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Periodic Solutions to the Navier–Stokes Equation for a Viscous Incompressible Fluid and Gas in Space \({\mathbb{R}}^{n}\)

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Abstract

Initial value problems are considered for equations of motion of a viscous incompressible fluid and gas in Lagrangian variables. It is shown that the incompressible fluid motion is not related to pressure. In the absence of external forces, the pressure is constant and allows the fluid to make free motion. This motion is purely turbulent and is described by quasi-linear equations of parabolic type. The existence and uniqueness of the classical periodic solution to the initial-value problem in the \({\mathbb{R}}^{n}\) at \(n\geqslant 2\) are shown. Equations of motion of fluid and gas in steady-state conditions are derived. The problem on a turbulent flow of a partially compressible fluid and gas is solved. It is established that there is no turbulent flow in the incompressible fluid. It is shown that spatially stable periodic structures appear as a result of synchronization of frequencies.

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Funding

This work was supported by the Ministry of Science and Higher Education of the Russian Federation part of the program of the Moscow Center of Fundamental and Applied Mathematics according to agreement no. 075-15-2022-284.

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  1. Department of Computational Mathematics and Cybernetics, Moscow State University, 119991, Moscow, Russia

    A. V. Baev

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  1. A. V. Baev
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Correspondence to A. V. Baev.

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Translated by D. Churochkin

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Baev, A.V. Periodic Solutions to the Navier–Stokes Equation for a Viscous Incompressible Fluid and Gas in Space \({\mathbb{R}}^{n}\). MoscowUniv.Comput.Math.Cybern. 47, 1–11 (2023). https://doi.org/10.3103/S0278641923010028

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