Analysis Seminar

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A constructive proof of finite time blowup of 3D incompressible Euler equations with smooth data and boundary

Speaker: Jiajie Chen, Courant Institute

Location: Warren Weaver Hall 1302

Date: Thursday, November 17, 2022, 11 a.m.

Synopsis:

Whether the 3D incompressible Euler equations can develop a finite-time singularity from smooth initial data is an outstanding open problem. In this talk, we will present a result inspired by the Hou-Luo scenario for a potential 3D Euler singularity, in which we prove finite time blowup of the 2D Boussinesq and 3D Euler equations with smooth initial data and boundary. To establish the blowup results, we develop a constructive proof strategy with computer assistance and prove the nonlinear stability of an approximate self-similar blowup profile. In the stability analysis, we decompose the linearized operator into the leading order operator and the remainder. We develop sharp functional inequalities using optimal transport and the symmetry properties of the velocity kernels to estimate the nonlocal terms from the velocity and use weighted energy estimates to establish the stability analysis of the leading order operator. The key role of computer assistance is to construct an approximate blowup profile and approximate space-time solutions with rigorous error control, which provides critical small parameters in the energy estimates for the stability analysis and allows us to control the remainder perturbatively. Joint work with Tom Hou.