A Critical one-component regularity criteria for the three-dimensional Navier--Stokes equations

Abstract: We establish two critical one-component regularity criteria for the three-dimensional incompressible Navier--Stokes equations. First, for divergence-free initial data $u_0\in H^1(\mathbb R<sup>3)$, we prove that a strong solution can be continued beyond a finite time $T$ provided that [ u^3\in L^2\bigl(0,T;\dot B^{3/2}_{2,r}(\mathbb R^3)\bigr) \qquad\text{for some }2<r\leq\infty. ] Second, for every $2&lt;p&lt;\infty$, if $u_0\in\dot H^{{1/2}(\mathbb} R<sup>3)$ and, for some fixed unit vector $\boldsymbol e\in\mathbb S<sup>2$, [ u\cdot\boldsymbol e \in L^p\bigl(0,T;\dot H^{1/2+2/p}(\mathbb R^3)\bigr), ] then the corresponding Fujita--Kato solution also extends beyond $T$, without any additional low-integrability assumption on the initial vorticity. These results extend the critical one-component regularity criteria of Han, Lei, Li, and Zhao in two different directions. At the time endpoint $p=2$, we combine a new vertical-vorticity estimate with coupled anisotropic energy estimates and an energy-dependent frequency decomposition to relax the dyadic square-summability condition [ \dot H^{3/2}=\dot B^{3/2}_{2,2} ] to [ \dot B^{3/2}_{2,r}, \qquad 2<r\leq\infty, ] with the endpoint case $r=\infty$ closed by an Osgood-type argument. In the nonendpoint range $2&lt;p&lt;\infty$, we use a positive-time heat-flow decomposition to separate the solution into a smooth linear part and a nonlinear remainder. The remainder has finite energy and acquires the required $L<sup>q$-integrability of the vorticity at positive times, which allows the anisotropic energy estimates to be applied while the additional transport, stretching, and pressure terms generated by the heat flow remain integrable. This removes the extra $L<sup>{q_0}$-integrability assumption on the initial vorticity while retaining the natural critical initial-data space $\dot H^{{1/2}(\mathbb} R<sup>3)$.