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 Uspekhi Mat. Nauk, 2000, Volume 55, Issue 4(334), Pages 25–58 (Mi umn313)

Recent results on mathematical and statistical hydrodynamics

W. Ea, Ya. G. Sinaibc

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

Abstract: This paper is a survey of recent results of the authors and their collaborators on stochastic partial differential equations in hydrodynamics. We discuss the stochastic Burgers equation, the stochastic Navier–Stokes equation, and the stochastic passive scalar transport equation. In contrast to previous publications on this subject (see, for example, [25], which is mainly devoted to existence problems for stochastic dynamics), the work surveyed here emphasizes qualitative properties of solutions, including the existence and uniqueness of an invariant measure under certain physical assumptions, the asymptotic behaviour of the statistics of these solutions, and so on. We also discuss new investigations concerning the deterministic Navier–Stokes equation.

DOI: https://doi.org/10.4213/rm313

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English version:
Russian Mathematical Surveys, 2000, 55:4, 635–666

Bibliographic databases:

UDC: 517.95+519.21
MSC: Primary 35R60, 35Q53, 35Q30; Secondary 35L67, 76D03, 76D05, 76N10, 60H15

Citation: W. E, Ya. G. Sinai, “Recent results on mathematical and statistical hydrodynamics”, Uspekhi Mat. Nauk, 55:4(334) (2000), 25–58; Russian Math. Surveys, 55:4 (2000), 635–666

Citation in format AMSBIB
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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. E. I. Dinaburg, Ya. G. Sinai, “Existence and Uniqueness of Solutions of a Quasilinear Approximation for the Three-Dimensional Navier–Stokes System”, Problems Inform. Transmission, 39:1 (2003), 47–50
2. Mikulevicius R., Rozovskii B.L., “Stochastic Navier–Stokes equations for turbulent flows”, SIAM J. Math. Anal., 35:5 (2003), 1250–1310
3. Zgliczynski P., “On smooth dependence on initial conditions for dissipative PDEs, an ODE-type approach”, J. Differential Equations, 195:2 (2003), 271–283
4. Sinai Ya.G., “Mathematical hydrodynamics”, Russ. J. Math. Phys., 11:3 (2004), 355–358
5. Afendikov A.L., Mielke A., “Dynamical properties of spatially non-decaying 2D Navier–Stokes flows with Kolmogorov forcing in an infinite strip”, J. Math. Fluid Mech., 7, suppl. 1 (2005), S51–S67
6. Wang Bin, Xiang Kainan, Yang Xiangqun, “On a class of measure-valued processes: singular cases”, Sci. China Ser. A, 49:10 (2006), 1315–1326
7. Nolen J., Xin Jack, “A variational principle for KPP front speeds in temporally random shear flows”, Comm. Math. Phys., 269:2 (2006), 493–532
8. Kim Hongjoong, “An efficient computational method for statistical moments of Burger's equation with random initial conditions”, Math. Probl. Eng., 2006, 17406, 21 pp.
9. Menon G., Pego R.L., “Universality classes in burgers turbulence”, Comm. Math. Phys., 273:1 (2007), 177–202
10. 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
11. 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
12. 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, part 2 (2008), 71–79
13. Cortissoz J., “Some elementary estimates for the Navier–Stokes system”, Proc. Amer. Math. Soc., 137:10 (2009), 3343–3353
14. Sango M., “Density Dependent Stochastic Navier–Stokes Equations With Non-Lipschitz Random Forcing”, Reviews in Mathematical Physics, 22:6 (2010), 669–697
15. Govind Menon, “Complete Integrability of Shock Clustering and Burgers Turbulence”, Arch Rational Mech Anal, 2011
16. Menon G., “Lesser Known Miracles of Burgers Equation”, Acta Math Sci Ser B Engl Ed, 32:1 (2012), 281–294
17. Cyranka J., “Existence of Globally Attracting Fixed Points of Viscous Burgers Equation With Constant Forcing. a Computer Assisted Proof”, 45, no. 2, 2015, 655–697
18. Cyranka J., Zgliczynski P., “Stabilizing effect of large average initial velocity in forced dissipative PDEs invariant with respect to Galilean transformations”, J. Differ. Equ., 261:8 (2016), 4648–4708
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