Eigenvalue blow-up conjecture for finite-time Navier–Stokes singularities

Let $u\in C\left(\left[0,T_{\max}\right);\dot{H}^1_{df}\right)$ be a smooth solution of the Navier–Stokes equation with finite maximal existence time $T_{\max}<+\infty$. Let $\lambda_2^+(x,t)$ denote the positive part of the second eigenvalue of the strain matrix associated with $u$. Eigenvalue blow-up conjecture. Then

$$\limsup_{t\to T_{\max}}\|\lambda_2^+(\cdot,t)\|_{L^{\frac{3}{2}}}=+\infty.$$

The conjecture proposes that blow-up of this critical strain-eigenvalue norm is necessary for a smooth Navier–Stokes solution to develop a finite-time singularity; the paper states that this remains open.