On the Dimensional Disparities of Incompressible Boussinesq Equations via Calderón-Zygmund Operators

In this paper, we investigate the fundamental structural disparities between twodimensional (2D) and three-dimensional (3D) incompressible Boussinesq equations through the lens of Calderón-Zygmund singular integral operators. Specifically, we analyze the pressure Hessian, ∂i∂jp, and its localized behavior by employing Riesz transforms, RiRj . By contrasting the regularizing thermal effects in 2D with the vortex stretching mechanism in 3D, we establish a comparative spectral framework using Littlewood-Paley decomposition. This methodological approach not only highlights the geometric constraints of the pressure gradient but also provides a rigorous basis for understanding high-frequency interactions in thermally driven fluid flows.