Abstract and Applied Analysis
Volume 2011 (2011), Article ID 369571, 15 pages
Research Article

Asymptotic Behavior of the Navier-Stokes Equations with Nonzero Far-Field Velocity

Department of Mathematics, Hallym University, Chuncheon 200-702, Republic of Korea

Received 18 June 2011; Revised 7 October 2011; Accepted 17 October 2011

Academic Editor: Weiqing Ren

Copyright © 2011 Jaiok Roh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Concerning the nonstationary Navier-Stokes flow with a nonzero constant velocity at infinity, the temporal stability has been studied by Heywood (1970, 1972) and Masuda (1975) in 𝐿 2 space and by Shibata (1999) and Enomoto-Shibata (2005) in 𝐿 𝑝 spaces for 𝑝 3 . However, their results did not include enough information to find the spatial decay. So, Bae-Roh (2010) improved Enomoto-Shibata's results in some sense and estimated the spatial decay even though their results are limited. In this paper, we will prove temporal decay with a weighted function by using 𝐿 𝑟 𝐿 𝑝 decay estimates obtained by Roh (2011). Bae-Roh (2010) proved the temporal rate becomes slower by ( 1 + 𝜎 ) / 2 if a weighted function is | 𝑥 | 𝜎 for 0 < 𝜎 < 1 / 2 . In this paper, we prove that the temporal decay becomes slower by 𝜎 , where 0 < 𝜎 < 3 / 2 if a weighted function is | 𝑥 | 𝜎 . For the proof, we deduce an integral representation of the solution and then establish the temporal decay estimates of weighted 𝐿 𝑝 -norm of solutions. This method was first initiated by He and Xin (2000) and developed by Bae and Jin (2006, 2007, 2008).