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 uh is an element of L20,T ;B infinity,infinity 0(R3)or backward difference huh is an element of L10,T ;B infinity,infinity 0R3,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} u_{h}\in L<^>{2}\left( 0,T;\dot{B}_{\infty ,\infty }<^>{0}({\mathbb {R}} <^>{3})\right) \quad \text {or}\quad \nabla _{h}u_{h}\in L<^>{1}\left( 0,T;\dot{B} _{\infty ,\infty }<^>{0}\left( {\mathbb {R}}<^>{3}\right) \right) , \end{aligned}$$\end{document}where uh\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_{h}$$\end{document} and backward difference h\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla _{h}$$\end{document} 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.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta .$$\end{document} 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.