Irreducibility of the 3-D stochastic Navier-Stokes equation
The author considers a three-dimensional Navier-Stokes equation with random body force. The author proves 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. This result relies on a controllability property and on the continuity of the mapping noise \(\mapsto\) solution along the controllers. The most relevant fact seems to be the latter property, taking into account that it is not known if the three-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 the data, in the class of weak solutions, around regular solutions. The author uses 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.
scientific article; zbMATH DE number 1078495
scientific article