Non-unique Ergodicity for Deterministic and Stochastic 3D Navier–Stokes and Euler Equations

We establish the existence of infinitely many statistically stationary solutions, as well as ergodic statistically stationary solutions, to the three dimensional Navier–Stokes and Euler equations in both deterministic and stochastic settings, driven by additive noise. These solutions belong to the regularity class C(R;Hϑ)∩Cϑ(R;L2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C({{\mathbb {R}}};H^{\vartheta })\cap C^{\vartheta }({{\mathbb {R}}};L^{2})$$\end{document} for some ϑ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vartheta >0$$\end{document} and satisfy the equations in an analytically weak sense. The solutions to the Euler equations are obtained as vanishing viscosity limits of statistically stationary solutions to the Navier–Stokes equations. Furthermore, regardless of their construction, every statistically stationary solution to the Euler equations within this regularity class, which satisfies a suitable moment bound, is a limit in law of statistically stationary analytically weak solutions to Navier–Stokes equations with vanishing viscosities. Our results are based on a novel stochastic version of the convex integration method, which provides uniform moment bounds in the aforementioned function spaces.