An Inverse Problem for Multi-Dimensional Piston Models with Large Velocity Variations
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| Format: | Preprint |
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2025
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| _version_ | 1866915288055808000 |
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| author | Hu, Dian Li, Qianfeng Zhang, Yongqian |
| author_facet | Hu, Dian Li, Qianfeng Zhang, Yongqian |
| contents | When a circular symmetric piston suddenly expands into a still gas, a leading shock wave is generated. This paper investigates an inverse problem of reconstructing the trajectory of the piston from the given leading shock front and the given initial flow conditions. We observe that in piston models, as the initial density goes to zero, the piston approaches the shock front; however, in the region between the piston and the shock front, the strict hyperbolicity of the system degenerates. By applying asymptotic analysis, we provide quantitative characterizations of the distance between the piston and the shock front, and the degeneration of strict hyperbolicity. Consequently, by designing appropriate a priori assumptions to balance the benefits and drawbacks arising as the initial density approaches zero, we employ the method of characteristics to prove the global-in-time existence of the piecewise smooth solution for this inverse problem. In particular, the resulting flow structure exhibits significant velocity variations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_10209 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An Inverse Problem for Multi-Dimensional Piston Models with Large Velocity Variations Hu, Dian Li, Qianfeng Zhang, Yongqian Analysis of PDEs 35L50, 35L65, 35Q31, 35R30, 76N10 When a circular symmetric piston suddenly expands into a still gas, a leading shock wave is generated. This paper investigates an inverse problem of reconstructing the trajectory of the piston from the given leading shock front and the given initial flow conditions. We observe that in piston models, as the initial density goes to zero, the piston approaches the shock front; however, in the region between the piston and the shock front, the strict hyperbolicity of the system degenerates. By applying asymptotic analysis, we provide quantitative characterizations of the distance between the piston and the shock front, and the degeneration of strict hyperbolicity. Consequently, by designing appropriate a priori assumptions to balance the benefits and drawbacks arising as the initial density approaches zero, we employ the method of characteristics to prove the global-in-time existence of the piecewise smooth solution for this inverse problem. In particular, the resulting flow structure exhibits significant velocity variations. |
| title | An Inverse Problem for Multi-Dimensional Piston Models with Large Velocity Variations |
| topic | Analysis of PDEs 35L50, 35L65, 35Q31, 35R30, 76N10 |
| url | https://arxiv.org/abs/2505.10209 |