Supercritical nonuniqueness and non-Leray-Hopf conjecture for the Navier–Stokes equations

Let $d\geq 2$ and let $p,q\in[1,\infty]$ satisfy

$$\frac{2}{p}+\frac{d}{q}>1.$$

Supercritical nonuniqueness and non-Leray-Hopf conjecture. Then there exist two weak solutions $u,v\in X^{p,q}([0,T];\mathbb{T}^d)$ of the Navier–Stokes equations with

$$u(0)=v(0)\quad\text{but}\quad v\ne u,$$

and there exists a weak solution $u\in X^{p,q}([0,T];\mathbb{T}^d)$ that is not Leray–Hopf. The conjecture proposes sharpness of the critical threshold $\frac{2}{p}+\frac{d}{q}=1$ for uniqueness and the Leray–Hopf property of weak solutions.