Sharp Nonuniqueness of Solutions to Stochastic Navier–Stokes Equations

. In this paper we establish a sharp nonuniqueness result for stochastic \ (d\) -dimensional (\ (d 2\) ) incompressible Navier–Stokes equations. First, for every divergence-free initial condition in \ (L²\) we show existence of infinitly many global-in-time probabilistically strong and analytically weak solutions in the class \ (L^ (, LᵖₜL^) \) for any \ (1 p 2, 1\). Second, we prove that the above result is sharp in the sense that pathwise uniqueness holds in the class of \ (LᵖₜLq\) for some \ (p 2, , q (2, ]\) such that \ (2p+dq 1\), which is a stochastic version of Ladyzhenskaya–Prodi–Serrin criteria. Moreover, for the stochastic \ (d\) -dimensional incompressible Euler equation, the existence of infinitely many global-in-time probabilistically strong and analytically weak solutions is obtained. 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 convex integration scheme and construct directly solutions on the whole time interval \ ([0, ) \). Keywordsstochastic Navier–Stokes equationsstochastic Euler equationsprobabilistically strong solutionssharp nonuniquenessconvex integrationMSC codes60H1535R6035Q30