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 arbitrary initial data in homogeneous Sobolev-Lorentz spaces (H) ovrt dot(Lq,r)(s) (R-d) := (-Delta)L--s/2(q,r) for d >= 2, q > 1, s >= 0, 1 <= r <= infinity, and d/q - 1 <= s < d/q. The obtained result improves the known ones for q > d,r = q, s = 0 (see Cannone (1995), Cannone and Meyer (1995)), for q = r = 2, d/2 - 1 < s < d/2 (see Cannone (1995), Chemin (1992)), and for s = 0, d < q < +infinity, 1 <= r <= + infinity (see Lemarie-Rieusset (2002)). In the case of critical indexes (s = d/q - 1), we prove global well-posedness for NSE provided the norm of the initial value is small enough. This result is also a generalization of the one in Cannone (1997) and Kozono and Yamazaki (1995) [27], Meyer (1999) [30] in which (q = r = d, s = 0) and (q = d, s = 0, r = +infinity), respectively. (C) 2016 Elsevier Ltd. All rights reserved.