Regularity Criteria for the 2d Boussinesq Equations with Supercritical Dissipation∗

This paper focuses on the 2D incompressible Boussinesq equations with fractional dissipation, given by Λαu in the velocity equation and by Λβθ in the temperature equation, where Λ= √−Δ denotes the Zygmund operator. Due to the vortex stretching and the lack of sufficient dissipation, the global regularity problem for the supercritical regime α+β<1 remains an outstanding problem. This paper presents several regularity criteria for the supercritical Boussinesq equations. These criteria are sharp and reflect the level of difficulty of the supercritical Boussinesq problem. In addition, these criteria are important tools in understanding some crucial properties of Boussinesq solutions such as the eventual regularity.