From Instability to Singularity Formation in Incompressible Fluids

We establish finite-time singularity formation for $C^{1,\alpha}$ solutions to the Boussinesq system that are compactly supported on $\mathbb{R}^2$ and infinitely smooth except in the radial direction at the origin. The solutions are smooth in the angular variable at the blow-up point, which was a fundamental obstruction in previous works. This is done by exploiting a second-order effect, related to the classical Rayleigh--B\'enard instability, that overcomes the regularizing effect of transport. A similar result is established for the 3d Euler system based on the Taylor--Couette instability.