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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2604.01868 |
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| _version_ | 1866912998427197440 |
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| author | Chen, Bojin Huang, De Li, Xiangyuan |
| author_facet | Chen, Bojin Huang, De Li, Xiangyuan |
| contents | We present novel self-similar finite-time blowup scenarios for the 1D Hou--Luo model. We numerically demonstrate that solutions that initially satisfy certain derivative degeneracy condition can develop asymptotically self-similar finite-time blowups with singular self-similar profiles that are unbounded at some point. Moreover, this blowup phenomenon exhibits a two-stage feature: the solution first undergoes a local $L^{\infty}$ blowup at some time $\tilde{T}$, then continues in the weak sense beyond $\tilde{T}$ and develops a local $L^p$ blowup at a later time $T>\tilde{T}$ for some $p>0$. A further numerical investigation indicates that both stages are asymptotically self-similar. Finally, we extend our numerical study to the 2D Boussinesq equations and discover similar self-similar finite-time blowups with singular profiles that also exhibit a two-stage feature. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_01868 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Novel Self-similar Finite-time Blowups with Singular Profiles of the 1D Hou-Luo Model and the 2D Boussinesq Equations: A Numerical Investigation Chen, Bojin Huang, De Li, Xiangyuan Analysis of PDEs Numerical Analysis 35A21, 35Q31, 35Q35 We present novel self-similar finite-time blowup scenarios for the 1D Hou--Luo model. We numerically demonstrate that solutions that initially satisfy certain derivative degeneracy condition can develop asymptotically self-similar finite-time blowups with singular self-similar profiles that are unbounded at some point. Moreover, this blowup phenomenon exhibits a two-stage feature: the solution first undergoes a local $L^{\infty}$ blowup at some time $\tilde{T}$, then continues in the weak sense beyond $\tilde{T}$ and develops a local $L^p$ blowup at a later time $T>\tilde{T}$ for some $p>0$. A further numerical investigation indicates that both stages are asymptotically self-similar. Finally, we extend our numerical study to the 2D Boussinesq equations and discover similar self-similar finite-time blowups with singular profiles that also exhibit a two-stage feature. |
| title | Novel Self-similar Finite-time Blowups with Singular Profiles of the 1D Hou-Luo Model and the 2D Boussinesq Equations: A Numerical Investigation |
| topic | Analysis of PDEs Numerical Analysis 35A21, 35Q31, 35Q35 |
| url | https://arxiv.org/abs/2604.01868 |