Zygmund-Calderón Lectures in Analysis (2026) - Lecture 1 - Thomas Y. Hou (Caltech)

Title: Potentially Singular Behavior in the 3D Navier–Stokes Equations and Related Models Abstract: Whether the three-dimensional incompressible Navier–Stokes equations can develop a finite-time singularity from smooth initial data is one of the seven Clay Millennium Prize Problems. In this talk, I will first review our recent work establishing a rigorous computer-assisted proof of singularity formation for the 3D Euler equations with smooth initial data in the presence of a boundary. I will then present numerical evidence suggesting potentially singular behavior in the 3D Navier–Stokes equations near the origin, with the maximum vorticity increasing by a factor of 10^7. Several blowup criteria are applied to assess this behavior. Finally, I will present new numerical evidence indicating that a generalized axisymmetric Navier–Stokes equation in a dimension slightly higher than three develops a tornado-like self-similar blowup, with the maximum vorticity increasing by a factor of 10^21. May 18, 2026