Remarks on the non-uniqueness in law of the Navier-Stokes equations up to the J.-L. Lions' exponent

Abstract

Lions (1959, Bull. Soc. Math. France, \textbf{87}, 245--273) introduced the Navier-Stokes equations with a viscous diffusion in the form of a fractional Laplacian; subsequently, he (1969, Dunod, Gauthiers-Villars, Paris) claimed the uniqueness of its solution when its exponent is not less than five quarters in case the spatial dimension is three. Following the work of Hofmanov$\acute{\mathrm{a}}$, Zhu and Zhu (2019, arXiv:1912.11841 [math.PR]), we prove the non-uniqueness in law for the three-dimensional stochastic Navier-Stokes equations with the viscous diffusion in the form of a fractional Laplacian with its exponent less than five quarters.

arXiv commentThis is the version accepted by Stochastic Processes and their Applications

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Cited by in corpus (3)

• Non-uniqueness in law for Boussinesq system forced by random noise
• On ill- and well-posedness of dissipative martingale solutions to stochastic 3D Euler equations
• Global-in-time probabilistically strong and Markov solutions to stochastic 3D Navier--Stokes equations: existence and non-uniqueness