Well-posedness for the Navier–Stokes equations with data in homogeneous Sobolev–Lorentz spaces. (January 2017)

• Record Type: Journal Article
• Authors: Khai, D.Q.; Tri, N.M.
• Is Part Of: Nonlinear analysis. Volume 149(2017)
• Page Start: 130
• Page End: 145
• Publication Date: 2017-01
• Subjects: primary 35Q30 -- secondary 76D05 76N10; Navier–Stokes equations -- Existence and uniqueness of local and global mild solutions -- Homogeneous Sobolev–Lorentz spaces; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2016.10.015
• Languages: English
• ISSNs: 0362-546X

Abstract

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.