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

TL;DR

<p>The paper proves finite-time shock formation for the isentropic compressible Euler equations on the 2-sphere, showing that curvature introduces new geometric terms that enrich the dynamics but do not prevent blow-up. The authors develop a modulation/geometry framework that rotates the sphere, applies a stereographic projection, and uses shock-adapted coordinates to recast the equations into a self-similar, Burgers-type system on $S^2$. A four-part strategy—geometric coordinate adaptation, symmetric hyperbolic reformulation, self-similar rescaling with modulation ODEs, and a bootstrap/energy framework—yields global well-posedness for the self-similar system and, upon transferring back to physical coordinates, finite-time blow-up of the gradient with $C^{1/3}$ regularity, while the density remains bounded away from vacuum. An equivariant variant is also provided, demonstrating shock formation for axisymmetric, no-swirl flows on $S^2$. These results illuminate how curvature can influence shock formation and lay groundwork for further geometric analyses of compressible flows on manifolds.</p>

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 $\mathbb{S}^2$.

Figures (1)

• Figure 1.1: Coordinate transformations

Theorems & Definitions (128)

• Theorem 1.1: Shock Formation for Compressible Euler Equations on $\mathbb{S}^2$
• Remark 1.2
• Remark 1.3
• Theorem 1.4: Equivariant Shock Formation for Compressible Euler Equations on $\mathbb{S}^2$
• Remark 1.5
• Proposition 2.1
• proof
• Proposition 2.2
• Remark 3.1
• Theorem 3.2