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隨機復雜結構與數據科學重點實驗室
學術報告


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Speaker:

朱蓉禪 副教授,北京理工大學

Inviter: 羅德軍 博士
Title:
NON-UNIQUENESS IN LAW OF STOCHASTIC 3D NAVIER-STOKES EQUATIONS
Time & Venue:

2020.1.17 10:00 N620

Abstract:

We consider the stochastic Navier--Stokes equations in three dimensions and prove that the law of analytically weak solutions is not unique. In particular, we focus on two iconic examples of a stochastic perturbation: either an additive or a linear multiplicative noise driven by a Wiener process. In both cases, we develop a stochastic counterpart of the convex integration method introduced recently by Buckmaster and Vicol. This permits to construct probabilistically strong and analytically weak solutions defined up to a suitable stopping time. In addition, these solutions fail the corresponding energy inequality at a prescribed time with a prescribed probability. Then we introduce a general probabilistic construction used to extend the convex integration solutions beyond the stopping time and in particular to the whole time interval $[0,\infty)$. Finally, we show that their law is distinct from the law of solutions obtained by Galerkin approximation. In particular, non-uniqueness in law holds on an arbitrary time interval $[0,T]$, $T>0$.

Affiliation:  

學術報告中國科學院數學與系統科學研究院應用數學研究所
地址 北京市海淀區中關村東路55號 思源樓6-7層 南樓5-6、8層 100190
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