Vorticity blowup in compressible Euler equations in $\mathbb{R}^d, d \geq 3$

TL;DR

This paper proves finite-time vorticity blowup in the compressible Euler equations for dimensions three and higher, extending previous two-dimensional results by leveraging axisymmetry and analyzing singularity formation.

Contribution

It extends vorticity blowup results from 2D to higher dimensions using axisymmetry and constructs initial data leading to singularity formation in $\

Findings

Vorticity blowup occurs on a sphere in $\\mathbb{R}^d$.

Non-radial implosion accompanies vorticity blowup.

Stable swirl velocity dominates initial non-radial components.

Abstract

We prove finite-time vorticity blowup in the compressible Euler equations in $\mathbb{R}^d$ for any $d \geq 3$, starting from smooth, localized, and non-vacuous initial data. This is achieved by lifting the vorticity blowup result from [CCSV24] in $\mathbb{R}^2$ to $\mathbb{R}^d$ and utilizing the axisymmetry in $\mathbb{R}^d$. At the time of the first singularity, both vorticity blowup and implosion occur on a sphere $S^{d-2}$. Additionally, the solution exhibits a non-radial implosion, accompanied by a stable swirl velocity that is sufficiently strong to initially dominate the non-radial components and to generate the vorticity blowup.