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That is, we prove that there exist two positive constants $c_0,C_0$ such that if \\begin{equation*}\n  \\|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. 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