Short communicationDerivation of the nonlocal pressure form of the fractional porous medium equation in the hydrological setting
Introduction
In [4] the following form of the fractional porous medium equation has been introducedwhere is a linear integral operator (for later results see for ex. [5], [13], [31], [32]). Specifically, in this particular case it is the inverse of the fractional Laplacian (the Riesz operator [2], [33])Not that Authors of the original paper use a different constant . The fractional Laplacian can be defined for example with the Fourier transform Plugging (2) into (1) we can see that the following operator arises and can be thought as a fractional gradientwhich is a pseudo-differential operator of order . It is also possible to define the fractional gradient via the singular integral of a smooth and bounded functions (see [14])where Cn,α is a known constant. In this paper we consider the following nonlocal nonlinear diffusion equationNotice that we allow the diffusivity to be a nonlinear function of both the dependent and independent variable. When D(u, x) ∝ um the above nonlocal PDE reduces to the porous medium equation with a nonlocal pressure.
Apart from the mentioned description of the porous media, generalizations or variations of (6) appear is different settings. In [16] a similar nonlocal equation has been used to model long-range interactions in the gas particle system. Moreover, some version of (6) has also been used to explain evolution of dislocations in crystal lattice [18]. Furthermore, the so-called hydrodynamic limit appears when in which the 1D Eq. (1) can be reduced to the Burgers equation exhibiting hyperbolic shock wave phenomena. Its multidimensional variant has been used for example in modeling vortex liquid in Ginzburg–Landau theory of superconductivity [38].
In this short note we give a phenomenological argument that the nonlocal equation of the form (6) arises naturally as a description of the moisture imbibition in a porous medium exhibiting superdiffusive jump phenomena. The latter mechanism is responsible for emergence of the spatially nonlocal character of the flux which is represented by the fractional gradient operator. On the other hand, a temporal nonlocality can also arise as a consequence of the waiting time phenomenon in which the water can be trapped in certain regions of the medium for prolonged periods of time. This produces the time-fractional derivative and brings the subdiffusive character of the evolution (for a derivation and related result see [25], [26], [27]).
In the literature there also exists another form of the fractional porous medium equation, namelyFor a relevant mathematical results see [11], [12] and for applications see [3], [20]. Notice that both (7) and (6) reduce to the classical porous medium equation in the limit . On the other hand, in the nonlocal setting, i.e. when α ∈ (0, 2) they also agree in the linear case. Hence, they are nonlocal generalizations of the porous medium equation, albeit it can be shown that they are not equivalent! Most notably, the solutions of (7) have infinite speed of propagation while those of (6) are compactly supported. Therefore, the nonlinearity of the equation is the main factor responsible for the lack of a unique generalization of the porous medium equation into the nonlocal setting. The interplay of nonlinearity and nonlocality produces a plethora of interesting phenomena. For a detailed comparison and summary see [37]. In what follows we will show that, at least in the hydrological setting, the nonlocal pressure form is preferred.
Section snippets
Classical case
Here we revisit the derivation of the classical porous medium equation in the hydrological setting. Let us consider a porous medium (for example a soil or brick) and a fluid (water) that penetrates it. We want to derive an equation that governs the evolution of moisture distribution in space and time. By denote the fluid concentration (with dimension ) at point x and time t. Moreover, let a vector be the flux (with dimension ), that is the amount and direction of
Acknowledgments
Author would like to express his utmost gratitude to Prof. Grzegorz Karch and Prof. Moritz Kassmann for a lot of prolific talks and discussions. They greatly motivated all the results appearing in this work.
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