Partial Regularity and Blowup for an Averaged Three-Dimensional Navier-Stokes Equation

We prove two results that together strongly suggest that obtaining a positive answer to the Navier-Stokes global regularity question requires more than a refinement of partial regularity theory. First we prove that there exists a class of bilinear operators $\mathfrak{B}$, which contains the Euler bilinear operator $\mathcal{E}(u,v):=\frac{1}{2}\mathbb{P}(u\cdot \mathrm{\nabla }v+v\cdot \mathrm{\nabla }u)$, such that for any $B\in \mathfrak{B}$, $n\ge 3$, $\alpha \in ((n+1)\mathrm{/}4,(n+2)\mathrm{/}4)$, and smooth solution $u$ of the pseudodifferential equation ${\mathrm{\partial }}_{t}u+(-\mathrm{\Delta }{)}^{\alpha }u+B(u,u)=0$ on ${\mathbb{R}}^n\times [0,T)$, we have that $u$ is also smooth at time $T$ away from a closed set of Hausdorff dimension at most $n+2-4\alpha$. Next we prove that, for the Euclidean space ${\mathbb{R}}^3$, there exists an operator $C(u,v)\in \mathfrak{B}$ that is an averaged version of $\mathcal{E}$, that formally allows the dissipation of energy by the “cancellation identity” $⟨C(u,u),u⟩=0$, and whose corresponding pseudodifferential equation ${\mathrm{\partial }}_{t}u+(-\mathrm{\Delta }{)}^{\alpha }u+C(u,u)=0$ admits a solution that blows up in finite time for all $\alpha \in (0,5\mathrm{/}4)$.