Finite-time blowup for the infinite dimensional vorticity equation
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 . 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 . 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.
Keywords
Cite
@article{arxiv.2508.03877,
title = {Finite-time blowup for the infinite dimensional vorticity equation},
author = {Evan Miller},
journal= {arXiv preprint arXiv:2508.03877},
year = {2026}
}
Comments
arXiv admin note: substantial text overlap with arXiv:2204.13406 Author note: The content of this article was previously a section in a preprint version of reference [7], which is arXiv:2204.13406. This section was removed from [7] and published as the standalone article above