Zygmund-Calderón Lectures in Analysis (2026) - Lecture 3 - Thomas Y. Hou (Caltech)

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. May 22, 2026