Global regularity of the 2D fractional Boussinesq equations with subcritical dissipation

Abstract

This paper studies the global regularity problem for the two-dimensional incompressible Boussinesq equations with fractional dissipation given by $(-\Delta {)}^{\frac{\alpha }{2}}u$ and $(-\Delta {)}^{\frac{\beta }{2}}\theta$. Attention is focused on the subcritical regime where $\alpha +\beta >1$. The case $\alpha >\frac{2}{3}$ was recently settled in a joint work of the authors [Math. Ann., \textbf{391} (2025), 5965-6012], which established global regularity under this condition. This paper addresses the remaining case $\alpha \le \frac{2}{3}$. We obtain the sharpest regularity result by minimizing assumptions on $\alpha$ and $\beta$. We derive nonlinear lower bounds for the fractional Laplacian operator and implement an iterative procedure.

43 pages