Rigorous results in space-periodic two-dimensional turbulence

Sergei Kuksin (1, 2) , Armen Shirikyan (3, 4)

1 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche
2 CNRS - Centre National de la Recherche Scientifique
3 AGM - UMR 8088 - Analyse, Géométrie et Modélisation
4 CRM - Centre de Recherches Mathématiques [Montréal]

We survey the recent advance in the rigorous qualitative theory of the 2d stochastic Navier–Stokes system that are relevant to the description of turbulence in two-dimensional fluids. After discussing briefly the initial-boundary value problem and the associated Markov process, we formulate results on the existence, uniqueness and mixing of a stationary measure. We next turn to various consequences of these properties: strong law of large numbers, central limit theorem, and random attractors related to a unique stationary measure. We also discuss the Donsker–Varadhan and Freidlin–Wentzell type large deviations, as well as the inviscid limit and asymptotic results in 3d thin domains. We conclude with some open problems.

Mots clés: 2d Navier–Stokes system; stationary measure; exponential mixing; strong law of large numbers; central limit theorem; random attractors; large deviations; inviscid limit

Domaines: Physique mathématique [math-ph]; Dynamique des Fluides [physics.flu-dyn]; Equations aux dérivées partielles [math.AP]; Probabilités [math.PR]

Sergei Kuksin, Armen Shirikyan. Rigorous results in space-periodic two-dimensional turbulence. Physics of Fluids, 2017, 29, pp.125106. ⟨hal-01673113⟩