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Open AccessArticle

Solvability of the p-Adic Analogue of Navier–Stokes Equation via the Wavelet Theory

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International Center for Mathematical Modelling in Physics and Cognitive Sciences, Mathematical Institute, Linnaeus University, SE-351 95 Växjö, Sweden
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Department of Mathematics, Iran University of Science and Technology, Narmak, Tehran 16846-13114, Iran
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Centro de Geociencias, Campus UNAM Juriquilla, Universidad Nacional Autonoma de Mexico (UNAM), Blvd. Juriquilla 3001, 76230 Queretaro, Mexico
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Edificio Piramide, Boulevard Adolfo Ruiz Cortines 1202, Oropeza, 86030 Villahermosa, Tabasco, Mexico
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Author to whom correspondence should be addressed.
Entropy 2019, 21(11), 1129; https://doi.org/10.3390/e21111129
Received: 9 October 2019 / Revised: 7 November 2019 / Accepted: 12 November 2019 / Published: 17 November 2019
(This article belongs to the Section Multidisciplinary Applications)
P-adic numbers serve as the simplest ultrametric model for the tree-like structures arising in various physical and biological phenomena. Recently p-adic dynamical equations started to be applied to geophysics, to model propagation of fluids (oil, water, and oil-in-water and water-in-oil emulsion) in capillary networks in porous random media. In particular, a p-adic analog of the Navier–Stokes equation was derived starting with a system of differential equations respecting the hierarchic structure of a capillary tree. In this paper, using the Schauder fixed point theorem together with the wavelet functions, we extend the study of the solvability of a p-adic field analog of the Navier–Stokes equation derived from a system of hierarchic equations for fluid flow in a capillary network in porous medium. This equation describes propagation of fluid’s flow through Geo-conduits, consisting of the mixture of fractures (as well as fracture’s corridors) and capillary networks, detected by seismic as joint wave/mass conducts. Furthermore, applying the Adomian decomposition method we formulate the solution of the p-adic analog of the Navier–Stokes equation in term of series in general form. This solution may help researchers to come closer and find more facts, taking into consideration the scaling, hierarchies, and formal derivations, imprinted from the analogous aspects of the real world phenomena. View Full-Text
Keywords: tree-like geometry; capillary networks; p-adic model of porous medium; fluid’s propagation; complex geological phenomena; p-adic analog of Navier–Stokes equation; pseudo-differential equations; p-adic wavelet basis; Schauder fixed point theorem; Vladimirov’s operator; existence of solution tree-like geometry; capillary networks; p-adic model of porous medium; fluid’s propagation; complex geological phenomena; p-adic analog of Navier–Stokes equation; pseudo-differential equations; p-adic wavelet basis; Schauder fixed point theorem; Vladimirov’s operator; existence of solution
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Pourhadi, E.; Khrennikov, A.; Saadati, R.; Oleschko, K.; Correa Lopez, M.J. Solvability of the p-Adic Analogue of Navier–Stokes Equation via the Wavelet Theory. Entropy 2019, 21, 1129.

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