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 $(-Δ)^{\frac\alpha2}u$ and $(-Δ)^{\frac\beta2} θ$. Attention is focused on the subcritical regime where $α+ β>1$. The case $α>\frac23$ 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 $α\leq \frac23$. We obtain the sharpest regularity result by minimizing assumptions on $α$ and $β$. We derive nonlinear lower bounds for the fractional Laplacian operator and implement an iterative procedure.