Convergence of Time Averages of Weak Solutions of the Three-Dimensional Navier–Stokes Equations

Abstract

Using the concept of stationary statistical solution, which generalizes the notion of invariant measure, it is proved that, in a suitable sense, time averages of almost every Leray–Hopf weak solution of the three-dimensional incompressible Navier–Stokes equations converge as the averaging time goes to infinity. This system of equations is not known to be globally well-posed, and the above result answers a long-standing problem, extending to this system a classical result from ergodic theory. It is also shown that, from a measure-theoretic point of view, the stationary statistical solution obtained from a generalized limit of time averages is independent of the choice of the generalized limit. Finally, any Borel subset of the phase space with positive measure with respect to a stationary statistical solution is such that for almost all initial conditions in that Borel set and for at least one Leray–Hopf weak solution starting with that initial condition, the corresponding orbit is recurrent to that Borel subset and its mean sojourn time within that Borel subset is strictly positive.