Global regularity of the two-dimensional fractional Boussinesq equations with critical dissipation

Abstract

In the present paper, our primary focus is on the Boussinesq equations with fractional velocity dissipation $Λ^αu$ and temperature diffusion $Λ^βθ$. On the critical balance line $α+β=1$, establishing global regularity for weak velocity dissipation has been obstructed by the quadratic convective-buoyancy feedback, and all available results below $α=\frac{10}{13}$ require a smallness assumption on the initial temperature. In this paper, we prove large-data global regularity across the extended regime $α>\frac{\sqrt{57}-5}{4} \approx 0.637$. This provides a step beyond the previous barrier $α=\frac{10}{13}$, unconditionally dismantling the temperature smallness assumption that has persisted in the literature for over a decade. The mathematical mechanism relies on a high-order Quadratic Normal Form transformation $H=ω-R_αθ-\Bop(θ,θ)$ tailored to eliminate non-resonant quadratic vorticity-temperature interactions. Crucially, by uncovering the exact physical scalings of the underlying multilinear commutators and maintaining sharp fractional conjugates in the $L^m$-energy framework, we ensure that the global energy closure is governed entirely by the quadratic thermal dissipation. Once an integrability threshold $m>\frac{2}α$ is attained, a fine Littlewood-Paley transport estimate drives a fractional regularity gain $H \in B_{r,\infty}^{3α-2}$, ensuring the velocity field achieves critical drift regularity $C^α$. By Silvestre-type drift regularization, this spontaneously generates positive Hölder continuity for the temperature, which in turn elevates the drift regularity strictly beyond the critical threshold, unlocking non-local Schauder estimates to prevent finite-time singularity formation.