We discuss recent results on the inviscid limits for the randomly forced 2D Navier-Stokes equation under periodic boundary conditions, their relevance for the theory of stationary space periodic 2D turbulence and some related conjectures.
@article{SEDP_2006-2007____A7_0,
author = {Kuksin, Sergei B.},
title = {Rigorous results and conjectures on stationary space-periodic {2D} turbulence},
journal = {S\'eminaire \'Equations aux d\'eriv\'ees partielles (Polytechnique) dit aussi "S\'eminaire Goulaouic-Schwartz"},
note = {talk:7},
publisher = {Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
year = {2006-2007},
mrnumber = {2385194},
language = {en},
url = {http://www.numdam.org/item/SEDP_2006-2007____A7_0/}
}
TY - JOUR AU - Kuksin, Sergei B. TI - Rigorous results and conjectures on stationary space-periodic 2D turbulence JO - Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" N1 - talk:7 PY - 2006-2007 DA - 2006-2007/// PB - Centre de mathématiques Laurent Schwartz, École polytechnique UR - http://www.numdam.org/item/SEDP_2006-2007____A7_0/ UR - https://www.ams.org/mathscinet-getitem?mr=2385194 LA - en ID - SEDP_2006-2007____A7_0 ER -
Kuksin, Sergei B. Rigorous results and conjectures on stationary space-periodic 2D turbulence. Séminaire Équations aux dérivées partielles (Polytechnique) dit aussi "Séminaire Goulaouic-Schwartz" (2006-2007), Exposé no. 7, 16 p. http://www.numdam.org/item/SEDP_2006-2007____A7_0/
[AK01] V. Arnold and B. Khesin, Topological Methods in Hydrodynamics, Springer-Verlag, Berlin, 2001. | MR 1612569 | Zbl 0902.76001
[Dud02] R. M. Dudley, Real Analysis and Probability, Cambridge University Press, Cambridge, 2002. | MR 1932358 | Zbl 1023.60001
[FW98] M. Freidlin and A. Wentzell, Random Perturbations of Dynamical Systems, 2nd ed., Springer-Verlag, New York, 1998. | MR 1652127 | Zbl 0922.60006
[HM06] M. Hairer and J. Mattingly, Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing, Annals of Mathematics 164 (2006), no. 3. | MR 2259251 | Zbl 1130.37038
[KP03] T. Kappeler and J. Pöschel, KAM & KdV, Springer, 2003.
[KP05] S. B. Kuksin and O. Penrose, A family of balance relations for the two-dimensional Navier–Stokes equations with random forcing, J. Stat. Physics 118 (2005), 437–449. | MR 2123643 | Zbl 1064.76027
[KP06] S. B. Kuksin and A. L. Piatnitski, Khasminskii - Whitham averaging for randomly perturbed KdV equation, Preprint, see mparc 06-313 (2006). | MR 2225710
[Kry80] N. V. Krylov, Controlled Diffusion Processes, Springer, 1980. | MR 601776 | Zbl 0459.93002
[KS04] S. B. Kuksin and A. Shirikyan, Randomly forced CGL equation: stationary measures and the inviscid limit, J. Phys. A: Math. Gen. 37 (2004), 1–18. | MR 2039838 | Zbl 1047.35061
[Kuk04] S. B. Kuksin, The Eulerian limit for 2D statistical hydrodynamics, J. Stat. Physics 115 (2004), 469–492. | MR 2070104 | Zbl 1157.76319
[Kuk06a] —, Randomly Forced Nonlinear PDEs and Statistical Hydrodynamics in 2 Space Dimensions, Europear Mathematical Society Publishing House, 2006, also see mp_arc 06-178. | MR 2225710 | Zbl 1099.35083
[Kuk06b] —, Remarks on the balance relations for the two-dimensional Navier–Stokes equation with random forcing, J. Stat. Physics 122 (2006), 101–114. | MR 2203784 | Zbl 1089.76013
[Kuk07a] —, Eulerian limit for 2D Navier-Stokes equation and damped/driven KdV equation as its model, preprint, see mparc 07-25 (2007).
[Kuk07b] —, On distribution of energy and vorticity for solutions of 2D Navier-Stokes equations with small viscosity, preprint, see mparc 07-60 (2007).
[MT76] H. McKean and E. Trubowitz, Hill’s operator and hyperelliptic function theory in the presence of infinitely many branching points, Comm. Pure Appl. Math. 29 (1976), 143–226. | Zbl 0339.34024
[Par77] K. R. Parthasarathy, Introduction to Probability and Measure, Macmillan, 1977. | MR 651012 | Zbl 0395.28001
