报告摘要： The nonuniqueness of Leray–Hopf solutions to the unforced, incompressible 3D Navier–Stokes equations is a longstanding open problem in mathematical fluid dynamics. In this talk I will present, to our knowledge, the first rigorous computer-assisted proof of such nonuniqueness. Building on ideas of Jia–Šverák, Guillod–Šverák, and Albritton–Brue–Colombo, we construct a self-similar Leray–Hopf solution and then, via a stability analysis of the linearized operator around this profile, show the existence of a second Leray–Hopf solution generated by an unstable perturbation. Our proof blends high-precision numerics with a fully rigorous a posteriori validation framework. We first decompose the linearized operator into a coercive part plus a compact perturbation. We then approximate this compact perturbation by a finite-rank operator with controlled error and rigorously verify invertibility on the image of this finite-rank operator. This certifies the existence of an unstable eigenpair and, consequently, yields a second solution—indeed, infinitely many Leray–Hopf solutions. This is a joint work with Yixuan Wang and Changhe Yang.
报告人简介：Professor Thomas Yizhao Hou is the Charles Lee Powell Professor of Applied and Computational Mathematics at California Institute of Technology. He is a distinguished applied and computational mathematician renowned for his exceptional strengths in both numeric and analysis, and he has made groundbreaking contributions to several frontier areas in applied mathematics, including computational fluid dynamics, multiscale analysis, and singularity formation of three-dimensional incompressible Euler and Navier-Stokes equations. Prof. Hou has received a number of awards and honors, including Alfred P. Sloan Research Fellowship in 1990, Feng Kang Prize in Scientific Computing in 1997, Francois Frenkiel Award from American Physical Society in 1998, James H. Wilkinson Prize in Numerical Analysis and Scientific Computing from SIAM in 2001, Morningside Gold Medal in Applied Mathematics in 2004, the Computational and Applied Sciences Award from United States Association of Computational Mechanics in 2005, SIAM Outstanding Paper Prize in 2018, SIAM Ralph E. Kleinman Prize in 2023, and William Benter Prize in Applied Mathematics in 2024. Prof. Hou was also an invited Speaker at International Congress of Mathematicians in 1998 and a plenary speaker at International Congress on Industrial and Applied Math in 2003. Prof. Hou was elected SIAM Fellow in 2009, Fellow of American Academy of Art and Sciences in 2011, Fellow of American Mathematical Society in 2012, and Member of National Academy of Sciences in 2024. He has served as the founding Editor-in-Chief of SIAM first interdisciplinary journal "Multiscale Modeling and Simulation" from 2002 to 2007.
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会议号: 863 8009 1584
密码: 912280
