Global regularity results for the 2D Boussinesq equations and micropolar equations with partial dissipation

This paper establishes the global regularity of the two-dimensional (2D) Boussinesq equations and micropolar equations with partial dissipation. Our first result is the global regularity of the 2D Boussinesq equations with fractional vertical dissipation in the horizontal velocity, horizontal dissipation in the vertical velocity and zero thermal diffusion, which is shown by taking advantage of the nice structure of the 2D Boussinesq equations and several refined commutator estimates. The second goal of this paper is to consider a system of the 2D incompressible micropolar equations with vertical dissipation in the horizontal velocity equation, horizontal dissipation in the vertical velocity equation and the fractional Lambda(alpha) dissipation in the micro-rotation velocity. In order to overcome the difficulty caused by the lack of full Laplacian diffusion in the velocity equations, we fully exploit the nice structure of the corresponding equations to show that this equations with arbitrarily small alpha > 0 always possesses a unique global classical solution. (C) 2019 Elsevier Inc. All rights reserved.