Sharpness conjecture for the enstrophy dissipation rate of measure vorticities

Let $\delta\in(0,1)$ and $p>1$. On the two-dimensional torus $\mathbb{T}^2$, let $u_0$ be a divergence-free vector field in $L^2(\mathbb{T}^2)$ with vorticity $\omega_0\in\mathcal M(\mathbb{T}^2)$ decomposed as

$$\omega_0=f_0+\mu_0,$$

where $f_0\in L^p(\mathbb{T}^2)$ and $\mu_0\geq0$. Let $\{\omega^\nu\}_\nu$ be the corresponding sequence of Leray solutions to the two-dimensional Navier–Stokes vorticity equation. Sharpness conjecture. There exists such initial data for which

$$\nu\int_\delta^1\|\omega^\nu(\tau)\|_{L^2}^2\,d\tau\gtrsim\frac{1}{\sqrt{|\log\nu|}}$$

for every $0<\nu\ll1$. The conjecture proposes that the previously obtained upper bound for the enstrophy dissipation rate is essentially sharp, although constructing initial vorticities whose Navier–Stokes evolution attains this rate remains open.