Sharp nonuniqueness for the Navier-Stokes equations

Abstract

In this paper, we prove a sharp nonuniqueness result for the incompressible Navier-Stokes equations in the periodic setting. In any dimension $d\ge 2$ and given any $p<2$, we show the nonuniqueness of weak solutions in the class $L_t^p{L}^{\infty }$, 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 $p<2$, $q<\infty$, and $\epsilon >0$, we construct non-Leray-Hopf weak solutions $u\in L_t^p{L}^{\infty }\cap L_t^1{W}^{1,q}$ that are smooth outside a set of singular times with Hausdorff dimension less than $\epsilon$. As a byproduct, examples of anomalous dissipation in the class $L_t^{3/2-\epsilon }{C}^{1/3}$ are given in both the viscous and inviscid case.

minor corrections, to appear in inventiones