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 Uspekhi Mat. Nauk, 2005, Volume 60, Issue 5(365), Pages 47–70 (Mi umn1642)

Diagrammatic approach to the 3D Navier–Stokes system

Ya. G. Sinaiab

a L. D. Landau Institute for Theoretical Physics, Russian Academy of Sciences
b Princeton University, Department of Mathematics

Abstract: This paper concerns the 3-dimensional Navier–Stokes system (NSS) on $\mathbb R^3$ which describes the dynamics of viscous incompressible fluids without external forcing. For bounded initial conditions with compact support a locally convergent series is constructed which gives the solution of the NSS and whose coefficients are multidimensional integrals called diagrams. Estimates are given for various classes of diagrams and it is shown in particular that simple diagrams decay faster than exponentially.

DOI: https://doi.org/10.4213/rm1642

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English version:
Russian Mathematical Surveys, 2005, 60:5, 849–873

Bibliographic databases:

UDC: 517.957
MSC: Primary 35Q30; Secondary 35C10, 76D07

Citation: Ya. G. Sinai, “Diagrammatic approach to the 3D Navier–Stokes system”, Uspekhi Mat. Nauk, 60:5(365) (2005), 47–70; Russian Math. Surveys, 60:5 (2005), 849–873

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/umn1642
• https://doi.org/10.4213/rm1642
• http://mi.mathnet.ru/eng/umn/v60/i5/p47

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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. Li Dong, Sinai Ya.G., “Complex singularities of solutions of some 1D hydrodynamic models”, Phys. D, 237:14-17 (2008), 1945–1950
2. Li Dong, Sinai Ya.G., “Blow ups of complex solutions of the 3D Navier–Stokes system and renormalization group method”, J. Eur. Math. Soc. (JEMS), 10:2 (2008), 267–313
3. Sinai Ya.G., “Mathematical results related to the Navier–Stokes system”, SPDE in hydrodynamic: recent progress and prospects, Lecture Notes in Math., 1942, Springer, Berlin, 2008, 151–164
4. Sinai Ya.G., Arnold M.D., “Global existence and uniqueness theorem for 3D-Navier–Stokes system on $\mathbb T^3$ for small initial conditions in the spaces $\Phi(\alpha)$”, Pure Appl. Math. Q., 4:1 (2008), 71–79
5. Li Dong, “Existence theorems for the 2D quasi-geostrophic equation with plane wave initial conditions”, Nonlinearity, 22:7 (2009), 1639–1651
6. Bardos C., Frisch U., Pauls W., Ray S.S., Titi E.S., “Entire solutions of hydrodynamical equations with exponential dissipation”, Comm. Math. Phys., 293:2 (2010), 519–543
7. Li D., Sinai Ya.G., “Blowups of complex-valued solutions for some hydrodynamic models”, Regul. Chaotic Dyn., 15:4-5 (2010), 521–531
8. W Pauls, “Some remarks on Li–Sinai-type solutions of the Burgers equation”, J. Phys. A: Math. Theor, 44:28 (2011), 285209
9. Gubinelli M., “Rough solutions for the periodic Korteweg–de Vries equation”, Commun. Pure Appl. Anal., 11:2 (2012), 709–733
10. Nikolai Chernov, Dong Li, “Decay of Fourier modes of solutions to the dissipative surface quasi-geostrophic equations on a finite domain”, Chaos, Solitons & Fractals, 45:9-10 (2012), 1192
11. 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
12. Pauls W. Ray S.S., “Analytic Structure of Solutions of the One-Dimensional Burgers Equation With Modified Dissipation”, J. Phys. A-Math. Theor., 53:11 (2020), 115702
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