Finite-time blowup for the infinite dimensional vorticity equation

TL;DR

This paper investigates a high-dimensional vorticity model that demonstrates finite-time blowup, providing insights into potential singularity formation in Euler equations as the dimension increases.

Contribution

The study introduces an infinite-dimensional vorticity model derived from the limit as dimension approaches infinity, revealing finite-time blowup phenomena.

Findings

The infinite-dimensional model exhibits finite-time blowup similar to Burgers shocks.

Finite-time blowup mechanisms in high dimensions may explain singularities in Euler equations.

Perturbation analysis suggests the full Euler equation can be approximated by the infinite-dimensional model.

Abstract

In a previous work with Tai-Peng Tsai, the author studied the dynamics of axisymmetric, swirl-free Euler equation in four and higher dimensions. One conclusion of this analysis is that the dynamics become dramatically more singular as the dimension increases. In particular, the barriers to finite-time blowup for smooth solutions which exist in three dimensions do not exist in higher dimensions $d\geq 4$. Motivated by this result, we will consider a model equation that is obtained by taking the formal limit of the scalar vorticity evolution equation as $d\to +\infty$. This model exhibits finite-time blowup of a Burgers shock type. The blowup result for the infinite dimensional model equation strongly suggests a mechanism for the finite-time blowup of smooth solutions of the Euler equation in sufficiently high dimensions. It is also possible to treat the full Euler equation as a perturbation of the infinite dimensional model equation, although this perturbation is highly singular.