Geodesic
Eulerian Limit for 2D Navier--Stokes Equation and Damped/Driven KdV Equation as Its Model
Informatics and Automation, Analysis and singularities. Part 2, Tome 259 (2007), pp. 134-142.

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We discuss the inviscid limits for the randomly forced 2D Navier–Stokes equation (NSE) and the damped/driven KdV equation. The former describes the space-periodic 2D turbulence in terms of a special class of solutions for the free Euler equation, and we view the latter as its model. We review and revise recent results on the inviscid limit for the perturbed KdV and use them to suggest a setup which could be used to make a next step in the study of the inviscid limit of 2D NSE. The proposed approach is based on an ergodic hypothesis for the flow of the 2D Euler equation on iso-integral surfaces. It invokes a Whitham equation for the 2D Navier–Stokes equation, written in terms of the ergodic measures. 
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S. B. Kuksin. Eulerian Limit for 2D Navier--Stokes Equation and Damped/Driven KdV Equation as Its Model. Informatics and Automation, Analysis and singularities. Part 2, Tome 259 (2007), pp. 134-142. http://geodesic.mathdoc.fr/item/TRSPY_2007_259_a8/

[1] Arnold V. I., Khesin B. A., Topological methods in hydrodynamics, Springer, New York, 1998 | MR

[2] Arnold V. I., Matematicheskie metody klassicheskoi mekhaniki, 2-e izd., Nauka, M., 1979 | MR

[3] Beale J. T., Kato T., Majda A., “Remarks on the breakdown of smooth solutions for the 3-D Euler equations”, Commun. Math. Phys., 94 (1984), 61–66 | DOI | MR | Zbl

[4] Bourbaki N., Élémentes de mathématique, XXV. Pt. 1. Livre 6: Intégration. Ch. 6: Intégration vectorielle, Hermann, Paris, 1959 | Zbl

[5] Dudley R. M., Real analysis and probability, Cambridge Univ. Press, Cambridge, 2002 | MR

[6] Kappeler T., Pöschel J., KdV KAM, Springer, Berlin, 2003 | MR

[7] Kuksin S. B., Piatnitski A. L., Khasminskii–Whitham averaging for randomly perturbed KdV equation, Preprint, 2006 ; http://www.ma.hw.ac.uk/~kuksin/rfpdef/kuk\textunderscore pia.pdf | MR

[8] Kuksin S. B., “The Eulerian limit for 2D statistical hydrodynamics”, J. Stat. Phys., 115 (2004), 469–492 | DOI | MR | Zbl

[9] Kuksin S. B., Randomly forced nonlinear PDEs and statistical hydrodynamics in 2 space dimensions, Eur. Math. Soc., Zürich, 2006 ; mp\textunderscore arc 06-178 | MR | Zbl

[10] McKean H. P., Trubowitz E., “Hill's operator and hyperelliptic function theory in the presence of infinitely many branch points”, Commun. Pure and Appl. Math., 29 (1976), 143–226 | DOI | MR | Zbl