In this paper, we establish a sharp nonuniqueness result for stochastic d-dimensional (d\geq 2) incompressible Navier–Stokes equations. First, for every divergence-free initial condition in L^2, we show the existence of infinitely many global-in-time probabilistically strong and analytically weak solutions in the class L^\alpha(\Omega, L^p_t L^\infty) for any 1 \leq p < 2, \alpha \geq 1. Second, we prove that the above result is sharp in the sense that pathwise uniqueness holds in the class of L^p_t L^q for some p \in [2,\infty], q \in (2,\infty] such that 2/p+d/q \leq 1, which is a stochastic version of the Ladyzhenskaya–Prodi–Serrin criteria. Moreover, for the stochastic d-dimensional incompressible Euler equation, we obtain the existence of infinitely many global-in-time probabilistically strong and analytically weak solutions. Compared to the stopping time argument used in Hoffmanová, Zhu, and Zhu [J. Eur. Math. Soc. (JEMS), to appear; Ann. Probab., 51 (2023), pp. 524–579], we developed a new stochastic version of the convex integration. More precisely, we introduce expectation during the convex integration scheme and construct solutions directly on the entire time interval [0,\infty).

Publication:

SIAM Journal on Mathematical Analysis ( Volume: 56, Issue: 2, 2024)

https://doi.org/10.1137/23M1563141

Author:

Weiquan Chen

Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China

Zhao Dong

Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China

Email: dzhao@amt.ac.cn

Xiangchan Zhu

Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China

Email: zhuxiangchan@amss.ac.cn