Eyink's anomalous-dissipation conjecture for viscosity solutions of 2D Euler

Let $\boldsymbol{u}^ u,\boldsymbol{ abla}\boldsymbol{\times}\boldsymbol{u}^ u=\boldsymbol{ abla}\times\boldsymbol{u}^ u$ be a sequence of 2D Navier–Stokes solutions satisfying the paper's estimate (u-besov), and let $\boldsymbol{ abla}\times\boldsymbol{u}$ be the limiting 2D Euler vorticity, as supplied by Theorem 3. For $h\in\mathcal{H}_2$, define the mollified defect $Z_{h,\varepsilon}(\omega)$ as in the paper. Anomalous-dissipation conjecture. The distributional limit

$$Z_h(\omega)=\lim_{\varepsilon\rightarrow 0}Z_{h,\varepsilon}(\omega)$$

exists for every $h\in\mathcal{H}_2$, and, for $h\in\mathcal{H}_2\cap C^2$, equals the vanishing-viscosity limit

$$Z_h(\omega)=\lim_{\nu\rightarrow 0}\nu h”(\omega^\nu)|\boldsymbol{\nabla}\omega^\nu|^2.$$

For every such convex $h$, $Z_h(\omega)$ is a nonnegative measure, and there exists a suitable 2D Euler solution for which $Z_h(\omega)>0$ for a convex $h\in\mathcal{H}_2$. This conjecture seeks a rigorous defect-measure formulation of enstrophy dissipation for distributional Euler solutions; the source does not establish these assertions.