A Critical Chemin--Lerner Regularity Criterion via One Velocity Component for the Three-Dimensional Navier--Stokes Equations

Abstract: We prove a scaling-critical regularity criterion involving only one velocity component for finite-energy suitable weak solutions of the three-dimensional incompressible Navier--Stokes equations. Let $2&lt;p&lt;\infty$ and $m=3p/(p-2)$, so that $2/p+3/m=1$. We show that a singularity cannot occur provided [ \sum_{j\in\mathbb Z} |\dotΔj u^3|{L^p(0,T;L^m(\mathbb R^3))}<\infty, ] that is, $$u^{3\in\widetilde} L^p(0,T;\dot B^{0_{m,1}(\mathbb} R<sup>3))$$. The assumption is a spatial-frequency $\ell<sup>1$ refinement of the still unresolved critical condition $u^3\in L<sup>p_tL<sup>m_x$ and is complementary to the Lorentz-in-time refinement $L<sup>{p,1}_tL<sup>m_x$ obtained by Wang, Wu, and Zhang. By Bernstein embedding, the result extends to $$u^{3\in\widetilde} L^p_t\dot B<sup>s_{q,1}$$ on the nonnegative-smoothness critical line $s=-1+2/p+3/q\ge0$. The principal innovation is a frequency--scale matching scheme embedded in the local energy inequality. Each dyadic block of $u<sup>3$ is retained until it is paired with the vertical scale selected by a one-dimensional backward heat kernel. Low frequencies gain from the slab thickness, high frequencies from transferring the projection to the localized flux and applying an inverse Bernstein estimate, and spatially separated pressure sources from harmonic decay. These mechanisms generate a two-sided $\ell<sup>1$ kernel, converting spatial-frequency summability into summability of physical-scale energy increments. Consequently, we obtain a uniform Type-I local energy bound without a Lorentz refinement in time; compactness and one-component rigidity then exclude singular blow-up limits whose third velocity component vanishes.