Singularity formation in 3D Euler equations with smooth initial data and boundary

Abstract

A long-standing fundamental open problem in mathematical fluid dynamics and nonlinear partial differential equations is to determine whether solutions of the 3D incompressible Euler equations can develop a finite-time singularity from smooth, finite-energy initial data. Leonhard Euler introduced these equations in 1757 [L. Euler, Mémoires de l’Académie des Sci. de Berlin 11 , 274–315 (1757).], and they are closely linked to the Navier–Stokes equations and turbulence. While the general singularity formation problem remains unresolved, we review a recent computer-assisted proof of finite-time, nearly self-similar blowup for the 2D Boussinesq and 3D axisymmetric Euler equations in a smooth bounded domain with smooth initial data. The proof introduces a framework for (nearly) self-similar blowup, demonstrating the nonlinear stability of an approximate self-similar profile constructed numerically via the dynamical rescaling formulation.