What does this research mean for the field?

The 2D generalized Boussinesq equations with critical anisotropic fractional dissipation exhibit unconditional global well-posedness and global uniqueness in highly singular endpoint Besov spaces, with finite-time singularities governed solely by the accumulation of the maximum spatial gradient of the velocity field. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

This research aims to explore the well-posedness and blow-up criteria of the 2D generalized Boussinesq equations with critical fractional dissipation. It focuses on understanding singular conditions and developing mathematical estimates to manage fluid dynamics.

June 8, 2026

Comprehensive analysis of the 2D generalized Boussinesq equations with critical anisotropic fractional dissipation: well-posedness and blow-up criteria

Key Points

This research aims to explore the well-posedness and blow-up criteria of the 2D generalized Boussinesq equations with critical fractional dissipation. It focuses on understanding singular conditions and developing mathematical estimates to manage fluid dynamics.
Utilized anisotropic fractional dissipation in endpoint Besov spaces, highlighting critical conditions.
Employed contraction mapping arguments in critical Besov spaces for local well-posedness confirmation.
Developed sharp commutator estimates and Calderon-Zygmund integral estimates to manage pressure field isolation.
Established local well-posedness under critical conditions through refinement of estimates.
Proved a blow-up criterion where finite-time singularities arise solely from the maximum spatial gradient of the velocity field.
Maintained global well-posedness by balancing derivatives and utilizing pointwise inequalities.

Abstract

This paper considers the mathematical framework for the two-dimensional generalized Boussinesq equations driven by strictly anisotropic fractional dissipation in highly singular endpoint Besov spaces. We investigate the critical fluid regime where the velocity equation exclusively possesses horizontal fractional dissipation, while the active scalar temperature equation exclusively possesses vertical fractional dissipation, subject precisely to the critical singular condition where the sum of their fractional derivative orders equals one. To surmount the profound degeneracy caused by the complete lack of full Laplacian smoothing, we systematically develop a novel, sharp anisotropic commutator estimate within the refined Littlewood-Paley framework. We strictly prove the local well-posedness via a delicate contraction mapping argument exclusively in critical Besov spaces, simultaneously incorporating Calderon-Zygmund singular integral estimates to rigorously isolate the internal pressure field. Furthermore, we establish a sharp Beale-Kato-Majda type blow-up criterion, demonstrating that any finite-time singularity is fundamentally governed solely by the accumulation of the maximum spatial gradient of the velocity field. Ultimately, by meticulously balancing the cross-directional derivatives and employing a generalized Cordoba-Cordoba pointwise inequality, we extract global a priori bounds, upgrading the local solutions to unconditional global well-posedness. The global uniqueness is proven via a severe perturbation analysis in negative-index function spaces.

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