Application of p-Adic Wavelets to Model Reaction–Diffusion Dynamics in Random Porous Media

Khrennikov, Andrei; Oleshko, Klaudia; López, María de J.

doi:10.1007/s00041-015-9433-y

Cited by 35 publications

(40 citation statements)

References 26 publications

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“…Concluding this introductory section we remark that novel applications of ultrametric methods are not restricted to our recent studies in geophysics [4][5][6]. We can also point to a series of studies performed in [30][31][32][33].…”

Section: Introductionmentioning

confidence: 99%

“…We point out that, although the ultrametric pseudo-differential equations studied in [4][5][6] describe some distinguishing features of propagation of fluids in capillary networks, they did not explicitly reflect coupling with the physical parameters of such networks. Roughly speaking, they were designed on the basis of abstract reasoning about the natural mathematical form of fluid dynamics on the tree-like configuration space.…”

Section: Introductionmentioning

confidence: 99%

“…[4][5][6]. On the other hand, by proceeding from the discrete tree-like dynamics, for which the physical meaning of dynamics' parameters are well defined, to "continuous" pseudo-differential equations, we also clarify the physical meaning of their parameters.…”

Section: Introductionmentioning

confidence: 99%

“…The tree-like geometry of percolation networks was always of special interest of Mandelbrot, the father of fractal geometry. In the set of our previous publications [4][5][6] we have discussed the advantages of using the scaling, as well as hierarchical representations of several complex systems. Some simplified models of oil, oil-in-water, and water-in-oil emulsion propagation in capillary networks in random porous media based on their physically real tree-like geometry were proposed [4][5][6].…”

Section: Introductionmentioning

confidence: 99%

“…Mathematically, such tree-like fractals are represented as ultrametric spaces and there exist well developed theories of fractional (pseudo-)differential equations on such spaces [7][8][9][10][11][12][13][14][15], which were applied in [4][5][6] to represent the dynamics of fluids in these tree-like capillary networks. The class of homogeneous trees with the constant number of vertices, p > 1, leaving each vertex, the so called p-adic trees, is especially useful for theoretical modeling as representing a mathematically simple model which, at the same time, represents the basic features of fluids' propagation through capillary/fractures networks.…”

Section: Introductionmentioning

confidence: 99%

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P-Adic Analog of Navier–Stokes Equations: Dynamics of Fluid’s Flow in Percolation Networks (from Discrete Dynamics with Hierarchic Interactions to Continuous Universal Scaling Model)

Oleschko

Khrennikov

López

2017

Entropy

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Abstract:Recently p-adic (and, more generally, ultrametric) spaces representing tree-like networks of percolation, and as a special case of capillary patterns in porous media, started to be used to model the propagation of fluids (e.g., oil, water, oil-in-water, and water-in-oil emulsion). The aim of this note is to derive p-adic dynamics described by fractional differential operators (Vladimirov operators) starting with discrete dynamics based on hierarchically-structured interactions between the fluids' volumes concentrated at different levels of the percolation tree and coming to the multiscale universal topology of the percolating nets. Similar systems of discrete hierarchic equations were widely applied to modeling of turbulence. However, in the present work this similarity is only formal since, in our model, the trees are real physical patterns with a tree-like topology of capillaries (or fractures) in random porous media (not cascade trees, as in the case of turbulence, which we will be discussed elsewhere for the spinner flowmeter commonly used in the petroleum industry). By going to the "continuous limit" (with respect to the p-adic topology) we represent the dynamics on the tree-like configuration space as an evolutionary nonlinear p-adic fractional (pseudo-) differential equation, the tree-like analog of the Navier-Stokes equation. We hope that our work helps to come closer to a nonlinear equation solution, taking into account the scaling, hierarchies, and formal derivations, imprinted from the similar properties of the real physical world. Once this coupling is resolved, the more problematic question of information scaling in industrial applications will be achieved.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

P-Adic Analog of Navier–Stokes Equations: Dynamics of Fluid’s Flow in Percolation Networks (from Discrete Dynamics with Hierarchic Interactions to Continuous Universal Scaling Model)

Oleschko

Khrennikov

López

2017

Entropy

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Boundedness of H ∞ Functional Calculus of Hodge-Dirac Operators

Arhancet¹,

Kriegler

2022

Lecture Notes in Mathematics

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For any non-archimedean local field K and any integer n ⩾ 1, we show that the Taibleson operator admits a bounded H ∞ (Σ θ ) functional calculus for any angle θ > 0 on the Banach space L p (K n ), where Σ θ = {z ∈ C * : | arg z| < θ} and 1 < p < ∞, and even a bounded Hörmander functional calculus of order 3 2 (with striking contrast to the Euclidean Laplacian on R n ). In our study, we explore harmonic analysis on locally compact Spector-Vilenkin groups establishing the R-boundedness of a family of convolution operators. Our results enhance the understanding of functional calculi of operators acting on L p -spaces associated to totally disconnected spaces and have implications for the maximal regularity of the fundamental evolution equations associated to the Taibleson operator, relevant in various physical models. Contents 1 Introduction 1 2 Preliminaries 5 3 Locally compact Spector-Vilenkin groups and radial functions 10 4 Maximal inequalities 15 5 R-boundedness of some family of convolution operators 17 6 Application to functional calculus 20 Bibliography 27 2020 Mathematics subject classification: 46S10, 47S10, 11S80.

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Image Segmentation with the Aid of the p-Adic Metrics

Khrennikov

Kotovich

2017

STEAM-H: Science, Technology, Engineering, Agriculture, Mathematics &Amp; Health

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Application of p-Adic Wavelets to Model Reaction–Diffusion Dynamics in Random Porous Media

Cited by 35 publications

References 26 publications

P-Adic Analog of Navier–Stokes Equations: Dynamics of Fluid’s Flow in Percolation Networks (from Discrete Dynamics with Hierarchic Interactions to Continuous Universal Scaling Model)

P-Adic Analog of Navier–Stokes Equations: Dynamics of Fluid’s Flow in Percolation Networks (from Discrete Dynamics with Hierarchic Interactions to Continuous Universal Scaling Model)

Boundedness of H ∞ Functional Calculus of Hodge-Dirac Operators

Image Segmentation with the Aid of the p-Adic Metrics

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