Asymptotic self-similar blow-up for the regularized Saint-Venant equations
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arXiv
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| Format: | Preprint |
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2026
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| author | Kim, Yunjoo Kwon, Bongsuk Shim, Wanyong |
| author_facet | Kim, Yunjoo Kwon, Bongsuk Shim, Wanyong |
| contents | We investigate singularity formation in the regularized Saint--Venant (rSV) equations, a conservative, non-dispersive shallow water system that is formally regarded as a Hamiltonian regularization of the isentropic Euler equations. While it is known that smooth solutions to the rSV system can develop gradient blow-up in finite time, the precise structure of such singularities has not been rigorously characterized. In this work, we establish stability of self-similar blow-up profiles of the Hunter--Saxton equation within the rSV framework, using a nonlinear bootstrap argument in dynamically rescaled coordinates. Our analysis captures the detailed space-time dynamics of solutions near the singularity, and proves their sharp $C^{3/5}$ Hölder regularity at the singular time. This regularity differs from the $C^{1/3}$ Hölder regularity of the cubic-root singularities found in the compressible Euler and inviscid Burgers equations. This contrast highlights the structural influence of the Hamiltonian regularization on singularity formation. To illuminate this effect, we also show that the same $C^{3/5}$ blow-up profile emerges in the regularized Burgers equation, a scalar analogue of the rSV system. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_03188 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Asymptotic self-similar blow-up for the regularized Saint-Venant equations Kim, Yunjoo Kwon, Bongsuk Shim, Wanyong Analysis of PDEs 35Q35, 35A21, 35C06, 76B15 We investigate singularity formation in the regularized Saint--Venant (rSV) equations, a conservative, non-dispersive shallow water system that is formally regarded as a Hamiltonian regularization of the isentropic Euler equations. While it is known that smooth solutions to the rSV system can develop gradient blow-up in finite time, the precise structure of such singularities has not been rigorously characterized. In this work, we establish stability of self-similar blow-up profiles of the Hunter--Saxton equation within the rSV framework, using a nonlinear bootstrap argument in dynamically rescaled coordinates. Our analysis captures the detailed space-time dynamics of solutions near the singularity, and proves their sharp $C^{3/5}$ Hölder regularity at the singular time. This regularity differs from the $C^{1/3}$ Hölder regularity of the cubic-root singularities found in the compressible Euler and inviscid Burgers equations. This contrast highlights the structural influence of the Hamiltonian regularization on singularity formation. To illuminate this effect, we also show that the same $C^{3/5}$ blow-up profile emerges in the regularized Burgers equation, a scalar analogue of the rSV system. |
| title | Asymptotic self-similar blow-up for the regularized Saint-Venant equations |
| topic | Analysis of PDEs 35Q35, 35A21, 35C06, 76B15 |
| url | https://arxiv.org/abs/2604.03188 |