Shock Formation for Compressible Euler Equations on ${\mathbb{S}}^2$

Abstract

In this paper, we prove the finite-time shock formation for the compressible Euler equations on the two-dimensional sphere ${\mathbb{S}}^2$. In contrast to the flat Euclidean case ${\mathbb{R}}^2$, the geometry of ${\mathbb{S}}^2$ imposes new difficulties, and the fluid dynamics are affected by the curved background. To overcome these challenges, we modify the existing modulation method and employ a set of carefully constructed, time-dependent coordinates that precisely track the shock formation on ${\mathbb{S}}^2$. In particular, we first perform a time-dependent rotation of ${\mathbb{S}}^2$, then apply the stereographic projection to the sphere, straighten the steepening shock front, and finally construct shock-adapted coordinates. In the shock-adapted coordinates, the compressible Euler equations on ${\mathbb{S}}^2$ can be recast into a form suitable for self-similar analysis. Within this framework, we implement a detailed bootstrap argument and establish global well-posedness for the self-similar system. After transferring these results back to the original physical system, we thereby demonstrate the finite-time shock formation on ${\mathbb{S}}^2$.

89 pages