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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 627813, 12 pages
http://dx.doi.org/10.1155/2012/627813
Research Article

Long-Time Decay to the Global Solution of the 2D Dissipative Quasigeostrophic Equation

Department of Mathematics, College of Science, King Saud University, Riyadh 11451, Saudi Arabia

Received 10 January 2012; Accepted 8 March 2012

Academic Editor: Muhammad Aslam Noor

Copyright © 2012 Jamel Benameur and Mongi Blel. 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 study the behavior at infinity in time of any global solution 𝜃 𝒞 ( + , ̇ 𝐻 2 2 𝛼 ( 2 ) ) of the surface quasigeostrophic equation with subcritical exponent 2 / 3 𝛼 1 . We prove that l i m 𝑡 𝜃 ( 𝑡 ) ̇ 𝐻 2 2 𝛼 = 0 . Moreover, we prove also the nonhomogeneous version of the previous result, and we prove that if 𝜃 𝒞 ( + , ̇ 𝐻 2 2 𝛼 ( 2 ) ) is a global solution, then l i m 𝑡 𝜃 ( 𝑡 ) 𝐻 2 2 𝛼 = 0 .