Potential singularity of the 3D Euler equations in the interior domain

TL;DR

This paper provides numerical evidence suggesting that the 3D axisymmetric Euler equations with smooth initial data may develop a finite time singularity at the origin, exhibiting nearly self-similar scaling properties similar to those of the Navier-Stokes equations.

Contribution

The study introduces a new simple initial condition leading to a potential finite time singularity, differing from previous boundary blow-up scenarios, and demonstrates the stability of the self-similar profile under perturbations.

Findings

Potential finite time singularity at the origin.

Nearly self-similar scaling properties observed.

Solution profile stable under small initial perturbations.

Abstract

Whether the 3D incompressible Euler equations can develop a finite time singularity from smooth initial data is one of the most challenging problems in nonlinear PDEs. In this paper, we present some new numerical evidence that the 3D axisymmetric incompressible Euler equations with smooth initial data of finite energy develop a potential finite time singularity at the origin. This potential singularity is different from the blow-up scenario revealed by Luo-Hou in \cite{luo2014potentially,luo2014toward}, which occurs on the boundary. Our initial condition has a simple form and shares several attractive features of a more sophisticated initial condition constructed by Hou-Huang in \cite{Hou-Huang-2021,Hou-Huang-2022}. One important difference between these two blow-up scenarios is that the solution for our initial data has a one-scale structure instead of a two-scale structure reported in \cite{Hou-Huang-2021,Hou-Huang-2022}. More importantly, the solution seems to develop nearly self-similar scaling properties that are compatible with those of the 3D Navier-Stokes equations. We will present numerical evidence that the 3D Euler equations seem to develop a potential finite time singularity. Moreover, the nearly self-similar profile seems to be very stable to the small perturbation of the initial data.