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 Uspekhi Mat. Nauk, 2004, Volume 59, Issue 6(360), Pages 55–72 (Mi umn795)

On solutions with infinite energy and enstrophy of the Navier–Stokes system

Yu. Yu. Bakhtina, E. I. Dinaburga, Ya. G. Sinaibc

a International Institute of Earthquake Prediction Theory and Mathematical Geophysics RAS
b L. D. Landau Institute for Theoretical Physics, Russian Academy of Sciences
c Princeton University, Department of Mathematics

Abstract: The Cauchy problem is considered for the Navier–Stokes system. Local and global existence and uniqueness theorems are given for initial data whose Fourier transform decays at infinity as a power-law function with negative exponent and has a power-law singularity at zero. The paper contains a survey of known facts and some new results.

DOI: https://doi.org/10.4213/rm795

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English version:
Russian Mathematical Surveys, 2004, 59:6, 1061–1078

Bibliographic databases:

UDC: 517.957
MSC: Primary 35Q30; Secondary 35A05, 35A07, 76D05

Citation: Yu. Yu. Bakhtin, E. I. Dinaburg, Ya. G. Sinai, “On solutions with infinite energy and enstrophy of the Navier–Stokes system”, Uspekhi Mat. Nauk, 59:6(360) (2004), 55–72; Russian Math. Surveys, 59:6 (2004), 1061–1078

Citation in format AMSBIB
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\paper On solutions with infinite energy and enstrophy of the Navier--Stokes system
\jour Uspekhi Mat. Nauk
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\vol 59
\issue 6(360)
\pages 55--72
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\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2138467}
\zmath{https://zbmath.org/?q=an:1083.76012}
\transl
\jour Russian Math. Surveys
\yr 2004
\vol 59
\issue 6
\pages 1061--1078
\crossref{https://doi.org/10.1070/RM2004v059n06ABEH000795}
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• http://mi.mathnet.ru/eng/umn795
• https://doi.org/10.4213/rm795
• http://mi.mathnet.ru/eng/umn/v59/i6/p55

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Sinai Ya., “Power series for solutions of the 3D-Navier–Stokes system on $R^3$”, J. Stat. Phys., 121:5-6 (2005), 779–803
2. M. D. Arnold, “Local existence theorem for the solutions of the $d$-dimensional system of Navier–Stokes equations”, Russian Math. Surveys, 60:3 (2005), 562–563
3. Sinai Ya., “Absence of the local existence theorem in the critical space for the 3D-Navier–Stokes system”, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 15:11 (2005), 3635–3637
4. Bakhtin Yu., “Existence and uniqueness of stationary solutions for 3D Navier–Stokes system with small random forcing via stochastic cascades”, J. Stat. Phys., 122:2 (2006), 351–360
5. Sinai Ya., “A new approach to the study of the 3D-Navier–Stokes system”, Prospects in Mathematical Physics, Contemporary Mathematics Series, 437, 2007, 223–229
6. Albeverio S., Ferrario B., “Some methods of infinite dimensional analysis in hydrodynamics: An introduction”, SPDE in hydrodynamic: recent progress and prospects, Lecture Notes in Math., 1942, Springer, Berlin, 2008, 1–50
7. Orum Ch., Ossiander M., “Exponent Bounds for a Convolution Inequality in Euclidean Space with Applications to the Navier–Stokes Equations”, Proc. Amer. Math. Soc., 141:11 (2013), 3883–3897
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