Speaker: Thomas Hou (Caltech)

Whether the 3D incompressible Euler equations can develop a finite-time singularity from smooth initial data remains one of the central open problems in nonlinear PDEs. In this talk, I will present recent joint work with Dr. Jiajie Chen, in which we rigorously prove finite-time blowup for the 2D Boussinesq equations and the 3D axisymmetric Euler equations with smooth initial data and smooth boundary. Our approach uses a dynamically rescaled formulation that reduces singularity formation to the long-time stability of an approximate self-similar blowup profile. A key difficulty is proving the linear stability of a numerically constructed profile. To address this, we decompose the solution operator into a leading-order part, which admits sharp stability estimates, and a finite-rank perturbation, which is controlled by a computer-assisted proof. I will also discuss recent joint work with Yixuan Wang and Changhe Yang on nonuniqueness of Leray–Hopf solutions to the unforced 3D incompressible Navier–Stokes equations. In this setting, the viscous term introduces several new ingredients but also greatly simplifies the analysis: standard $H^1$ estimates suffice, without the singular weights needed in the inviscid case. A central step is to establish the existence of a self-similar Leray–Hopf solution and then prove the existence of a second solution by analyzing the stability of the linearized operator around this profile and showing that it admits an unstable mode. These results highlight the fruitful interplay among analysis, computation, and rigorous validation in nonlinear PDEs.