H.B. Keller Colloquium

OpenAI's recently announced construction of a finite-time singularity for the Navier–Stokes equations relies crucially on C∞C^\infty external forcing. After brief remarks on this development, I will present two computer-assisted proofs for unforced incompressible flows. The first establishes finite-time, nearly self-similar blowup for the three-dimensional axisymmetric Euler equations with smooth, finite-energy initial data in a domain with boundary. It uses nonlinear stability around an approximate self-similar profile. The second establishes nonuniqueness of Leray–Hopf solutions to the unforced three-dimensional Navier–Stokes equations from a common nonsmooth, finite-energy initial velocity. Key ingredients include the construction of an exact forward self-similar profile and rigorous verification of an unstable eigenmode of the associated linearized operator. Both analyses share a central strategy: decomposing the linearized operator into a coercive part, controlled through sharp functional-analytic estimates in weighted spaces, and a finite-rank perturbation, rigorously estimated with computer assistance. The talk will highlight the complementary roles of stability and instability and how numerical observations can be transformed into rigorous results. The Euler work is joint with Jiajie Chen, and the Navier–Stokes work is joint with Yixuan Wang and Changhe Yang.