Sharp Nonuniqueness of Solutions to Stochastic Navier–Stokes Equations

.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 existence of infinitly many global-in-time probabilistically strong and analytically weak solutions in the class \(L^\alpha (\Omega,L^p_tL^\infty )\) for any \(1\leq p\lt 2,\alpha \geq 1\). Second, we prove that the above result is sharp in the sense that pathwise uniqueness holds in the class of \(L^p_tL^q\) for some \(p\in [2,\infty ],q\in (2,\infty ]\) such that \(\frac 2{p}+\frac{d}{q}\leq 1\), 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 \([0,\infty )\).Keywordsstochastic Navier–Stokes equationsstochastic Euler equationsprobabilistically strong solutionssharp nonuniquenessconvex integrationMSC codes60H1535R6035Q30