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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 682436, 7 pages
http://dx.doi.org/10.1155/2012/682436
Research Article

A Regularity Criterion for the Navier-Stokes Equations in the Multiplier Spaces

1College of Physics and Electronic Information Engineering, Wenzhou University, Zhejiang, Wenzhou 325035, China
2The Key Laboratory of Low-voltage Apparatus Intellectual Technology of Zhejiang, Wenzhou 325035, China

Received 16 February 2012; Accepted 23 April 2012

Academic Editor: Benchawan Wiwatanapataphee

Copyright © 2012 Xiang'ou Zhu. 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 exhibit a regularity condition concerning the pressure gradient for the Navier-Stokes equations in a special class. It is shown that if the pressure gradient belongs to 𝐿 2 / ( 2 𝑟 ) ̇ 𝐻 ( ( 0 , 𝑇 ) ; ( 𝑟 ( 3 ̇ 𝐻 ) 𝑟 ( 3 ) ) ) , where ̇ 𝐻 ( 𝑟 ( 3 ̇ 𝐻 ) 𝑟 ( 3 ) ) is the multipliers between Sobolev spaces whose definition is given later for 0 < 𝑟 < 1 , then the Leray-Hopf weak solution to the Navier-Stokes equations is actually regular.