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

Abstract

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 $\mathcal{E}(u,v):=\frac{1}{2}\mathbb{P}(u\cdot\nabla v + v\cdot\nabla u)$, such that for any $B\in \mathfrak{B}$, $n\geq3$, $\alpha \in ((n+1)/4, (n+2)/4)$, and smooth solution $u$ of the pseudodifferential equation $\partial_t u +(-\Delta)^\alpha u +B(u,u)=0$ 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" $\langle C(u,u), u\rangle =0$, and whose corresponding pseudodifferential equation $\partial_t u +(-\Delta)^\alpha u +C(u,u)=0$ admits a solution that blows up in finite time for all $\alpha \in (0,5/4)$.