Speaker

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　　Time

　　Venue

Zoom 会议号：863 6187 9778
https://us06web.zoom.us/j/86361879778

　　Abstract

　　Whether the 3D incompressible Euler equations can develop a finite time singularity from smooth initial data is one of the most challenging problems in nonlinear PDEs. In this talk, we will present a new exciting result with Dr. Jiajie Chen in which we prove finite time blowup of the 2D Boussinesq and 3D Euler equations with smooth initial data and boundary. There are several essential difficulties in establishing such blowup results. We overcome these difficulties by establishing a constructive proof strategy. We first construct an approximate self-similar blowup profile using the dynamic rescaling formulation. To establish the stability of the approximate blowup profile, we decompose the linearized operator into a leading order operator plus a finite rank perturbation operator. We use sharp functional inequalities and optimal transport to establish the stability of the leading order operator. To estimate the finite rank operator, we use energy estimates and space-time numerical solutions with rigorous error control. This enables us to establish nonlinear stability of the approximate self-similar profile and prove stable nearly self-similar blowup of the 2D Boussinesq and 3D Euler equations with smooth initial data. This provides the first rigorous justification of the Hou-Luo blowup scenario.

　　Affiliation

　　侯一钊，加州理工学院应用数学系Charles Lee Powell讲座教授，美国艺术与科学院院士。在流体力学、多尺度问题、非线性偏微分方程等方面有许多独创性成果。

　　侯一钊教授曾获冯康科学计算奖（1997），美国工业与应用数学学会 James H. Wilkinson 数值分析和科学计算奖（2001），晨兴数学金奖（2004），美国应用与计算力学会计算与应用科学奖（2005），美国工业与应用数学学会杰出论文奖（2018）和 SIAM Review SIGEST Award（2019）等奖励。

　　侯一钊教授1998年在国际数学家大会做邀请报告，2003年国际工业与应用数学大会做大会特邀报告。