Tran–Dritschel conjecture on maximum enstrophy dissipation

Let $\varphi$ be the initial condition for the two-dimensional Navier–Stokes system, let $T$ be the length of the time window, and let $\nu$ denote the viscosity. Write $\chi_\nu$ for the enstrophy dissipation in solutions of the system.

Tran–Dritschel conjecture. The enstrophy dissipation satisfies

$$\chi_\nu \leq C\left[-\ln(\nu)\right]^{-\frac{1}{2}},$$

for some constant $C>0$ depending on the initial condition $\varphi$ and the length $T$ of the time window.

This proposed upper bound concerns the vanishing-viscosity behavior of enstrophy dissipation in two-dimensional Navier–Stokes flows. The source describes it as relying on assumptions about the form of the solution spectrum, so its resolution depends on establishing or refuting those spectral assumptions.