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

We establish two critical one-component regularity criteria for the three-dimensional incompressible Navier--Stokes equations. First, for divergence-free initial data $u_0\in H^1(\mathbb R^3)$, we prove that a strong solution can be continued beyond a finite time $T$ provided that \[ u^3\in L^2\bi