ON BEHAVIOR OF TRAJECTORIES OF WEAK SOLUTIONS OF N-DIMENSIONAL STOCHASTIC NAVIER-STOKES EQUATIONS

Abstract

The research paper study a behavior of trajectories of weak solutions of n-dimensional (n≥2) Navier-Stokes equations, perturbed by an additive white noise. It is shown that at any given moment t trajectories in its subsequent motion along the phase space of the system: a) inevitably leave any bounded subset of the phase space; b) inevitably return to some compact set K, depending on the viscosity and on the external forces acting on the system. Thus, it is established that the trajectories alternately go away arbitrarily far from the mentioned set K, then again return to it, i.e. the recurrence of trajectories in relation to the set K and infinity. These results are obtained by estimating the mathematical expectation of the moments of the first exit of trajectory after t from the corresponding subsets of the phase space.