Global well-posedness to the 3-D incompressible inhomogeneous Navier-Stokes equations with a class of large velocity

In this article, we consider the global well-posedness to the 3-D incompressible inhomogeneous Navier-Stokes equations with a class of large velocity. More precisely, assuming $a_0 \in \dot{B}{q,1}^{\frac{3}{q}}($\mathbb{R}$^3)$and$u_0=(u_0^h,u_0^3)\in \dot{B}{p,1}^{-1+\frac{3}{p}}($\mathbb{R}$^3)$for$p,q \in (1,6)$with$\sup(\frac{1}{p}, \frac{1}{q})\leq\frac{1}{3}+ \inf (\frac{1}{p}, \frac{1}{q})$,weprovethatif$C|a_0|{\dot{B}{q,1}^{\frac{3}{q}}}^{\alpha}(|u_0^3|{\dot{B}{p,1}^{-1+\frac{3}{p}}}/{\mu}+1)\leq1$,$\frac{C}{\mu}(|u_0^h|{\dot{B}{p,1}^{-1+\frac{3}{p}}}+|u_0^3|{\dot{B}{p,1}^{-1+\frac{3}{p}}}^{1-\alpha}|u_0^h|{\dot{B}{p,1}^{-1+\frac{3}{p}}}^{\alpha})\leq 1$,thenthesystemhasauniqueglobalsolution$a\in\widetilde{$\mathcal{C}$}([0,\infty);\dot{B}{q,1}^{\frac{3}{q}}($\mathbb{R}$^3))$,$u\in\widetilde{$\mathcal{C}$}([0,\infty);\dot{B}{p,1}^{-1+\frac{3}{p}}($\mathbb{R}$^3))\cap L^1($\mathbb{R}$^+;\dot{B}_{p,1}^{1+\frac{3}{p}}($\mathbb{R}$^3))$. It improves the recent result of M. Paicu, P. Zhang (J. Funct. Anal. 262 (2012) 3556-3584), where the exponent form of the initial smallness condition is replaced by a polynomial form.