Asymptotic self-similar blow-up profile for three-dimensional axisymmetric Euler equations using neural networks
Abstract
Whether there exist finite time blow-up solutions for the 2-D Boussinesq and the 3-D Euler equations are of fundamental importance to the field of fluid mechanics. We develop a new numerical framework, employing physics-informed neural networks (PINNs), that discover, for the first time, a smooth self-similar blow-up profile for both equations. The solution itself could form the basis of a future computer-assisted proof of blow-up for both equations. In addition, we demonstrate PINNs could be successfully applied to find unstable self-similar solutions to fluid equations by constructing the first example of an unstable self-similar solution to the C\'ordoba-C\'ordoba-Fontelos equation. We show that our numerical framework is both robust and adaptable to various other equations.
Keywords
Cite
@article{arxiv.2201.06780,
title = {Asymptotic self-similar blow-up profile for three-dimensional axisymmetric Euler equations using neural networks},
author = {Yongji Wang and Ching-Yao Lai and Javier Gómez-Serrano and Tristan Buckmaster},
journal= {arXiv preprint arXiv:2201.06780},
year = {2023}
}
Comments
Main paper: 6 pages, 3 figures. Supplementary material: 15 pages, 12 figures. To appear in Physical Review Letters