Derivation of the nonlocal pressure form of the fractional porous medium equation in the hydrological setting

Łukasz Płociniczak

doi:10.1016/j.cnsns.2019.04.014

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arXiv · 1812.08058

Derivation of the nonlocal pressure form of the fractional porous medium equation in the hydrological setting

Łukasz Płociniczak

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Abstract

In this short note we consider a nonlinear and spatially nonlocal PDE modelling moisture evolution in a porous medium. We then show that it naturally arises as a description of superdiffusive jump phenomenon occurring in the medium. We provide a deterministic derivation which allows us to naturally incorporate the nonlinear effects. This reasoning shows that in our setting the so-called nonlocal pressure form of the porous medium equation is preferred as a description of the evolution. In that case the governing nonlocal operator is the fractional gradient rather than the fractional Laplacian.

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Łukasz Płociniczak. 2018-12-19. Derivation of the nonlocal pressure form of the fractional porous medium equation in the hydrological setting. https://doi.org/10.1016/j.cnsns.2019.04.014

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