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 Uspekhi Mat. Nauk, 2002, Volume 57, Issue 4(346), Pages 151–166 (Mi umn536)

Analyticity of solutions for randomly forced two-dimensional Navier–Stokes equations

A. R. Shirikyan

Heriot Watt University

Abstract: A study is made of randomly forced two-dimensional Navier–Stokes equations with periodic boundary conditions. Under the assumption that the random forcing is analytic in the spatial variables and is a white noise in the time, it is proved that a large class of solutions, which contains all stationary solutions with finite energy, admits analytic continuation to a small complex neighbourhood of the torus. Moreover, a lower bound is obtained for the radius of analyticity in terms of the viscosity $\nu$, and it is shown that the Kolmogorov dissipation scale can be asymptotically estimated below by $\nu^{2+\delta}$ for any $\delta>0$.

DOI: https://doi.org/10.4213/rm536

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English version:
Russian Mathematical Surveys, 2002, 57:4, 785–799

Bibliographic databases:

UDC: 517.95
MSC: Primary 35Q30, 35R60; Secondary 60H15, 35B65, 76D05

Citation: A. R. Shirikyan, “Analyticity of solutions for randomly forced two-dimensional Navier–Stokes equations”, Uspekhi Mat. Nauk, 57:4(346) (2002), 151–166; Russian Math. Surveys, 57:4 (2002), 785–799

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/umn536
• https://doi.org/10.4213/rm536
• http://mi.mathnet.ru/eng/umn/v57/i4/p151

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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. Kuksin S., Shirikyan A., “Some limiting properties of randomly forced two-dimensional Navier–Stokes equations”, Proc. Roy. Soc. Edinburgh Sect. A, 133:4 (2003), 875–891
2. Kuksin S., Shirikyan A., “Randomly forced CGL equation: stationary measures and the inviscid limit”, J. Phys. A, 37:12 (2004), 3805–3822
3. Lord G.J., Rougemont J., “A numerical scheme for stochastic PDEs with Gevrey regularity”, IMA J. Numer. Anal., 24:4 (2004), 587–604
4. Kuksin S.B., “The Eulerian limit for 2D statistical hydrodynamics”, J. Statist. Phys., 115:1-2 (2004), 469–492
5. Shirikyan A., “Ergodicity for a class of Markov processes and applications to randomly forced PDE's. I”, Russ. J. Math. Phys., 12:1 (2005), 81–96
6. Shirikyan A., “Law of large numbers and central limit theorem for randomly forced PDE's”, Probab. Theory Related Fields, 134:2 (2006), 215–247
7. Odasso C., “Spatial smoothness of the stationary solutions of the 3D Navier–Stokes equations”, Electron. J. Probab., 11 (2006), 686–699
8. Odasso C., “Exponential mixing for the 3D stochastic Navier–Stokes equations”, Comm. Math. Phys., 270:1 (2007), 109–139
9. Matthias Morzfeld, Xuemin Tu, Ethan Atkins, Alexandre J. Chorin, “A random map implementation of implicit filters”, Journal of Computational Physics, 2011
10. Hieber M., Stannat W., “Stochastic Stability of the Ekman Spiral”, Ann. Scuola Norm. Super. Pisa-Cl. Sci., 12:1 (2013), 189–208
11. Shao J., Guo B., Duan L., “Analytical Study of the Two-Dimensional Time-Fractional Navier-Stokes Equations”, J. Appl. Anal. Comput., 9:5 (2019), 1999–2022
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