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

TL;DR

This paper demonstrates non-uniqueness in law for 3D stochastic Navier-Stokes equations with fractional Laplacian diffusion when the exponent is below the critical five quarters, challenging previous uniqueness claims.

Contribution

It proves non-uniqueness in law for stochastic Navier-Stokes equations with fractional diffusion exponents below the Lions' critical value.

Findings

Non-uniqueness in law established for sub-critical fractional exponents.

Extends previous deterministic results to stochastic setting.

Highlights limitations of Lions' uniqueness result in stochastic context.

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.