Zygmund-Calderón Lectures in Analysis (2026) - Lecture 2 - Thomas Y. Hou (Caltech)

Title: Stable Nearly Self-Similar Blowup for the 2D Boussinesq and 3D Euler Equations with Smooth Data and Boundary Abstract: Whether the 3D incompressible Euler equations can develop a finite-time singularity from smooth initial data is one of the most challenging open problems in nonlinear PDEs. In joint work with Jiajie Chen, we prove stable nearly self-similar finite-time blowup for the 2D Boussinesq and 3D Euler equations with smooth initial data and boundary. Our proof is based on a nonlinear stability analysis of an approximate blowup profile in the dynamic rescaling formulation. A central ingredient is the decomposition of the solution operator into a leading-order part and a compact perturbation. For the leading-order operator, we derive sharp stability estimates using singularly weighted L^\infty and C^{1/2} norms, an optimal transport argument, and an analytic low-rank correction technique. The compact perturbation is approximated by a finite-rank operator and controlled through space-time numerical solutions with rigorous error bounds. This gives the first rigorous justification of the Hou–Luo blowup scenario. May 20, 2026