Finite-time singularity via multi-layer degenerate pendula for the 2D Boussinesq equation with uniform $C^{1,\sqrt{\frac{4}{3}}-1-\epsilon }\cap L^2$ force

We establish the existence of compactly supported solutions of the inviscid incompressible 2D Boussinesq equation with $C^{1,\sqrt{\frac{4}{3}}-1-\epsilon }\cap L^2$ force that develop a singularity in finite time. Importantly, the force preserves this regularity at the blow-up time. Moreover, the forces in the vorticity and density equations have compact support. The mechanism behind the blow-up is an accumulated hysteresis effect on the vorticity caused by an infinite chain of “degenerate” pendula and flickering density.