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

This article is cited in 12 scientific papers (total in 12 papers)

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.


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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
Received: 18.08.2005

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
\by Ya.~G.~Sinai
\paper Diagrammatic approach to the 3D Navier--Stokes system
\jour Uspekhi Mat. Nauk
\yr 2005
\vol 60
\issue 5(365)
\pages 47--70
\jour Russian Math. Surveys
\yr 2005
\vol 60
\issue 5
\pages 849--873

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    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  crossref  mathscinet  zmath  isi  elib
    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  crossref  mathscinet  zmath  isi  elib
    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  crossref  mathscinet  zmath  isi  elib
    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  crossref  mathscinet  zmath  isi  elib
    5. Li Dong, “Existence theorems for the 2D quasi-geostrophic equation with plane wave initial conditions”, Nonlinearity, 22:7 (2009), 1639–1651  crossref  mathscinet  zmath  isi  elib
    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  crossref  mathscinet  zmath  adsnasa  isi
    7. Li D., Sinai Ya.G., “Blowups of complex-valued solutions for some hydrodynamic models”, Regul. Chaotic Dyn., 15:4-5 (2010), 521–531  crossref  mathscinet  zmath  adsnasa  isi  elib
    8. W Pauls, “Some remarks on Li–Sinai-type solutions of the Burgers equation”, J. Phys. A: Math. Theor, 44:28 (2011), 285209  crossref  mathscinet  zmath  isi
    9. Gubinelli M., “Rough solutions for the periodic Korteweg–de Vries equation”, Commun. Pure Appl. Anal., 11:2 (2012), 709–733  crossref  mathscinet  zmath  isi  elib
    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  crossref  mathscinet  zmath  isi
    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  crossref  mathscinet  zmath  isi  elib
    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  crossref  mathscinet  isi
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