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This article is cited in 38 scientific papers (total in 38 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.
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”, Russian Math. Surveys, 58:2 (2003), 287–318
Linking options:
https://www.mathnet.ru/eng/rm611https://doi.org/10.1070/RM2003v058n02ABEH000611 https://www.mathnet.ru/eng/rm/v58/i2/p79
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