Mathematics Colloquium

Whether the 3D incompressible Euler equations can develop a finite-time singularity from smooth initial data remains one of the most challenging problems in nonlinear PDEs. In this talk, I will give an overview of my recent joint work with Dr. Jiajie Chen, in which we rigorously prove finite-time blowup for the 2D Boussinesq and 3D Euler equations with smooth initial data. Our approach is based on the dynamic rescaling formulation, which transforms the singularity problem into one of establishing the long-time stability of an approximate self-similar blowup profile. A key difficulty lies in proving linear stability and controlling several nonlocal terms. To address this, we decompose the solution operator into a leading-order part that admits sharp stability estimates using techniques from optimal transport, and a finite-rank perturbation that we estimate using space-time numerical solutions with rigorous error control. This framework allows us to establish the nonlinear stability of the approximate blowup profile and to prove stable, nearly self-similar blowup of the 3D Euler equations. The result highlights a fruitful synergy between analysis, numerical computation, and rigorous validation in tackling challenging problems in nonlinear PDEs.