Electron. J. Diff. Equ., Vol. 2015 (2015), No. 62, pp. 1-9.

Blow-up criterion for the zero-diffusive Boussinesq equations via the velocity components

Weihua Wang

Abstract:
This article concerns the blow up for the smooth solutions of the three-dimensional Boussinesq equations with zero diffusivity. It is shown that if any two components of the velocity field u satisfy
$$ \int_0^T \frac{ \||u_1|+|u_2|\|^q_{L^{p,\infty}} } {1+\ln ( e+\|\nabla u\|^2_{L^2}) } ds<\infty,\quad \frac{2}{q}+\frac{3}{p}=1,\quad 3<p<\infty, $$
then the local smooth solution $(u,\theta)$ can be continuously extended to $(0,T_1)$ for some $T_1>T$.

Submitted September 29, 2014. Published March 11, 2015.
Math Subject Classifications: 35Q35, 76D05.
Key Words: Zero-diffusive Boussinesq equations; blow up criterion; Lorentz spaces.

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Weihua Wang
School of Mathematics and Statistics
Hubei University, Wuhan 430062, China
email: wwh73@hubu.edu.cn

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