Abstract and Applied Analysis
Volume 2011 (2011), Article ID 856896, 15 pages
http://dx.doi.org/10.1155/2011/856896
Research Article

𝐿 𝑟 - 𝐿 𝑝 Stability of the Incompressible Flows with Nonzero Far-Field Velocity

Jaiok Roh

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

Received 16 June 2011; Accepted 15 September 2011

Academic Editor: Ibrahim Sadek

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.

Abstract

We consider the stability of stationary solutions 𝐰 for the exterior Navier-Stokes flows with a nonzero constant velocity 𝐮 ∞ at infinity. For 𝐮 ∞ = 0 with nonzero stationary solution 𝐰 , Chen (1993), Kozono and Ogawa (1994), and Borchers and Miyakawa (1995) have studied the temporal stability in 𝐿 𝑝 spaces for 1 < 𝑝 and obtained good stability decay rates. For the spatial direction, we recently obtained some results. For 𝐮 ∞ ≠ 0 , Heywood (1970, 1972) and Masuda (1975) have studied the temporal stability in 𝐿 2 space. Shibata (1999) and Enomoto and Shibata (2005) have studied the temporal stability in 𝐿 𝑝 spaces for 𝑝 ≥ 3 . Then, Bae and Roh recently improved Enomoto and Shibata's results in some sense. In this paper, we improve Bae and Roh's result in the spaces 𝐿 𝑝 for 𝑝 > 1 and obtain 𝐿 𝑟 - 𝐿 𝑝 stability as Kozono and Ogawa and Borchers and Miyakawa obtained for 𝐮 ∞ = 0 .