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.
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
\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.
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\zmath{https://zbmath.org/?q=an:1161.35461}
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\transl
\jour Proc. Steklov Inst. Math.
\yr 2007
\vol 259
\pages 128--136
\crossref{https://doi.org/10.1134/S0081543807040098}
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Linking options:
https://www.mathnet.ru/eng/tm573
https://www.mathnet.ru/eng/tm/v259/p134
This publication is cited in the following 8 articles:
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
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
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
Nathan Glatt-Holtz, Vladimír Šverák, Vlad Vicol, “On Inviscid Limits for the Stochastic Navier–Stokes Equations and Related Models”, Arch Rational Mech Anal, 217:2 (2015), 619
Jonathan C. Mattingly, Etienne Pardoux, “Invariant measure selection by noise. An example”, DCDS-A, 34:10 (2014), 4223
Kuksin S.B., “Damped-driven KdV and effective equations for long-time behaviour of its solutions”, Geom. Funct. Anal., 20:6 (2010), 1431–1463
Kuksin S.B., “Dissipative Perturbations of KdV”, Xvith International Congress on Mathematical Physics, 2010, 323–327
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