  In this paper, we study local well-posedness for the Navier-Stokes \linebreak
equations with arbitrary initial data in homogeneous Sobolev spaces
$\dot{H}^s_p(\mathbb{R}^d)$ for $d \geq 2, p > \frac{d}{2},\ {\rm and}\
\frac{d}{p} - 1 \leq s < \frac{d}{2p}$. The obtained result improves the known
ones for $p > d$ and $s = 0$ M. Cannone and Y. Meyer (1995). In the case of
critical indexes $s=\frac{d}{p}-1$, we prove global well-posedness for
Navier-Stokes equations when the norm of the initial value is small enough.
This result is a generalization of the one in M. Cannone (1997) in which $p =
d$ and $s = 0$.
