Gradient catastrophes and an infinite hierarchy of Hölder cusp-singularities for 1D Euler
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866916546592374784 |
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| author | Neal, Isaac Shkoller, Steve Vicol, Vlad |
| author_facet | Neal, Isaac Shkoller, Steve Vicol, Vlad |
| contents | We establish an infinite hierarchy of finite-time gradient catastrophes for smooth solutions of the 1D Euler equations of gas dynamics with non-constant entropy. Specifically, for all integers $n\geq 1$, we prove that there exist classical solutions, emanating from smooth, compressive, and non-vacuous initial data, which form a cusp-type gradient singularity in finite time, in which the gradient of the solution has precisely $C^{0,\frac{1}{2n+1}}$ Hölder-regularity. We show that such Euler solutions are codimension-$(2n-2)$ stable in the Sobolev space $W^{2n+2,\infty}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_21040 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Gradient catastrophes and an infinite hierarchy of Hölder cusp-singularities for 1D Euler Neal, Isaac Shkoller, Steve Vicol, Vlad Analysis of PDEs 35L67 (Primary) 76N99 (Secondary) We establish an infinite hierarchy of finite-time gradient catastrophes for smooth solutions of the 1D Euler equations of gas dynamics with non-constant entropy. Specifically, for all integers $n\geq 1$, we prove that there exist classical solutions, emanating from smooth, compressive, and non-vacuous initial data, which form a cusp-type gradient singularity in finite time, in which the gradient of the solution has precisely $C^{0,\frac{1}{2n+1}}$ Hölder-regularity. We show that such Euler solutions are codimension-$(2n-2)$ stable in the Sobolev space $W^{2n+2,\infty}$. |
| title | Gradient catastrophes and an infinite hierarchy of Hölder cusp-singularities for 1D Euler |
| topic | Analysis of PDEs 35L67 (Primary) 76N99 (Secondary) |
| url | https://arxiv.org/abs/2412.21040 |