On the convergence of trajectory statistical solutions

TL;DR

This paper investigates the convergence of trajectory statistical solutions within a general framework, providing conditions for convergence and applying results to fluid dynamics models like Navier-Stokes and Euler equations.

Contribution

It establishes new convergence criteria for trajectory statistical solutions and applies these to key fluid dynamics problems, including inviscid limits and Galerkin approximations.

Findings

Convergence conditions depend on a priori estimates and compactness assumptions.

Existence of trajectory statistical solutions for 2D and 3D Euler equations.

Application to Galerkin approximations of 3D Navier-Stokes solutions.

Abstract

In this work, a recently introduced general framework for trajectory statistical solutions is considered, and the question of convergence of families of such solutions is addressed. Conditions for the convergence are given which rely on natural assumptions related to a priori estimates for the individual solutions of typical approximating problems. The first main result is based on the assumption that the superior limit of suitable families of compact subsets of carriers of the family of trajectory statistical solutions be included in the set of solutions of the limit problem. The second main result is a version of the former in the case in which the approximating family is associated with a well-posed system. These two results are then applied to the inviscid limit of incompressible Navier-Stokes system in two and three spatial dimensions, showing, in particular, the existence of trajectory statistical solutions to the two- and three-dimensional Euler equations, in the context of weak and dissipative solutions, respectively. Another application of the second main result is on the Galerkin approximations of statistical solutions of the three-dimensional Navier-Stokes equations.