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Global regularity for the 2D Boussinesq equations with temperature-dependent viscosity. (English) Zbl 1433.35265
Summary: 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 [H. Abidi and P. Zhang, Adv. Math. 305, 1202–1249 (2017; Zbl 1353.35220)] and [X. Zhai et al., J. Differ. Equations 267, No. 1, 364–387 (2019; Zbl 1414.35153)]. In addition, a regularity criterion via the velocity is also obtained for this system without the above assumption on the viscosity coefficient.
##### MSC:
 35Q35 PDEs in connection with fluid mechanics 35B65 Smoothness and regularity of solutions to PDEs 76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids 35R11 Fractional partial differential equations 42B25 Maximal functions, Littlewood-Paley theory
##### Keywords:
Boussinesq equations; variable viscosity; global regularity
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##### References:
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