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

TL;DR

This paper proves finite-time shock formation for the compressible Euler equations on the curved surface of the sphere $\

Contribution

It introduces a novel coordinate system and modulation method tailored to $\

Findings

Shock formation occurs in finite time on $\

The adapted coordinates enable self-similar analysis of the Euler equations on $\

The method overcomes geometric difficulties posed by the sphere's curvature

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 $\mathbb{S}^2$.