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Derivation of the nonlocal pressure form of the fractional porous medium equation in the hydrological setting. (English) Zbl 1509.35357

Summary: In this short note we consider a nonlinear and spatially nonlocal PDE modeling 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.

MSC:

35R11 Fractional partial differential equations
35K59 Quasilinear parabolic equations
35R09 Integro-partial differential equations
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