A remark on the regularity criterion for the 3D Navier–Stokes equations in terms of two vorticity components. (June 2023) - O, Chol-Jun
Abstract: In this paper, we establish a new regularity criterion for the 3D Navier–Stokes equations in terms of only two vorticity components. By making use of a logarithmic Sobolev inequality in the Besov spaces with negative indices and well known commutator estimate, we prove that a unique local strong solution does not blow-up at time T if two components of vorticity belong to L 2 ( 0, T ; B ̇ ∞, ∞ − 1 ) . This result is the further extension of the previous works by Guo et al. (2018) and by O (2021), and becomes the improvement of the work by Gala (2011). Highlights: A new regularity criterion via two vorticity components in the scaling invariant space. Giving the satisfactory answer to the open question posed in previous paper. Very simple and self-contained proof of the regularity.