Well-posedness for the Navier-Stokes equations with datum in the Sobolev spaces

In this paper, we study local well-posedness for the Navier-Stokes \linebreak equations with arbitrary initial data in homogeneous Sobolev spaces ${\dot{H}}_p^s({\mathbb{R}}^d)$ 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$s = 0$.