A 3-dimensional Navier Stokes equation with random force is investigated. A form of irreducibility, of interest in ergodic theory, is proved, under a full noise assumption. The basic tool is the fact that, even if the equation is a priori non-well-posed, the solutions depend continuously on the noise around regular solutions. 1997 Academic Press
1. Introduction
In this paper we consider a 3-dimensional Navier Stokes equation with random body force. The aim is to prove that, if the noise affects all the different modes, then the probability distribution of any weak solution at any positive time is full in the energy space H; i.e., its support in the H-topology is H itself. This is a property of irreducibility, in the language of ergodic theory. We can prove this result only if the initial condition is in the Sobolev space H 1 , but perhaps this technical condition can be avoided with more clever estimates.
This result relies on a controllability property and on the continuity of the mapping noise [ solution along the controllers. The most relevant fact seems to be the latter property, taking into account that it is not known if the 3-dimensional Navier Stokes equation is well posed. It is well known that regular solutions are unique also in the class of weak solutions; similarly, the solution depends continuously on data, in the class of weak solutions, around regular solutions. We use this fact, along with the regularity of the trajectories involved in the controllability argument. In view of the previous remarks, it seems that the irreducibility result proved in this paper is typical of equations which are well posed, in a sense, around regular solutions, while it does not hold for any differential equation just as a consequence of the assumption that the noise affects all modes.
The present paper may be related conceptually to some investigations of Fursikov (see for instance ), although we cannot do any more precise comparison.