Non-isentropic cavity flow for the multi-d compressible Euler system
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908774743146496 |
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| author | Jenssen, Helge Kristian Tsikkou, Charis |
| author_facet | Jenssen, Helge Kristian Tsikkou, Charis |
| contents | We rigorously construct non-isentropic and self-similar multi-d Euler flows in which a central cavity (vacuum region) collapses. While isentropic flows of this type have been analyzed earlier by Hunter \cite{hun_60} and others, the non-isentropic setting introduces additional complications, in particular with respect to the behavior along the fluid-vacuum interface. The flows we construct satisfy the physical boundary conditions: the interface is a material surface along which the pressure vanishes, and it propagates with a non-vanishing and finite acceleration until collapse.
We introduce a number of algebraic conditions on the parameters in the problem (spatial dimension, adiabatic index, similarity parameters). With these conditions satisfied, a simple argument based on trapping regions for the associated similarity ODEs yields the existence of non-isentropic cavity flows. We finally verify that the conditions are all met for several physically relevant cases in both two and three dimensions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_16435 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Non-isentropic cavity flow for the multi-d compressible Euler system Jenssen, Helge Kristian Tsikkou, Charis Analysis of PDEs Mathematical Physics Dynamical Systems 35L45, 35L67, 76N10, 35Q31 We rigorously construct non-isentropic and self-similar multi-d Euler flows in which a central cavity (vacuum region) collapses. While isentropic flows of this type have been analyzed earlier by Hunter \cite{hun_60} and others, the non-isentropic setting introduces additional complications, in particular with respect to the behavior along the fluid-vacuum interface. The flows we construct satisfy the physical boundary conditions: the interface is a material surface along which the pressure vanishes, and it propagates with a non-vanishing and finite acceleration until collapse. We introduce a number of algebraic conditions on the parameters in the problem (spatial dimension, adiabatic index, similarity parameters). With these conditions satisfied, a simple argument based on trapping regions for the associated similarity ODEs yields the existence of non-isentropic cavity flows. We finally verify that the conditions are all met for several physically relevant cases in both two and three dimensions. |
| title | Non-isentropic cavity flow for the multi-d compressible Euler system |
| topic | Analysis of PDEs Mathematical Physics Dynamical Systems 35L45, 35L67, 76N10, 35Q31 |
| url | https://arxiv.org/abs/2509.16435 |