Shock Formation for Compressible Euler Equations on $\mathbb{S}^2$
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866917169588076544 |
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| author | An, Xinliang Chen, Haoyang Qi, Fulin Su, Wenze |
| author_facet | An, Xinliang Chen, Haoyang Qi, Fulin Su, Wenze |
| contents | 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 $\mathbb{S}^2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_21548 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Shock Formation for Compressible Euler Equations on $\mathbb{S}^2$ An, Xinliang Chen, Haoyang Qi, Fulin Su, Wenze Analysis of PDEs Mathematical Physics 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 $\mathbb{S}^2$. |
| title | Shock Formation for Compressible Euler Equations on $\mathbb{S}^2$ |
| topic | Analysis of PDEs Mathematical Physics |
| url | https://arxiv.org/abs/2512.21548 |