2022/08/17 by Weiquan Chen, Chen, Weiquan, Zhao Yang Dong +3
Decision Sciences · Economics, Econometrics and Finance · Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Probability (math.PR) #Risk and Portfolio Optimization #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2208.08321
openalex publication_date 2022/08/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we establish a sharp non-uniqueness result for stochastic d-dimensional (d≥2) incompressible Navier-Stokes equations. First, for every divergence free initial condition in L2 we show existence of infinite many global in time probabilistically strong and analytically weak solutions in the class Lα(Ω,LptL^∞) for any 1≤ p<2,α≥1. Second, we prove 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 \frac2p+(d)/(q)≤1, which is a stochastic version of Ladyzhenskaya-Prodi-Serrin criteria. Moreover, for stochastic d-dimensional incompressible Euler equation, existence of infinitely many global in time probabilistically strong and analytically weak solutions is obtained. Compared to the stopping time argument used in \citeHZZ19, HZZ21a, 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,∞).