A Blowup Criteria of Smooth Solutions to the 3D Boussinesq Equations

In this work, we are concerned with the main mechanism for possible blow-up criteria of smooth solutions to the 3D incompressible Boussinesq equations. The main results state that the finite-time blowup/global existence of smooth solutions to the Boussinesq equation is controlled by either of the criteria \(\begin{aligned} u_{h}\in L^{2}\left( 0,T;\dot{B}_{\infty ,\infty }^{0}({\mathbb {R}} ^{3})\right) \quad \text {or}\quad \nabla _{h}u_{h}\in L^{1}\left( 0,T;\dot{B} _{\infty ,\infty }^{0}\left( {\mathbb {R}}^{3}\right) \right) , \end{aligned}\) $$\begin{array}{c}\hfill u_h\in L^2\left(0,,,T,;,{\dot{B}}_{\infty ,\infty }^0,({\mathbb{R}}^3)\right)\text{or}{\mathrm{\nabla }}_h{u}_h\in L^1\left(0,,,T,;,{\dot{B}}_{\infty ,\infty }^0,\left({\mathbb{R}}^3\right)\right),\end{array}$$ where \(u_{h}\) $u_h$ and \(\nabla _{h}\) ${\mathrm{\nabla }}_h$ denote the horizontal components of the velocity field and partial derivative with respect to the horizontal variables, respectively. We present a new simple proof for the regularity of this system without using the higher-order energy law and without any assumptions on the temperature \(\theta .\) $\theta .$ Our results extend the Navier–Stokes equations results in Dong and Zhang (Nonlinear Anal Real World Appl 11:2415–2421, 2010), Dong and Chen (J Math Anal Appl 338:1–10, 2008) and Gala and Ragusa (Electron J Qual Theory Differ Equ, 2016a) to Boussinesq equations.