Eyink's enstrophy-defect conjecture for two-dimensional Euler flows

Let $\omega$ be a weak solution of the incompressible two-dimensional Euler equations obtained by the vanishing-viscosity method, with

$$\omega\in L^2((0,T);B^0_{2,\infty}(\mathbb{R}^2)).$$

Assume that there exists a sequence $\omega_{\nu_k}$ of solutions of the incompressible Navier–Stokes equations such that

$$\omega_{\nu_k}\rightharpoonup\omega\quad\text{in weak-* }L^2((0,T);B^0_{2,\infty}(\mathbb{R}^2)).$$

Eyink's conjecture. Both limits

$$\lim_{\nu\to0^+}Z^\nu(\omega_\nu)\qquad\text{and}\qquad\lim_{\epsilon\to0^+}Z_\epsilon(\omega)$$

exist and are equal, so that $Z(\omega)=Z^V(\omega)=Z^T(\omega)$. Furthermore, $\omega$ is a dissipative solution, and there exists such an $\omega$ with $Z(\omega)>0$.

The conjecture expresses the expectation that the transport enstrophy defect accounts for the residual viscous enstrophy dissipation in the vanishing-viscosity limit for two-dimensional turbulence. The paper states that one of its main results gives an example showing that this equality of defects is not necessarily true, so the proposed claim is refuted.