Blow-up Criteria of Boussinesq Equations Without Thermal Diffusion Involving the Middle Eigenvalue of the Strain Tensor

In this paper, we consider blow-up criteria for classical solutions to the 3D Boussinesq equations without thermal diffusion via the middle eigenvalue of the strain tensor. It proved that if the middle eigenvalue of the strain tensor satisfies \( \lambda _2^{+}=\max \{\lambda _2,0\}\in L^p(0,T;L^q(\mathbb {R}^3))\quad \text { with }\quad \frac{2}{p}+\frac{3}{q}=2~\text { and }~\frac{3}{2}<q \le \infty , \) $${\lambda }_2^{+}=max\{{\lambda }_2,0\}\in L^p(0,T;L^q({\mathbb{R}}^3))\text{with}\frac{2}{p}+\frac{3}{q}=2\text{and}\frac{3}{2}<q\le \infty ,$$ then the local classical solution can be extended beyond the time T.