Well-posedness for the Navier-Stokes equations with data in homogeneous Sobolev-Lorentz spaces
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- 作者：D.Q. Khai khaitoantin@gmail.com ; N.M. Tri ; triminh@math.ac.vn
- 关键词：primary ; 35Q30 ; secondary ; 76D05 ; 76N10
- 刊名：Nonlinear Analysis: Theory, Methods & Applications
- 出版年：2017
- 出版时间：January 2017
- 年：2017
- 卷：149
- 期：Complete
- 页码：130-145
- 全文大小：717 K
- 卷排序：149
文摘
In this paper, we study local well-posedness for the Navier–Stokes equations (NSE) with arbitrary initial data in homogeneous Sobolev–Lorentz spaces ḢLq,rs(Rd):=(−Δ)−s/2Lq,r for d≥2,q>1,s≥0d≥2,q>1,s≥0, 1≤r≤∞1≤r≤∞, and dq−1≤s<dq. The obtained result improves the known ones for q>d,r=q,s=0q>d,r=q,s=0 (see Cannone (1995), Cannone and Meyer (1995)), for q=r=2,d2−1<s<d2 (see Cannone (1995), Chemin (1992)), and for s=0,d<q<+∞,1≤r≤+∞s=0,d<q<+∞,1≤r≤+∞ (see Lemarie-Rieusset (2002)). In the case of critical indexes (s=dq−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)(q=r=d,s=0) and (q=d,s=0,r=+∞)(q=d,s=0,r=+∞), respectively.