Remarks on the global large solution to the three-dimensional incompressible Navier-Stokes equations

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 ${\text{B}}_{\mathrm{\infty },\mathrm{\infty }}^{-1}({\mathbb{R}}^3)$ norm can be arbitrarily large.