  We study the Cauchy problem for the (generalized) incompressible
Navier-Stokes equations \begin{align} u_t+(-\Delta)^{\alpha}u+u\cdot \nabla u
+\nabla p=0, \ \ {\rm div} u=0, \ \ u(0,x)= u_0. \nonumber \end{align} We show
the analyticity of the local solutions of the Navier-Stokes equation
($\alpha=1$) with any initial data in critical Besov spaces
$\dot{B}^{n/p-1}_{p,q}(\mathbb{R}^n)$ with $1< p<\infty, \ 1\le q\le \infty $
and the solution is global if $u_0$ is sufficiently small in
$\dot{B}^{n/p-1}_{p,q}(\mathbb{R}^n)$. In the case $p=\infty$, the analyticity
for the local solutions of the Navier-Stokes equation ($\alpha=1$) with any
initial data in modulation space $M^{-1}_{\infty,1}(\mathbb{R}^n)$ is obtained.
We prove the global well-posedness for a fractional Navier-stokes equation
($\alpha=1/2$) with small data in critical Besov spaces
$\dot{B}^{n/p}_{p,1}(\mathbb{R}^n) \ (1\leq p\leq\infty)$ and show the
analyticity of solutions with small initial data either in
$\dot{B}^{n/p}_{p,1}(\mathbb{R}^n) \ (1\leq p<\infty)$ or in
$\dot{B}^0_{\infty,1} (\mathbb{R}^n)\cap {M}^0_{\infty,1}(\mathbb{R}^n)$.
Similar results also hold for all $\alpha\in (1/2,1)$.
