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

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