

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  1. Departamento de Matemáticas, Universidad Autónoma de Madrid, Madrid, Spain

    Matteo Bonforte

  2. Dipartimento di Matematica, Politecnico di Milano, Milano, Italy

    Gabriele Grillo

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Vázquez, J.L. (2017). The Mathematical Theories of Diffusion: Nonlinear and Fractional Diffusion. In: Bonforte, M., Grillo, G. (eds) Nonlocal and Nonlinear Diffusions and Interactions: New Methods and Directions. Lecture Notes in Mathematics(), vol 2186. Springer, Cham. https://doi.org/10.1007/978-3-319-61494-6_5

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