Whether the 3D incompressible Euler equations can develop a finite-time singularity from smooth initial data is an outstanding open problem. In this talk, we will present a result inspired by the Hou-Luo scenario for a potential 3D Euler singularity, in which we prove finite time blowup of the 2D Boussinesq and 3D Euler equations with smooth initial data and boundary. There are several essential difficulties in establishing such blowup results. We overcome these difficulties by establishing a constructive proof strategy with computer assistance. Our strategy is to prove the nonlinear stability of an approximate self-similar blowup profile. To establish stability, we decompose the linearized operator into the leading order operator and the remainder. We use weighted energy estimates and sharp, functional inequalities based on optimal transport to establish the stability of the leading order operator. The key role of computer assistance is to construct an approximate blowup profile and approximate space-time solutions with rigorous error control, which provides critical small parameters in the energy estimates for the stability analysis and allows us to control the remainder perturbatively.

This is joint work with Tom Hou.