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

Abstract

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 $1-\alpha <\beta <\min \{3-3\alpha ,\frac{\alpha }{2},\frac{3{\alpha }^2+4\alpha -4}{8(1-\alpha )}\}$ 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.

19 pages. This version adds some details and makes some changes. arXiv admin note: text overlap with arXiv:1506.08993

References in corpus (2)

• An incompressible 2D didactic model with singularity and explicit solutions of the 2D Boussinesq equations
• Sufficient and Necessary Conditions for the fractional Gagliardo-Nirenberg Inequalities and applications to Navier-Stokes and generalized boson equations