On the initial value problem for the Navier-Stokes equations with the initial datum in the Sobolev spaces

Abstract

In this paper, we study local well-posedness for the Navier-Stokes equations with arbitrary initial data in homogeneous Sobolev spaces ${\dot{H}}_p^s({\mathbb{R}}^d)$ for $d\ge 2,p>\frac{d}{2},\mathrm{and}\frac{d}{p}-1\le s<\frac{d}{2p}$. The obtained result improves the known ones for $p>d$ and $s=0$ (see M. Cannone (1995), 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 ones in Cannone (1999) and P. G. Lemarie-Rieusset (2002) in which $(p=d,s=0)$ and $(p>d,s=\frac{d}{p}-1)$, respectively.

18pages. arXiv admin note: text overlap with arXiv:1603.01896