Global Regularity for the 2D Boussinesq Equations with Temperature-Dependent Viscosity

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.