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 dimen- sional 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. The uniqueness of statistical steady states for the Navier-Stokes equations is a less known but nevertheless important problem in the mathematical theory of turbulence. The question is completely open in dimension three, mainly because, due to the lack of uniqueness of the equations, there is no way yet to give meaning to the mathematical objects involved in the subject. In the present paper the property of ergodicity is proved for the finite dimensional approxi- mations of the three-dimensional Navier-Stokes equations, driven by a random force. The same problem has been solved in two dimensions by E and Mattingly (2). Such result can have a qualitative interest for the statistical behaviour of an incompressible fluid. Indeed, if the Kolmogorov theory of turbulence is taken into account, one can believe that the cascade of energy, responsible of the transport of the energy through the scales, is effective in the inertial range so that at smaller scales only the dissipation ends up to be relevant. Hence the long-time statistical properties of the fluid can be sufficiently depicted by the low modes of the velocity field. In some sense, if the ultraviolet cut-off is sufficiently large, in order to capture all the important modes, the corresponding invariant measure gives the real behaviour of the fluid. In view of these considerations, the conclusions of the paper can give also both a hint and a possible starting point for the analysis of the infinite dimensional case. We consider a finite dimensional truncation of the three dimensional Navier-Stokes equations, driven by a random force, with periodic boundary conditions. The proof of ergodicity is classical and it is developed in two steps. Firstly we prove that the transition probability densities are regular, by checking that the diffusion operator is hypoelliptic (the Hormander condition). Then we show that the Markov process is irreducible, in the sense that each open set is visited with positive probability at each time. For this aim we study the associated control problem (see Section 6). Irreducibility for the infinite dimensional equations was firstly proved by Flandoli (4), under the assumption that the noise acts on all modes.