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

In this paper, we study local well-posedness for the Navier-Stokes equations (NSE) with the arbitrary initial value in homogeneous Sobolev-Lorentz spaces ${\dot{H}}_{L^{q,r}}^s({\mathbb{R}}^d):=(-\mathrm{\Delta }{)}^{-s\mathrm{/}2}{L}^{q,r}$ for $d \geq 2, q

1, s \geq 0$,$1 \leq r \leq \infty$,and$ \frac{d}{q}-1 \leq s < \frac{d}{q}$,thisresultimprovestheknownresultsfor$q > d,r=q, s = 0$(seeM\mathrm{.}Cannone(1995)andM\mathrm{.}CannoneandY\mathrm{.}Meyer(1995))andfor$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 $q = r=d, s=0$.