  We consider Cauchy problem of the incompressible Navier-Stokes equations with
initial data $u_0\in L^1(\mathbb{R}^3)\cap L^{\infty}(\mathbb{R}^3)$. There
exist a maximum time interval $[0,T_{max})$ and a unique solution $u\in
C\big([0,T_{max}); L^2(\mathbb{R}^3) \cap L^p(\mathbb{R}^3)\big)$ ($\forall
p>3$). We find one of function class $S_{regular}$ defined by scaling invariant
norm pair such that $T_{max}=\infty$ provided $u_0\in S_{regular}$. Especially,
$\|u_0\|_{L^p}$ is arbitrarily large for any $u_0\in S_{regular}$ and $p>3$. On
the other hand, the alternative theorem is proved. It is that either $T_{max}=
\infty$ or $T_{max}\in(T_l,T_r]$. Especially, $T_r<T_{max}<\infty$ is
disappearing. Here the explicit expressions of $T_l$ and $T_r$ are given. This
alternative theorem is one kind of regular criterion which can be verified by
computer. If $T_{max}=\infty$, the solution $u$ is regular for any
$(t,x)\in(0,\infty) \times \mathbb{R}^3$. As $t\rightarrow\infty$, the solution
is decay. On the other hand, lower bound of blow up rate of $u$ is obtained
again provided $T_{max}\in (T_l,T_r]$.
