A Computational Investigation of the Finite-Time Blow-Up of the 3D Incompressible Euler Equations Based on the Voigt Regularization

TL;DR

This paper uses the Euler-Voigt regularization to computationally investigate potential finite-time blow-up in 3D Euler equations, providing new criteria that are easier to simulate and can indicate singularity formation indirectly.

Contribution

It introduces and tests new blow-up criteria based on Euler-Voigt equations, offering a computationally efficient way to study singularities in 3D Euler equations.

Findings

New blow-up criteria are sufficient but not necessary.

Simulations of Euler-Voigt require less resolution than Euler.

Criteria successfully tested on Burgers equation.

Abstract

We report the results of a computational investigation of two blow-up criteria for the 3D incompressible Euler equations. One criterion was proven in a previous work, and a related criterion is proved here. These criteria are based on an inviscid regularization of the Euler equations known as the 3D Euler-Voigt equations, which are known to be globally well-posed. Moreover, simulations of the 3D Euler-Voigt equations also require less resolution than simulations of the 3D Euler equations for fixed values of the regularization parameter $\alpha>0$. Therefore, the new blow-up criteria allow one to gain information about possible singularity formation in the 3D Euler equations indirectly; namely, by simulating the better-behaved 3D Euler-Voigt equations. The new criteria are only known to be sufficient for blow-up. Therefore, to test the robustness of the inviscid-regularization approach, we also investigate analogous criteria for blow-up of the 1D Burgers equation, where blow-up is well-known to occur.