Global smooth solution to the 2D Boussinesq equations with fractional dissipation

Zhuan Ye

Published 2015-10-12Version 1

In this paper, we consider the two-dimensional (2D) incompressible Boussinesq system with fractional Laplacian dissipation and thermal diffusion. Based on the previous works and some new observations, we show that the condition $1-\alpha <\beta<\min\Big\{\frac{\alpha}{2},\,\,\frac{3\alpha-2}{2(2-\alpha)(1-\alpha)},\,\, \frac{3\alpha^{2}+4\alpha-4}{8(1-\alpha)}\Big\}$ with $0.7351\approx\frac{10-2\sqrt{10}}{5}<\alpha\leq \frac{4}{5}$ suffices in order for the solution pair of velocity and temperature to remain smooth for all time.