Asymptotic self-similar blow-up profile for three-dimensional axisymmetric Euler equations using neural networks

Department of Mathematics, Princeton University, Princeton, NJ 08544, USA (Dated: May 9, 2023) 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 Córdoba-Córdoba-Fontelos equation. We show that our numerical framework is both robust and adaptable to various other equations. A celebrated open question in fluids is whether or not cylindrical boundary is intrinsically linked to the same from smooth initial data the 3-D Euler equations may problem for the 2-D Bousinessq equations (cf. The mechanism for blowup self-similar blow-up was proven in the groundbreaking for the two equations is believed to be identical.