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

This paper studies the global regularity problem for the two-dimensional incompressible Boussinesq equations with fractional dissipation given by $(-\Delta)^{\frac\alpha2}u$ and $(-\Delta)^{\frac\beta2} \theta$. Attention is focused on the subcritical regime where $\alpha+ \beta>1$. The case $\alpha >\frac23$ 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 $\alpha \leq \frac23$. 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.