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Porous Medium Equation with Nonlocal Pressure

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Current Research in Nonlinear Analysis

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 135))

Abstract

We provide a rather complete description of the results obtained so far on the nonlinear diffusion equation u t  = ∇⋅ (u m−1∇(−Δ)−s u), which describes a flow through a porous medium driven by a nonlocal pressure. We consider constant parameters m > 1 and 0 < s < 1, we assume that the solutions are non-negative, and the problem is posed in the whole space. We present a theory of existence of solutions, results on uniqueness, and relation to other models. As new results of this paper, we prove the existence of self-similar solutions in the range when N = 1 and m > 2, and the asymptotic behavior of solutions when N = 1. The cases m = 1 and m = 2 were rather well known.

Dedicated to Profs. Haim Brezis and Louis Nirenberg with deep admiration

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Acknowledgements

This work was partially supported by Spanish Project MTM2014-52240-P. D. Stan is partially supported by the MEC-Juan de la Cierva postdoctoral fellowship number FJCI-2015-25797, by the ERCEA Advanced Grant 2014 669689—HADE, by the MINECO project MTM2014-53850-P, by Basque Government project IT-641-13 and also by the Basque Government through the BERC 2014–2017 program, and by Spanish Ministry of Economy and Competitiveness MINECO: BCAM Severo Ochoa excellence accreditation SEV-2013-0323. F.d.Teso is partially supported by the Toppforsk (research excellence) project Waves and Nonlinear Phenomena (WaNP), grant no. 250070 from the Research Council of Norway and by the ERCIM “Alain Bensoussan” Fellowship programme.

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Authors and Affiliations

  1. Basque Center for Applied Mathematics, Bilbao, Spain

    Diana Stan

  2. Norwegian University of Science and Technology, Trondheim, Norway

    Félix del Teso

  3. Department of Mathematics, Universidad Autónoma de Madrid, Madrid, Spain

    Juan Luis Vázquez

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  1. Diana Stan
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  2. Félix del Teso
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  3. Juan Luis Vázquez
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Correspondence to Juan Luis Vázquez .

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  1. Department of Mathematics, National Technical University of Athens, Athens, Greece

    Themistocles M. Rassias

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Stan, D., del Teso, F., Vázquez, J.L. (2018). Porous Medium Equation with Nonlocal Pressure. In: Rassias, T. (eds) Current Research in Nonlinear Analysis. Springer Optimization and Its Applications, vol 135. Springer, Cham. https://doi.org/10.1007/978-3-319-89800-1_12

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