March 26, 2026
Thomas Y. Hou, the Charles Lee Powell Professor of Applied and Computational Mathematics at Caltech, will give the 2026 Zygmund-Calderón lectures during the week of May 18 - May 22.
Lecture 1:
Monday, May 18, 4:00PM - 5:00PM, Eckhart 202
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}
Lecture 2:
Wednesday, May 20, 4:00PM - 5:00PM, Eckhart 202
Title: Stable Nearly Self-Similar Blowup for the 2D Boussinesq and 3D Euler Equations with Smooth Data and Boundary
Abstract:
Whether the 3D incompressible Euler equations can develop a finite-time singularity from smooth initial data is one of the most challenging open problems in nonlinear PDEs. In joint work with Jiajie Chen, we prove stable nearly self-similar finite-time blowup for the 2D Boussinesq and 3D Euler equations with smooth initial data and boundary. Our proof is based on a nonlinear stability analysis of an approximate blowup profile in the dynamic rescaling formulation. A central ingredient is the decomposition of the solution operator into a leading-order part and a compact perturbation. For the leading-order operator, we derive sharp stability estimates using singularly weighted L^\infty and C^{1/2} norms, an optimal transport argument, and an analytic low-rank correction technique. The compact perturbation is approximated by a finite-rank operator and controlled through space-time numerical solutions with rigorous error bounds. This gives the first rigorous justification of the Hou–Luo blowup scenario.
Lecture 3:
Friday, May 22, 4:00PM - 5:00PM, Eckhart 202
Title: Nonuniqueness of Leray–Hopf Solutions for the Unforced 3D Incompressible Navier–Stokes Equations
Abstract:
The nonuniqueness of Leray–Hopf solutions to the unforced 3D incompressible Navier–Stokes equations is a central open problem in mathematical fluid dynamics. In joint work with Yixuan Wang and Changhe Yang, we develop the first rigorous computer-assisted proof of such nonuniqueness. Inspired by earlier work in this area, we construct a Leray–Hopf solution in a self-similar setting and then prove the existence of a second solution by studying the linearized operator around this profile and showing that it admits an unstable perturbation. Our approach combines a novel high-precision numerical method for computing candidate solutions with a rigorous framework for establishing exact solutions in their neighborhood. A key ingredient is the decomposition of the linearized operator into a coercive part and a compact perturbation, followed by a finite-rank approximation of the compact part up to a small error. We then rigorously verify, by computer-assisted proof, the invertibility of the linearized operator restricted to the image of this finite-rank approximation. This yields a certified unstable eigenpair and, consequently, a second solution—indeed, infinitely many Leray–Hopf solutions.