A Regularity Criterion for Solutions to the 3D NSE in `Dynamically Restricted' Local Morrey Spaces

It is shown that a local-in-time strong solution $u$ to the 3D Navier-Stokes equations remains regular on an interval $(0,T)$ provided a smallness ${ϵ}_0$-condition on $u$ in a lower time-restricted local Morrey space is stipulated; more precisely, $$\underset{t\in (0,T)}{\sup }\underset{x\in {\mathbb{R}}^3,\eta (t)\le r\le 1}{\sup }\frac{1}{r^{\alpha }}{\int }_{B_r(x)}\mathrm{\mid }u(y,t){\mathrm{\mid }}^{p}dy\le {ϵ}_0$$ where $\eta$ is a dynamic dissipation scale consistent with the turbulence phenomenology and $\alpha$ and $p$ are suitable parameters. Such regularity criterion guarantees the volumetric sparseness of local spatial structure of intense vorticity components, preventing the formation of the finite-time blow up at $T$ under the framework of $Z_{\alpha }$-sparseness classes introduced in [Bradshaw, Farhat and Grujic, ARMA, 2018].