A regularity criterion for 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 ${ϵ}_0$-condition on u in a dynamically restricted local Morrey space is stipulated; more precisely, $\underset{t\in (0,T)}{sup}\underset{x\in {\mathbb{R}}^3,\eta (t)\le r\le 1}{sup}\frac{1}{r^{\alpha }}{\int }_{B_r(x)}|u(y,t){|}^p\text{d}y\le {ϵ}_0,$where η is a dynamic dissipation scale consistent with the turbulence phenomenology and α 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 et al. (An algebraic reduction of the ‘Scaling Gap’ in the Navier–Stokes regularity problem. Arch Ration Mech Anal. 2019;231(3):1983–2005).

AMS Classifications:

• 35
• 76

Disclosure statement

No potential conflict of interest was reported by the author(s).