Non Existence of the Stable 2D Boussinesq Equations in $H^2$ via Density-Vorticity Feedback

Abstract

We establish instantaneous $H^2$ norm explosion for the two-dimensional Boussinesq equations around the linearly stable background profile $-x_2$. While previous ill-posedness results for this system relied on $L^{\infty }$-type spaces, this work provides the first strong ill-posedness result in the energy-critical space, where stable background stratification acts to suppress deformation growth. We adapt an idea from our previous work on the IPM equation \cite{BCMZ2025} and select an initial density perturbation vanishing near the origin to neutralize the stable stratification. Crucially, unlike the IPM setting, the evolution is here driven by the non-linear coupling between density and vorticity. Since the initial vorticity vanishes identically, the instantaneous loss of regulari\-ty is driven by density-vorticity feedback - a mechanism fundamentally distinct from both $H^1$ Euler ill-posedness of Bourgain \& Li and the IPM evolution. Through an iterative scheme yielding a sequence of approximations with strong deformation at the origin, we construct a strong solution $(\rho (t),\omega (t))$ originating from data with arbitrarily small $H^2\times (H^1\cap {\dot{H}}^{-1})$ norm, which immediately exhibits $H^2\times H^1$ norm explosion for all $t>0$ while maintaining uniqueness within the energy class.