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Global smooth solution to the 2D Boussinesq equations with fractional dissipation. (English) Zbl 1369.35072
Summary: In this paper, we consider the 2D incompressible Boussinesq system with fractional Laplacian dissipation and thermal diffusion. On the basis of the previous works and some new observations, we show that the condition $$1-\alpha < \beta < \min \left\{3-3 \alpha, \frac {\alpha}{2}, \frac {3 \alpha^2+4 \alpha-4}{8(1-\alpha)}\right\}$$ with $$0.7351 \approx \frac {10-2\sqrt{10}}{5} < \alpha < 1$$ suffices in order for the solution pair of velocity and temperature to remain smooth for all time.

##### 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
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