Potential blow-up scenario for the 3D Navier–Stokes equations

Let $\alpha(\{\delta_j\}_{j=1}^n)$ be the exponent associated with the multi-level logarithmically improved criteria, and let $s$ denote the Sobolev regularity exponent. Potential blow-up scenario. If there exists an exponent $\alpha_* > 0$ such that

$$\lim_{n \to \infty} \lim_{\delta_1,\delta_2,\ldots,\delta_n \to \infty} \alpha(\{\delta_j\}_{j=1}^n) = \alpha_*,$$

then there exists initial data in $\dot{H}^s(\mathbb{R}^3)$ for $s \in (1/2,1/2+\alpha_*)$ that potentially leads to finite-time blow-up. This is presented as a possible mechanism for constructing blow-up scenarios below the regularity range covered by the preceding global-regularity result; the source does not establish that such data actually produce blow-up.