Vorticity Blowup in Compressible Euler Equations in \(\mathbb{R}^d, d \geq 3\)

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 (Chen arXiv preprint arXiv: 2407.06455, 2024) 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.