Global-in-time probabilistically strong and Markov solutions to stochastic 3D Navier--Stokes equations: existence and non-uniqueness

2021/04/20 by Martina Hofmanová, Hofmanová, Martina, Rongchan Zhu +3 · 2 citations
Mathematics · Economics, Econometrics and Finance · Engineering · #Navier-Stokes equation solutions #Stochastic processes and financial applications #Fluid Dynamics and Turbulent Flows

paper · pdf · doi:10.48550/arxiv.2104.09889

Abstract

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 particular implies non-uniqueness in law. Second, we prove non-uniqueness 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 non-uniqueness of the associated semiflows.

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• Pathwise uniqueness by noise for singular stochastic PDEs
• Sharp Non-uniqueness in Law for Stochastic Differential Equations on the Whole Space

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