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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.

DOI: https://doi.org/10.4213/rm611

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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
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    2. J. Math. Sci. (N. Y.), 136:2 (2006), 3768–3777  mathnet  crossref  mathscinet  zmath
    3. Bellout H., Neustupa J., Penel P., “On the Navier–Stokes equation with boundary conditions based on vorticity”, Math. Nachr., 269 (2004), 59–72  crossref  mathscinet  zmath  isi
    4. Mahalov A., Nicolaenko B., Bardos C., Golse F., “Regularity of Euler equations for a class of three-dimensional initial data”, Trends in Partial Differential Equations of Mathematical Physics, Progress in Nonlinear Differential Equations and their Applications, 61, 2005, 161–185  mathscinet  zmath  isi
    5. Saal J., “Maximal regularity for the Stokes system on noncylindrical space-time domains”, J. Math. Soc. Japan, 58:3 (2006), 617–641  crossref  mathscinet  zmath  isi
    6. Chemin J.-Y., Gallagher I., “On the global wellposedness of the 3-D Navier–Stokes equations with large initial data”, Ann. Sci. École Norm. Sup. (4), 39:4 (2006), 679–698  crossref  mathscinet  zmath  isi
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    10. Mucha P.B., “Stability of 2D incompressible flows in $\mathbf R^3$”, J. Differential Equations, 245:9 (2008), 2355–2367  crossref  mathscinet  zmath  adsnasa  isi  elib
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    12. Saks R.S., Khaybullin A.G., “One method for the numerical solution of the Cauchy problem for the Navier–Stokes equations and Fourier series of the curl operator”, Dokl. Math., 80:3 (2009), 800–805  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    13. Saks R.S., “Explicit global solutions to the Navier–Stokes equations and periodic eigenfunctions of the curl operator”, Dokl. Math., 79:1 (2009), 35–40  mathnet  crossref  mathscinet  zmath  isi  elib  elib
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