SHARP NONUNIQUENESS OF SOLUTIONS TO 2D NAVIER-STOKES EQUATIONS WITH SPACE-TIME WHITE NOISE

• 成果类型: Article
• 署名作者: Lu, Huaxiang; Zhu, Xiangchan
• 署名单位: Chinese Academy of Sciences; Academy of Mathematics & System Sciences, CAS
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/25-AAP2164
• 发表日期: 2025
• 页码: 1980-2030
• 关键词: euler equations; ill-posedness; uniqueness; driven; SYSTEM

摘要

In this paper we are concerned with the 2D incompressible Navier- Stokes equations driven by space-time white noise. We establish existence of solutions u for every divergence free initial condition u(0) is an element of L-p boolean OR C-1+delta, p is an element of (1, 2), delta > 0. More precisely, there exist infinitely many solutions such that u - z is an element of C([0, infinity); L-p) fl L-loc(2)([0, infinity); H-zeta) fl L-loc(1)([0, oo); W-1/3.1) for some zeta E (0, 1), where z is the solution to the linear equation. This result in particular implies nonuniqueness in law. Our result is sharp in the sense that the solution satisfying u - z is an element of C([0, infinity); L-2) boolean AND L-loc (2)([0, infinity); H-zeta) for some zeta is an element of (0, 1) is unique.