Trajectory Statistical Solutions for three-dimensional Navier-Stokes-like systems

TL;DR

This paper develops a general abstract framework for statistical solutions of trajectory spaces applicable to various incompressible viscous flow equations, extending the theory beyond Navier-Stokes to other evolution problems.

Contribution

It introduces a broad, abstract approach to statistical solutions on trajectory spaces, applicable to a wide class of evolution equations including Navier-Stokes and related systems.

Findings

Framework applies to 3D Navier-Stokes and similar equations.

Existence of statistical solutions under general topological conditions.

Illustrated with the Bénard convection problem.

Abstract

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 current theory of statistical solutions to an abstract level that applies to other evolution equations with properties similar to those of the three-dimensional Navier-Stokes equations. The applicability of the theory is illustrated with the B\'enard problem of convection in fluids.