The Boussinesq approximation is a fundamental model for fluid motion, with a mathematical structure closely related to the Navier–Stokes and Euler equations. For large-scale geophysical flows, such as those in the ocean and atmosphere, it can be simplified under the hydrostatic balance, yielding the primitive equations. In the viscous setting, global regularity was established by Cao and Titi, while the inviscid case presents major challenges, including instabilities and possible finite-time singularities.

This mini-course focuses on the inviscid regime, addressing well- and ill-posedness, the formation and stability of singularities, and the effect of strong Coriolis forces on solution lifespan. It also discusses the justification of the hydrostatic balance.

References: 

Ghoul-Ibrahim-Lin-Titi: On the effect of rotation on the lifespan of analytic solutions to the 3D inviscid primitive equations, ARMA 2022

Collot-Ibrahim-Lin: Stable singularity formation for the Inviscid Primitive Equations”.  Ann. Henri Poincaré 2024

Bianchini, Coti-Zelati-Ertzbischoff: Ill-posedness of the hydrostatic Euler-Boussinesq equations and failure of hydrostatic limit. Comm. Math. Phys. 2025.