3D Euler equations and ideal MHD mapped to regular systems: probing the finite-time blowup hypothesis

TL;DR

This paper introduces a method to map 3D Euler and similar fluid equations to regular systems, enabling numerical investigation of finite-time blowup hypotheses by analyzing solutions of the regularized system.

Contribution

The authors construct an explicit bijective mapping from 3D Euler solutions to regular systems, allowing for numerical analysis of singularity formation.

Findings

Mapping preserves key quantities like energy and circulation.

The method can be extended to other fluid equations with Beale-Kato-Majda criteria.

Numerical simulations of the regular system can inform about Euler singularities.

Abstract

We prove by an explicit construction that solutions to incompressible 3D Euler equations defined in the periodic cube can be mapped bijectively to a new system of equations whose solutions are globally regular. We establish that the usual Beale-Kato-Majda criterion for finite-time singularity (or blowup) of a solution to the 3D Euler system is equivalent to a condition on the corresponding \emph{regular} solution of the new system. In the hypothetical case of Euler finite-time singularity, we provide an explicit formula for the blowup time in terms of the regular solution of the new system. The new system is amenable to being integrated numerically using similar methods as in Euler equations. We propose a method to simulate numerically the new regular system and describe how to use this to draw robust and reliable conclusions on the finite-time singularity problem of Euler equations, based on the conservation of quantities directly related to energy and circulation. The method of mapping to a regular system can be extended to any fluid equation that admits a Beale-Kato-Majda type of theorem, e.g. 3D Navier-Stokes, 2D and 3D magnetohydrodynamics, and 1D inviscid Burgers. We discuss briefly the case of 2D ideal magnetohydrodynamics. In order to illustrate the usefulness of the mapping, we provide a thorough comparison of the analytical solution versus the numerical solution in the case of 1D inviscid Burgers equation.