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Uspekhi Mat. Nauk, 2003, Volume 58, Issue 2(350), Pages 79–110 (Mi umn611)  

This article is cited in 32 scientific papers (total in 32 papers)

Global solubility of the three-dimensional Navier–Stokes equations with uniformly large initial vorticity

A. S. Makhalov, V. P. Nikolaenko

Arizona State University

Abstract: This paper is a survey of results concerning the three-dimensional Navier–Stokes and Euler equations with initial data characterized by uniformly large vorticity. The existence of regular solutions of the three-dimensional Navier–Stokes equations on an unbounded time interval is proved for large initial data both in $\mathbb R^3$ and in bounded cylindrical domains. Moreover, the existence of smooth solutions on large finite time intervals is established for the three-dimensional Euler equations. These results are obtained without additional assumptions on the behaviour of solutions for $t>0$. Any smooth solution is not close to any two-dimensional manifold. Our approach is based on the computation of singular limits of rapidly oscillating operators, non-linear averaging, and a consideration of the mutual absorption of non-linear oscillations of the vorticity field. The use of resonance conditions, methods from the theory of small divisors, and non-linear averaging of almost periodic functions leads to the limit resonant Navier–Stokes equations. Global solubility of these equations is proved without any conditions on the three-dimensional initial data. The global regularity of weak solutions of three-dimensional Navier–Stokes equations with uniformly large vorticity at $t=0$ is proved by using the regularity of weak solutions and the strong convergence.


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English version:
Russian Mathematical Surveys, 2003, 58:2, 287–318

Bibliographic databases:

UDC: 517.9
MSC: Primary 35Q30, 35A05; Secondary 35B65, 35B34, 35D10, 76D05, 76D06
Received: 15.02.2003

Citation: A. S. Makhalov, V. P. Nikolaenko, “Global solubility of the three-dimensional Navier–Stokes equations with uniformly large initial vorticity”, Uspekhi Mat. Nauk, 58:2(350) (2003), 79–110; Russian Math. Surveys, 58:2 (2003), 287–318

Citation in format AMSBIB
\by A.~S.~Makhalov, V.~P.~Nikolaenko
\paper Global solubility of the three-dimensional Navier--Stokes equations with uniformly large initial vorticity
\jour Uspekhi Mat. Nauk
\yr 2003
\vol 58
\issue 2(350)
\pages 79--110
\jour Russian Math. Surveys
\yr 2003
\vol 58
\issue 2
\pages 287--318

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