Remarks on the blow-up  criterion  for smooth solutions of the  Boussinesq equations  with zero  diffusion

Yan Jia; Xingwei Zhang; Bo-Qing Dong

doi:10.3934/cpaa.2013.12.923

Article Contents

2013, Volume 12, Issue 2: 923-937. Doi: 10.3934/cpaa.2013.12.923

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Remarks on the blow-up criterion for smooth solutions of the Boussinesq equations with zero diffusion

1.
School of Mathematical Sciences, Anhui University, Hefei 230601
2.
School of Mathematical Science, Anhui University, Hefei 230039

Received: December 31, 2011

Revised: December 31, 2011

Published: February 2013

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Abstract

This article is concerned with the blow-up criterion for smooth solutions of three-dimensional Boussinesq equations with zero diffusion. It is shown that if the velocity field $u(x,t)$ satisfies \begin{eqnarray*} u\in L^p(0,T_1;B^r_{q,\infty}(R^3)),\quad \frac{2}{p}+\frac{3}{q}=1+r,\quad \frac{3}{1+r}< q \leq \infty, \quad -1 < r \leq 1, \end{eqnarray*} then the solution can be continually extended to the interval $(0,T)$ for some $T>T_1$.

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Mathematics Subject Classification: Primary: 35Q35; Secondary: 76D05.

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