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

Abstract

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 $\epsilon_0$-condition on $u$ in a lower time-restricted local Morrey space is stipulated; more precisely, $$\sup_{t\in(0,T)} \ \sup_{x \in \mathbb{R}^3, \ \eta(t) \le r \le 1} \ \frac{1}{r^\alpha} \int_{B_r(x)} |u(y,t)|^p dy \le \epsilon_0$$ 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].