Non-unique ergodicity for deterministic and stochastic 3D Navier--Stokes and Euler equations

TL;DR

This paper proves the existence of infinitely many stationary and ergodic solutions to 3D Navier--Stokes and Euler equations, including stochastic cases, using a new stochastic convex integration approach that links solutions across viscosity limits.

Contribution

It introduces a stochastic convex integration method to construct multiple stationary solutions for 3D Navier--Stokes and Euler equations, including in stochastic settings.

Findings

Existence of infinitely many stationary solutions for Navier--Stokes and Euler equations.

Construction of ergodic stationary solutions with specific regularity.

Stationary solutions to Euler are limits of Navier--Stokes solutions as viscosity vanishes.

Abstract

We establish the existence of infinitely many stationary solutions, as well as ergodic 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 $C(\mathbb{R};H^{\vartheta})\cap C^{\vartheta}(\mathbb{R};L^{2})$ for some $\vartheta>0$ and satisfy the equations in an analytically weak sense. The solutions to the Euler equations are obtained as vanishing viscosity limits of stationary solutions to the Navier--Stokes equations. Furthermore, regardless of their construction, every stationary solution to the Euler equations within this regularity class, which satisfies a suitable moment bound, is a limit in law of 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 locally in the aforementioned function spaces.