Asymptotic self-similar blow-up profile for three-dimensional axisymmetric Euler equations using neural networks
Published in Physical Review Letters, 2023
Whether there exist finite-time blow-up solutions for the 2D Boussinesq and the 3D Euler equations are of fundamental importance to the field of fluid mechanics. We develop a new numerical framework, employing physics-informed neural networks, 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 physics-informed neural networks 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 Córdoba-Córdoba-Fontelos equation. We show that our numerical framework is both robust and adaptable to various other equations.
Recommended citation: Y. Wang, C.-Y. Lai, J.Gómez-Serrano, and T. Buckmaster. (2023). "Asymptotic self-similar blow-up profile for three-dimensional axisymmetric Euler equations using neural networks." Physical Review Letters, 130, 244002.
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