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

In this paper, we derive a new smallness hypothesis of initial data for the threedimensional incompressible Navier-Stokes equations. That is, we prove that there exist two positive constants c0, C0 such that if ‖u0 + u0, u0‖ Ḃ −1+ 3 p p,1 ‖u0, u0‖ Ḃ −1+ 3 p p,1 exp{C0(‖u0‖2Ḃ−1 ∞,2 + ‖u0‖Ḃ−1 ∞,1 )} ≤ c0, then (1.1) has a unique global solution. As an application we construct two family of smooth solutions to the Navier-Stokes equations whose Ḃ ∞,∞(R ) norm can be arbitrarily large.