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Archive for Rational Mechanics and Analysis

, Volume 233, Issue 1, pp 451–496 | Cite as

Existence of Weak Solutions for a General Porous Medium Equation with Nonlocal Pressure

  • Diana Stan
  • Félix del Teso
  • Juan Luis VázquezEmail author
Article
  • 154 Downloads

Abstract

We study the general nonlinear diffusion equation \({u_t=\nabla\cdot (u^{m-1}\nabla (-\Delta)^{-s}u)}\) that describes a flow through a porous medium which is driven by a nonlocal pressure. We consider constant parameters \({m > 1}\) and \({0 < s < 1}\), we assume that the solutions are non-negative and that the problem is posed in the whole space. In this paper we prove the existence of weak solutions for all integrable initial data \({u_0 \ge 0}\) and for all exponents \({m > 1}\) by developing a new approximation method that allows one to treat the range \({m\geqq 3}\), which could not be covered by previous works. We also extend the class of initial data to include any non-negative measure \({\mu}\) with finite mass. In passing from bounded initial data to measure data we make strong use of an L1-\({L^\infty}\) smoothing effect and other functional estimates. Finite speed of propagation is established for all \({m \geqq 2}\), and this property implies the existence of free boundaries. The authors had already proved that finite propagation does not hold for \({m < 2}\).

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Notes

Acknowledgements

The authors are partially supported by the Spanish ProjectMTM2014-52240-P.Diana Stan and Félix del Teso are partially supported by theMEC-Juan de la Cierva postdoctoral fellowships number FJCI-2015-25797 and FJCI-2016-30148 respectively, and by the BCAM Severo Ochoa accreditation SEV-2017-0718. Félix del Teso is partially supported by the Toppforsk (research excellence) project Waves and Nonlinear Phenomena (WaNP), Grant 250070 from the Research Council of Norway. Juan Luis Vázquez has been a Visiting Professor at Univ. Complutense de Madrid during the academic year 2017–2018. The authors want to thank the anonymous referee for accurate suggestions that allowed them to improve the original text.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Departamento de Matemáticas, Estadística y ComputaciónUniversidad de CantabriaSantanderSpain
  2. 2.Basque Center for Applied MathematicsBilbaoSpain
  3. 3.Departamento de MatemáticasUniversidad Autónoma de MadridMadridSpain

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