115 research outputs found
Convergence of time averages of weak solutions of the three-dimensional Navier-Stokes equations
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 showed 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.Comment: Version 2: fixed some typos; added some references; and expanded some
sentences and some remarks for the sake of clarit
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