H.B. Keller Colloquium Series

Whether the 3D incompressible Euler equations can develop a finite time singularity from smooth initial data is one of the most challenging problems in nonlinear PDEs. In this talk, I will present a new exciting result with Dr. Jiajie Chen in which we prove finite time blowup of the 2D Boussinesq and 3D Euler equations with smooth initial data. There are several essential difficulties in establishing such blowup result. We overcome these difficulties by decomposing the solution operator into a leading order operator that enjoys sharp stability estimates plus a finite rank perturbation operator that can be estimated by using energy estimates and space-time numerical solutions with rigorous error control. This enables us to establish nonlinear stability of the approximate self-similar profile and prove stable nearly self-similar blowup of the 2D Boussinesq and 3D Euler equations with smooth initial data and boundary. This provides the first rigorous justification of the Luo-Hou blowup scenario.