Ergodicity of the Finite Dimensional Approximation of the 3D Navier–Stokes Equations Forced by a Degenerate Noise

Abstract

We prove ergodicity of the finite dimensional approximations of the three dimensional Navier-Stokes equations, driven by a random force. The forcing noise acts only on a few modes and some algebraic conditions on the forced modes are found that imply the ergodicity. The convergence rate to the unique invariant measure is shown to be exponential.2000 Mathematics Subject Classification. Primary 76D05; Secondary 35Q30, 76M35, 76F55. 1 2 M. ROMITOBoth these properties, strong Feller and irreducibility, are implied by an algebraic condition on the set of indices corresponding to the modes forced by the noise. The condition essentially means that it is possible to obtain any index as a sum of some of the forced indices. One can see this mechanism as a geometrical realisation of the cascade of energy, since the non-linear term transmits the random forcing from the few forced modes to all the other modes. As an example we show that the algebraic condition is satisfied if the three lowest modes are forced.Recently, many authors have applied the techniques we have used here, such us the hypoellipticity for degenerate diffusions, or the general theory for Markov chains and Markov processes collected and developed by Meyn and Tweedie (one can see their book [10]). Among many others we quote the papers by E and Mattingly [2], Eckmann and Hairer [3], Hairer [6], Rey-Bellet and Thomas [13], and some of the references therein.The paper is organised as follows. In the first section the main definitions are given, together with the statements of the main results and an outline of their proofs. The technical computations and the precise statement of some hypotheses are postponed in the following sections. The aim is to give a light presentation of the main ideas, without all the technicalities, which are then reserved to the interested readers.Acknowledgements. The author wish to thanks R. Bianchini and S. Dolfi for the helpful bibliographical suggestions on the control theory part in Section 6 and on the algebraic part in Section 5, and F. Flandoli for the many helpful conversations. This paper is dedicated to the memory of my father, who died whilst I was writing it. The main theoremWe consider the stochastic Navier-Stokes equations with additive noise du = (ν∆u − (u · ∇)u − ∇P ) dt + dB t div u = 0,

On recent progress for the stochastic Navier Stokes equations

“…A theorem similar to Theorem 9, but for the three dimensional Galerkin approximation, is proven in [Rom02]. There he proves even more; he shows that the system is actually globally controllable.…”

“…Namely that the off-diagonal nature of the nonlinearity leaves the system globally consolable even though its nonlinearity is even powered. We refer the reader to [Rom02] and [AS03] for the precise statement of the results.…”

Some Rigorous Results on a Stochastic GOY Model

“…In view of our application to K41 theory it would be interesting to prove a more difficult result, namely ergodicity when only very few modes are activated directly by the noise. There is hope to get such a result either for the finite dimensional Galerkin approximations, following [8], [26], or even for the infinite dimensional problem following [15]. However, especially the second result, that would fit with our framework, requires a long preparatory work of Malliavin calculus (see [24,23]) that requires careful investigation beyond the scope of the present work.…”

Navier?Stokes Equations: Controllability by Means of Low Modes Forcing

“…, which is not saturating according to the Denition 11, but still is sucient for guaranteeing global controllability (see Remark 7.1) of Galerkin approximations of 3D NS systems, has been provided by M.Romito in [16]. He has proven that the set of controlled forcing modes…”