Well-posedness for the Navier–Stokes equations with data in homogeneous Sobolev–Lorentz spaces. (January 2017) - Khai, D.Q.; Tri, N.M.
Abstract: In this paper, we study local well-posedness for the Navier–Stokes equations (NSE) with arbitrary initial data in homogeneous Sobolev–Lorentz spaces H ̇ L q, r s ( R d ) : = ( − Δ ) − s / 2 L q, r for d ≥ 2, q > 1, s ≥ 0, 1 ≤ r ≤ ∞, 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 < + ∞, 1 ≤ r ≤ + ∞ (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 = + ∞ ), respectively.