GLOBAL-IN-TIME PROBABILISTICALLY STRONG AND MARKOV SOLUTIONS TO STOCHASTIC 3D NAVIER-STOKES EQUATIONS: EXISTENCE AND NONUNIQUENESS

• 成果类型: Article
• 署名作者: Hofmanova, Martina; Zhu, Rongchan; Zhu, Xiangchan
• 署名单位: University of Bielefeld; Beijing Institute of Technology; Chinese Academy of Sciences; Academy of Mathematics & System Sciences, CAS
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/22-AOP1607
• 发表日期: 2023
• 页码: 524-579
• 关键词: ill-posedness; driven; selections

摘要

We are concerned with the three-dimensional incompressible Navier- Stokes equations driven by an additive stochastic forcing of trace class. First, for every divergence free initial condition in L2 we establish existence of infinitely many global-in-time probabilistically strong and analytically weak solutions, solving one of the open problems in the field. This result, in partic-ular, implies nonuniqueness in law. Second, we prove nonuniqueness of the associated Markov processes in a suitably chosen class of analytically weak solutions satisfying a relaxed form of an energy inequality. Translated to the deterministic setting, we obtain nonuniqueness of the associated semiflows.