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 Trudy Mat. Inst. Steklova, 2007, Volume 259, Pages 134–142 (Mi tm573)

Eulerian Limit for 2D Navier–Stokes Equation and Damped/Driven KdV Equation as Its Model

S. B. Kuksinab

a Steklov Mathematical Institute, Russian Academy of Sciences
b Department of Mathematics, Heriot Watt University

Abstract: 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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Proceedings of the Steklov Institute of Mathematics, 2007, 259, 128–136

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Citation: S. B. Kuksin, “Eulerian Limit for 2D Navier–Stokes Equation and Damped/Driven KdV Equation as Its Model”, Analysis and singularities. Part 2, Collected papers. Dedicated to academician Vladimir Igorevich Arnold on the occasion of his 70th birthday, Trudy Mat. Inst. Steklova, 259, Nauka, MAIK «Nauka/Inteperiodika», M., 2007, 134–142; Proc. Steklov Inst. Math., 259 (2007), 128–136

Citation in format AMSBIB
\Bibitem{Kuk07} \by S.~B.~Kuksin \paper Eulerian Limit for 2D Navier--Stokes Equation and Damped/Driven KdV Equation as Its Model \inbook Analysis and singularities. Part~2 \bookinfo Collected papers. Dedicated to academician Vladimir Igorevich Arnold on the occasion of his 70th birthday \serial Trudy Mat. Inst. Steklova \yr 2007 \vol 259 \pages 134--142 \publ Nauka, MAIK «Nauka/Inteperiodika» \publaddr M. \mathnet{http://mi.mathnet.ru/tm573} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2433681} \zmath{https://zbmath.org/?q=an:1161.35461} \elib{https://elibrary.ru/item.asp?id=9572732} \transl \jour Proc. Steklov Inst. Math. \yr 2007 \vol 259 \pages 128--136 \crossref{https://doi.org/10.1134/S0081543807040098} \elib{https://elibrary.ru/item.asp?id=13565182} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-38849147161} 

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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.B., “On distribution of energy and vorticity for solutions of 2D Navier–Stokes equation with small viscosity”, Comm. Math. Phys., 284:2 (2008), 407–424
2. Kuksin S.B., “Damped-driven KdV and effective equations for long-time behaviour of its solutions”, Geom. Funct. Anal., 20:6 (2010), 1431–1463
3. Kuksin S.B., “Dissipative Perturbations of KdV”, Xvith International Congress on Mathematical Physics, 2010, 323–327
4. Bakhtin Yu., “Ergodic Theory of the Burgers Equation”, Probability and Statistical Physics in St. Petersburg, Proceedings of Symposia in Pure Mathematics, 91, eds. Sidoravicius V., Smirnov S., Amer Mathematical Soc, 2016, 1–49
5. Ekren I., Kukavica I., Ziane M., “Existence of Invariant Measures For the Stochastic Damped KdV Equation”, Indiana Univ. Math. J., 67:3 (2018), 1221–1254
6. Bakhtin Yu., Li L., “Zero Temperature Limit For Directed Polymers and Inviscid Limit For Stationary Solutions of Stochastic Burgers Equation”, J. Stat. Phys., 172:5 (2018), 1358–1397
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