On strong solutions of Navier-Stokes equations with two velocity components

In this paper, we established a continuation criterion for local strong solutions to the three-dimensional incompressible Navier-Stokes equations based on partial velocity components. More precisely, we showed that a unique local strong solution $u$ does not blow up at time $T$ provided that the two horizontal velocity components $u_h$ belong to the Banach spaces ${\dot{V}}_{p,q,\theta }^s$ and ${\dot{U}}_{p,\beta ,\sigma }^s$. These functional spaces strictly contained the homogeneous Besov space ${\dot{B}}_{p,q}^s$, and thus allowed a wider admissible class than those considered in earlier works. Our results can therefore be viewed as an extension and refinement of previously known criteria.