A regularity criterion for 3D NSE in 'dynamically restricted' local Morrey spaces. Issue 16 (2nd November 2022) - Grujić, Zoran; Xu, Liaosha
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, 1 sup t ∈ ( 0, T ) sup x ∈ R 3, η ( t ) ≤ r ≤ 1 1 r α ∫ B r ( x ) | u ( y, t ) | p d y ≤ 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 α -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).