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On the Cauchy problems for certain Boussinesq-α equations

Published online by Cambridge University Press:  30 March 2010

Yong Zhou  and
Jishan Fan
Yong Zhou
Affiliation:
Department of Mathematics, Zhejiang Normal University, Jinhua 321004, Zhejiang, People's Republic of China (yzhoumath@zjnu.edu.cn)
Jishan Fan
Affiliation:
Department of Applied Mathematics, Nanjing Forestry University, Nanjing 210037, Jiangsu, People's Republic of China (fanjishan@njfu.com.cn)
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Abstract

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We study the Cauchy problem of certain Boussinesq-α equations in n dimensions with n = 2 or 3. We establish regularity for the solution under ▽u ∈ L1 (0, T; Ḃ0∞,∞(ℝn)). As a corollary, the smooth solution of the Leray-α–Boussinesq system exists globally, when n = 2. For the Lagrangian averaged Boussinesq equations, a regularity criterion ▽θ ∈ L1(0, T;L∞(ℝ2)) is established. Other Boussinesq systems with partial viscosity are also discussed in the paper.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 140 , Issue 2 , April 2010 , pp. 319 - 327
Copyright
Copyright © Royal Society of Edinburgh 2010