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

Abstract

We prove finite-time vorticity blowup in the compressible Euler equations in ${\mathbb{R}}^d$ for any $d\ge 3$, 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.