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

Bulletin of the Brazilian Mathematical Society, New Series Pub Date : 2024-02-02 DOI:10.1007/s00574-024-00383-x

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Abstract

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 $$\begin{aligned} u_{h}\in L^{2}\left( 0,T;\dot{B}_{\infty ,\infty }^{0}({\mathbb {R}} ^{3})\right) \quad \text {or}\quad \nabla _{h}u_{h}\in L^{1}\left( 0,T;\dot{B} _{\infty ,\infty }^{0}\left( {\mathbb {R}}^{3}\right) \right) , \end{aligned}$$ where $u_{h}$ and $\nabla _{h}$ 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 $\theta .$ 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.

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三维布森斯克方程光滑解的吹胀标准

摘要在这项工作中，我们关注的是三维不可压缩布辛斯方程光滑解的可能炸毁标准的主要机制。主要结果表明，Boussinesq方程光滑解的有限时间炸毁/全局存在性受$$begin{aligned} u_{h}\in L^{2}\left( 0,T.)或$$begin{aligned} u_{h}$0,T.)标准控制；\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}其中 \(u_{h}$ 和 $\nabla _{h}$ 分别表示速度场的水平分量和关于水平变量的偏导数。我们的结果将董和张（Nonlinear Anal Real World Appl 11:2415-2421，2010）、董和陈（J Math Anal Appl 338:1-10，2008）以及加拉和拉古萨（Electron J Qual Theory Differ Equ，2016a）中的纳维-斯托克斯方程结果扩展到了布西内斯克方程。

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Bulletin of the Brazilian Mathematical Society, New Series

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