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

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.