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) for some ϑ>0 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.

Publication:

JOURNAL OF COMPUTATIONAL PHYSICS

http://dx.doi.org/10.1016/j.jcp.2025.113911

Author:

Yong Liu

State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science

School of Mathematical Science, University of Chinese Academy of Sciences, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China

yongliu@lsec.cc.ac.cn

Jianfang Lu

School of Mathematics, South China University of Technology, Canton, Guangdong 510641, People’s Republic of China

jflu@scut.edu.cn

Chi-Wang Shu

Division of Applied Mathematics, Brown University, Providence, RI 02912, USA

chi-wang_shu@brown.edu