Well-posedness for the Navier-Stokes equations with data in homogeneous Sobolev-Lorentz spaces

Abstract

In this paper, we study local well-posedness for the Navier-Stokes equations (NSE) with the arbitrary initial value in homogeneous Sobolev-Lorentz spaces ${\dot{H}}_{L^{q,r}}^s({\mathbb{R}}^d):=(-\Delta {)}^{-s/2}{L}^{q,r}$ for $d\ge 2,q>1,s\ge 0$, $1\le r\le \infty$, and $\frac{d}{q}-1\le s<\frac{d}{q}$, this result improves the known results for $q>d,r=q,s=0$ (see M. Cannone (1995) and M. Cannone and Y. Meyer (1995)) and for $q=r=2,\frac{d}{2}-1<s<\frac{d}{2}$ (see M. Cannone (1995, J. M. Chemin (1992)). In the case of critical indexes ($s=\frac{d}{q}-1$), we prove global well-posedness for NSE provided the norm of the initial value is small enough. The result that is a generalization of the result of M. Cannone (1997) for $q=r=d,s=0$.

arXiv admin note: text overlap with arXiv:1601.01441, arXiv:1601.01726