Trajectory Statistical Solutions for Three-Dimensional Navier--Stokes-Like Systems

A general framework for the theory of statistical solutions on trajectory spaces is constructed for a wide range of equations involving incompressible viscous flows. This framework is constructed with a general Hausdorff topological space as the phase space of the system and with the corresponding set of trajectories belonging to the space of continuous paths in that phase space. A trajectory statistical solution is a Borel probability measure defined on the space of continuous paths and carried by a certain subset which is interpreted, in the applications, as the set of solutions of a given problem. The main hypotheses for the existence of a trajectory statistical solution concern the topology of that subset of “solutions,” along with conditions that characterize those solutions within a certain larger subset (a condition related to the assumption of strong continuity at the origin for the Leray--Hopf weak solutions in the case of the Navier--Stokes and related equations). The aim here is to raise the cu...