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 Λαu and temperature diffusion Λβθ. 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=10/13 require a smallness assumption on the initial temperature. In this paper, we prove large-data global regularity across the extended regime α>(√(57)−5)/4 ≈ 0.637. This provides a step beyond the previous barrier α=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−Rαθ−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 Lm-energy framework, we ensure that the global energy closure is governed entirely by the quadratic thermal dissipation. Once an integrability threshold m>2/α is attained, a fine Littlewood-Paley transport estimate drives a fractional regularity gain H ∈ Br,∞3α−2, ensuring the velocity field achieves critical drift regularity Cα. 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.
Related
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BibTeX
@misc{li2026global,
title = {{Global regularity of the two-dimensional fractional Boussinesq equations with critical dissipation}},
author = {Qian Li and Jiahong Wu and Xiaojing Xu and Zhuan Ye},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.08765},
eprint = {2611.08765},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.08765},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Global regularity of the two-dimensional fractional Boussinesq equations with critical dissipation
AU - Qian Li
AU - Jiahong Wu
AU - Xiaojing Xu
AU - Zhuan Ye
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.08765
UR - https://arxiv.org/abs/2611.08765
AB - 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=10/13 require a smallness assumption on the initial temperature. In this paper, we prove large-data global regularity across the extended regime α>(√(57)−5)/4 ≈ 0.637. This provides a step beyond the previous barrier α=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>2/α is attained, a fine Littlewood-Paley transport estimate drives a fractional regularity gain H ∈ B_r,∞^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.
ER -