Sharp Non-uniqueness of Solutions to 2D Navier-Stokes Equations with Space-Time White Noise

2023/04/13 by Huaxiang Lü, Lü, Huaxiang, Xiangchan Zhu +1 · 1 citation
Economics, Econometrics and Finance · Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Probability (math.PR) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2304.06526

openalex publication_date 2023/04/13 · openalex created_date 2023/04/15 · openalex updated_date 2026/07/28

Abstract

In this paper we are concerned with the 2D incompressible Navier-Stokes equations driven by space-time white noise. We establish existence of infinitely many global-in-time probabilistically strong and analytically weak solutions u for every divergence free initial condition u₀∈ L^p∪ C^-1+δ, p∈(1,2),δ>0. More precisely, there exist infinitely many solutions such that u-z∈ C([0,∞);L^p)∩ L²_\rmloc([0,∞);H^ζ)∩ L¹_\rmloc([0,∞);W^\frac13,1) for some ζ∈(0,1), where z is the solution to the linear equation. This result in particular implies non-uniqueness in law. Our result is sharp in the sense that the solution satisfying u-z∈ C([0,∞);L²)∩ L²_\rmloc([0,∞);H^ζ) for some ζ∈(0,1) is unique.

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• Sharp Non-uniqueness in Law for Stochastic Differential Equations on the Whole Space

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