On periodic solutions of the Navier–Stokes equation for a viscous incompressible liquid and gas in the space Rn

Recieved: 10/28/2022

Accepted: 11/08/2022

Published: 02/13/2023

Keywords: Lagrange variables, steady-state, turbulence, frequency synchronization

To cite this article

Baev A.V. On periodic solutions of the Navier–Stokes equation for a viscous incompressible liquid and gas in the space Rn. // Moscow University Journal. Series 15. Computational Mathematics and Cybernetics. 2023. N 1, p.3-13 https://doi.org/10.55959/MSU/0137–0782–15–2023–1–3–13.

This work is licensed under a Сreative Commons Atribiution - NonCommercial 4.0 International (CC BY-NC 4.0)

Baev A.V. (Lomonosov Moscow State University (MSU))

Abstract

Initial problems for the equations of motion of a viscous incompressible fluid and gas in Lagrangian variables are considered. It is shown that the motion of an incompressible fluid is not related to pressure. The pressure in the absence of external forces is constant, what allows the fluid to move freely. This motion is purely vortex in nature and is described by quasi-linear equations of the parabolic type. The existence and uniqueness of the classical periodic solution of the initial problem in Rn at n > 2 is proved. The equations of motion of liquid and gas in steady-state mode are obtained. The problem of turbulent flow of partially compressible liquid and gas is solved. It is established that there is no turbulent flow in an incompressible fluid. It is shown that as a result of frequency synchronization, spatially stable periodic structures arise.