One component optimal regularity for the Navier–Stokes equations with almost zero differentiability degree

Abstract

We study the conditional regularity for the incompressible Navier–Stokes equations in the whole three dimensional space in terms of one component of the velocity field $u=(u_1,u_2,u_3)$. Let $\alpha \in (0,\infty )$. For $f\in L^2\left({\mathbb{R}}^d\right)$, $d\in \mathbb{N}$, we define: ${\Vert f\Vert }_{2,lo{g}_{\alpha }}^2={\int }_{{\mathbb{R}}^d}{log}^{\alpha }(e+\left|\xi \right|){\left|\hat{f}\left(\xi \right)\right|}^{2}d\xi$, where $\hat{f}$ denotes the Fourier transform of $f$, and prove the following regularity criterion: $u$ is regular on $(0,T]$, $T>0$, if $u_3\in L^{\infty }(0,T;L_v^{\infty }{L}_{h,lo{g}_{\alpha }}^2)$ for some $\alpha \in (1,\infty )$, where $v$ and $h$ denote the vertical and horizontal components, respectively. This criterion possesses two substantial properties: it is almost optimal from the scaling point of view and does not almost require any information on the derivative of $u_3$. The relation of the presented criterion to the kin criteria published in the literature is discussed throughout the paper.