By means of Fourier analysis and the structure of the equation, for the 3D Boussinesq system with partial viscosity, we show the blow-up criterion of the smooth solution via a Beale-Kato-Majda type of ${sup}_{j\in \mathbb{Z}}{\int }_0^T{\Vert {\dot{\Delta }}_j\nabla \times u\Vert }_{\infty }dt$ and a Serrin type of ${\Vert u\Vert }_{C([0,T),B_{\infty ,\infty }^{-1})}$ or especially its oscillation in the $B_{\infty ,\infty }^{-1}$ norm with small amplitude.

中文翻译：

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限制情况下具有部分粘性项的 3D Boussinesq 方程的正则性准则

通过傅里叶分析和方程的结构，对于具有部分粘度的 3D Boussinesq 系统，我们通过 Beale-Kato-Majda 类型展示了光滑解的爆破准则${支持}_{j\in \mathbb{Z}}{\int }_0^{吨}{\Vert {\dot{\Delta }}_j\nabla \times 你\Vert }_{\infty }d吨$和 Serrin 类型的${\Vert 你\Vert }_{C([0,吨),{乙}_{\infty ,\infty }^{-1})}$或者特别是它的振荡${乙}_{\infty ,\infty }^{-1}$幅度较小的范数。