A specifically nonlinear property of the operator semigroup of the Navier-Stokes equations

Foias (1974) and Foias and Temam (1975, 1979) have found that for the two-dimensional Navier-Stokes equations the actual possible flows constitute very thin sets in the usual function phase space associated with the corresponding Cauchy problem. However, the same is true for the Stokes equations. The present investigation is concerned with a probabilistic property of thinness which is valid for the Navier-Stokes equations, but not for the Stokes equations. It is shown that for generic body forces the set of all possible flows at time t greater than zero is of null measure with respect to a large class of Gaussian measures on the function phase space, while for the Stokes equations this is not possible for any body force.