The Fourth Workshop on Fluids and PDE was held at the National Institute of Pure and Applied Mathematics (IMPA) in Rio de Janeiro, Brazil, from Monday 26 May to Friday 30 May 2014. This workshop is held every two to three years in Brazil. The fourth edition of the workshop was the closing event of a Thematic Program on Incompressible Fluids Dynamics, to be held at IMPA next Spring. Hence, the focus of the workshop will be incompressible fluid mechanics .

Using the concept of stationary statistical solution, we prove 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. In particular, this implies that, from a measure-theoretic point of view, the stationary statistical solution obtained from a generalized limit of time averages of a Leray-Hopf weak solution is, in general, independent of the choice of the generalized limit. We also show that for 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 and for at least one Leray-Hopf weak solution starting with that initial condition, the orbit is recurrent to that Borel subset and is such that the corresponding mean sojourn time of that solution within that Borel subset is strictly positive. This is a joint work with Ciprian Foias and Roger Temam .