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arXiv · 2606.03680

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

Atanas Stefanov·Jiahong Wu·Xiaojing Xu·Zhuan Ye
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Abstract

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.

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Atanas Stefanov, Jiahong Wu, Xiaojing Xu, Zhuan Ye. 2026-06-02. Global regularity of the 2D fractional Boussinesq equations with subcritical dissipation. https://arxiv.org/abs/2606.03680

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