  In this paper, we derive a new smallness hypothesis of initial data for the
three-dimensional incompressible Navier-Stokes equations. That is, we prove
that there exist two positive constants $c_0,C_0$ such that if
\begin{equation*}
  \|u_0^1+u^2_0,u^3_0\|_{\dot{B}_{p,1}^{-1+\frac{3}{p}}}
\|u^1_0,u^2_0\|_{\dot{B}_{p,1}^{-1+\frac{3}{p}}} \exp\{C_0
(\|u_0\|^{2}_{\dot{B}_{\infty,2}^{-1}}+\|u_0\|_{\dot{B}_{\infty,1}^{-1}})\}
\leq c_0, \end{equation*} then \eqref{NS} has a unique global solution. As an
application we construct two family of smooth solutions to the Navier-Stokes
equations whose $\B^{-1}_{\infty,\infty}(\mathbb{R}^3)$ norm can be arbitrarily
large.
