We consider the stochastic Navier–Stokes equations in three dimensions and prove that the law of analytically weak solutions is not unique. In particular, we focus on three examples of a stochastic perturbation: an additive, a linear multiplicative and a nonlinear noise of cylindrical type, all driven by a Wiener process. In these settings, we develop a stochastic counterpart of the convex integration method introduced recently by Buckmaster and Vicol. This permits us to construct probabilistically strong and analytically weak solutions defined up to a suitable stopping time. In addition, these solutions fail to satisfy the corresponding energy inequality at a prescribed time with a prescribed probability. Then we introduce a general probabilistic construction used to extend the convex integration solutions beyond the stopping time and in particular to the whole time interval [0,\infty). Finally, we show that their law is distinct from the law of solutions obtained by Galerkin approximation. In particular, nonuniqueness in law holds on an arbitrary time interval [0,T], T>0.

Publication:

Journal of the European Mathematical Society (Volume: 26, 2024)

https://doi.org/10.4171/JEMS/1360

Author:

Martina Hofmanová

Bielefeld University, Germany

Rongchan Zhu

Beijing Institute of Technology, China

Xiangchan Zhu

Chinese Academy of Sciences, Beijing, China

Email: zhuxiangchan@amss.ac.cn