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

Abstract

Lions (1959), 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 Hofmanová et al. (2019), 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.