Anisotropic Dissipation and Stability for the 2D Boussinesq Equations: Mechanisms, Methods and Open Problems
DOI:
https://doi.org/10.70112/ajsat-2026.15.2.4362Keywords:
Anisotropic Boussinesq Equations, Partial Dissipation, Mean–Oscillation Decomposition, Poincaré Inequality, Enhanced Dissipation, Singularity Formation, SurveyAbstract
This paper surveys the theory of global regularity, singularity formation and stability near equilibrium for the two-dimensional Boussinesq equations with anisotropic and partial dissipation. The survey is organised around the identification of three distinct mechanisms by which an incompletely dissipative stratified fluid can nevertheless be stabilised. Geometric coercivity operates on domains with a periodic direction, where a Poincaré inequality renders the surviving dissipation coercive on all mean-free modes; it yields exponential decay of the oscillation together with a persistent horizontal mean, so that the solution converges to its own horizontal average. Dispersion operates on the whole plane, where no Poincaré inequality exists: the velocity–temperature coupling generates internal gravity waves with phase ±|ξ_1 |/|ξ|, and their dispersive decay supplies the stabilisation that the geometry cannot. Mixing operates near a shear background, where the transport operator produces enhanced dissipation on a timescale far shorter than the viscous one. We present the mean oscillation decomposition in detail as the representative technique of the first mechanism, tabulate the known stability results across dissipation configurations and domains, and close with open problems, revised to reflect the reported resolution during 2025–2026 of two questions long regarded as central.
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