Global Regularity of the 4D Restricted Euler Equations

TL;DR

This paper investigates the critical threshold phenomena in 3D and 4D Restricted Euler equations, revealing that 3D solutions typically blow up while 4D solutions are generally globally regular, indicating a fundamental difference in their behaviors.

Contribution

It identifies the critical thresholds for 3D and 4D RE equations using spectral and trace dynamics, showing 4D regularity is generic unlike the 3D case.

Findings

3D RE solutions blow up for most initial conditions

4D RE solutions are globally regular for a broad set of initial spectra

The set of initial eigenvalues leading to bounded solutions is much richer in 4D

Abstract

We are concerned with the critical threshold phenomena in the Restricted Euler (RE) equations. Using the spectral and trace dynamics we identify the critical thresholds for 3D and the 4D restricted Euler equations. It is well known that the 3D RE solutions blow up. Projected on the 3-sphere, the set of initial eigenvalues which give rise to bounded stable solutions is reduced to a single point, which confirms that 3D RE blowup is generic. In contrast, we identify a surprisingly rich set of the initial spectrum on the 4-sphere which yields global smooth solutions; thus, 4D regularity is generic.