This paper is devoted to the global regularity for the Cauchy problem of the two-dimensional Boussinesq equations with the temperature-dependent viscosity. We prove the global solutions for this system with any positive power of the fractional Laplacian for temperature under the assumption that the viscosity coefficient is sufficiently close to some positive constant. Our obtained result improves considerably the recent results in Abidi and Zhang (Adv Math 305:1202–1249, 2017) and Zhai et al. (J Differ Equ 267:364–387, 2019). In addition, a regularity criterion via the velocity is also obtained for this system without the above assumption on the viscosity coefficient.

中文翻译：

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具有温度相关粘度的二维Boussinesq方程的整体正则性

本文致力于二维温度随粘度变化的Boussinesq方程的Cauchy问题的整体正则性。在粘度系数足够接近某个正常数的假设下，我们用分数拉普拉斯算术对温度的任何正幂证明了该系统的整体解。我们获得的结果大大改善了Abidi和Zhang（Adv Math 305：1202-1249，2017）和Zhai等人的最新结果。（J Differ Equ 267：364–387，2019）。另外，在不对粘度系数进行上述假设的情况下，也通过该系统获得了通过速度的规则性标准。