In this paper, we study local well-posedness for the Navier-Stokes equations (NSE) with the arbitrary initial value in homogeneous Sobolev-Lorentz spaces $\dot{ H}^s_{ L^{ q, r}}(\mathbb{ R}^d):= (-\Delta)^{ -s/2}L^{ q,r}$ for $d \geq 2, q > 1, s \geq 0$, $1 \leq r \leq \infty$, and $ \frac{ d}{ q}-1 \leq s < \frac{ d}{ q}$, 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 $q = r=d, s=0$.
Categories: math.AP