Sharp nonuniqueness for the Navier–Stokes equations

In this paper, we prove a sharp nonuniqueness result for the incompressible Navier–Stokes equations in the periodic setting. In any dimension d≥ 2 and given any p< 2 , we show the nonuniqueness of weak solutions in the class LtpL∞, which is sharp in view of the classical Ladyzhenskaya–Prodi–Serrin criteria. The proof is based on the construction of a class of non-Leray–Hopf weak solutions. More specifically, for any p< 2 , q< ∞, and ε> 0 , we construct non-Leray–Hopf weak solutions u∈LtpL∞∩Lt1W1,q that are smooth outside a set of singular times with Hausdorff dimension less than ε. As a byproduct, examples of anomalous dissipation in the class Lt3/2-εC1/3 are given in both the viscous and inviscid case.