Electron. J. Diff. Equ., Vol. 2016 (2016), No. 73, pp. 1-11.

Blow-up criterion for the 2D Euler-Boussinesq system in terms of temperature

Chenyin Qian

Abstract:
In this article, we study the blow-up slutions for the 2D Euler-Boussinesq equation. In particular, it is shown that if
$$ \int_0^{T^*} \sup_{r\geq 2}\frac{\|\Lambda^{1-\alpha} \theta(t)\|_{L^{r}}} {\sqrt{r\log r}}\,dt<\infty \quad \text{or}\quad \int_0^{T^*} \|\Lambda^{1-\alpha} \theta\|_{\dot{B}^0_{\infty,\infty}}\, dt <\infty, $$
then the local solution can be continued to the global one. This is an improvement of classical Lipschitz-type blow-up criterion ($\|\nabla\theta\|_{L^1_tL^{\infty}}$) in terms of the temperature $\theta$.

Submitted December 18, 20915. Published March 15, 2016.
Math Subject Classifications: 35Q35, 35B35, 35B65, 76D05.
Key Words: 2D Boussinesq equation; blow-up criterion; Besov space; transport equation.

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Chenyin Qian
Department of Mathematics
Zhejiang Normal University
Jinhua 321004, China
email: qcyjcsx@163.com

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