Homogenization of trajectory attractors of 3D Navier-Stokes system with randomly oscillating force

Kuanysh A. Bekmaganbetov; Gregory A. Chechkin; Vladimir V. Chepyzhov; Andrey Yu. Goritsky

doi:10.3934/dcds.2017103

Article Contents

2017, Volume 37, Issue 5: 2375-2393. Doi: 10.3934/dcds.2017103

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Homogenization of trajectory attractors of 3D Navier-Stokes system with randomly oscillating force

1.
Kazakhstan Branch of M.V.Lomonosov Moscow State University, Kazhymukan st., 11, Astana, 010010, Kazakhstan
2.
M.V.Lomonosov Moscow State University, Moscow, 119991, Russia
3.
Institute for Information Transmission Problems, Russian Academy of Sciences, Bolshoy Karetniy 19, Moscow 127051, Russia
4.
National Research University Higher School of Economics, Myasnitskaya Street 20, Moscow 101000, Russia

* Corresponding author: G.A.Chechkin
* Corresponding author: G.A.Chechkin

Received: December 31, 2015

Revised: November 30, 2016

Published: May 2017

Work of KAB is partially supported by the Committee of Science of the Ministry of Education and Science of the Republic of Kazakhstan (grant no. 0816/GF4). Work of GAC, VVC, and AYG is partially supported by the Russian Foundation for Basic Research (projects 15-01-07920 and 14-01-00346).

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Abstract

We consider the 3D Navier-Stokes systems with randomly rapidly oscillating right-hand sides. Under the assumption that the random functions are ergodic and statistically homogeneous in space variables or in time variables we prove that the trajectory attractors of these systems tend to the trajectory attractors of homogenized 3D Navier-Stokes systems whose right-hand sides are the average of the corresponding terms of the original systems. We do not assume that the Cauchy problem for the considered 3D Navier-Stokes systems is uniquely solvable.

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Mathematics Subject Classification: Primary: 35B40, 35B41, 35B45, 35Q30.

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