Supersonic Gravitational Collapse for Non-Isentropic Gaseous Stars
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866915198769561600 |
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| author | Alexander, Christopher Hadžić, Mahir Schrecker, Matthew |
| author_facet | Alexander, Christopher Hadžić, Mahir Schrecker, Matthew |
| contents | We show the existence of a new class of initially smooth spherically symmetric self-similar solutions to the non-isentropic Euler-Poisson system. These solutions exhibit supersonic gravitational implosion in the sense that the density blows-up in finite time while the fluid velocity remains supersonic. In particular, they occupy a portion of the phase space that is far from the recently constructed isentropic self-similar implosion. At the heart of our proof is the presence of a two-parameter scaling invariance and the reduction of the problem to a non-autonomous system of ordinary differential equations. We use the requirement of smoothness of the flow as a selection principle that constrains the choice of scaling indices. An important consequence of our analysis is that for all the solutions we construct, the polytropic index $γ$ is strictly bigger than $\frac{4}{3}$, which is in sharp contrast to the known results in the isentropic case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_18795 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Supersonic Gravitational Collapse for Non-Isentropic Gaseous Stars Alexander, Christopher Hadžić, Mahir Schrecker, Matthew Analysis of PDEs Mathematical Physics Dynamical Systems 35Q85, 35B44, 34C05 We show the existence of a new class of initially smooth spherically symmetric self-similar solutions to the non-isentropic Euler-Poisson system. These solutions exhibit supersonic gravitational implosion in the sense that the density blows-up in finite time while the fluid velocity remains supersonic. In particular, they occupy a portion of the phase space that is far from the recently constructed isentropic self-similar implosion. At the heart of our proof is the presence of a two-parameter scaling invariance and the reduction of the problem to a non-autonomous system of ordinary differential equations. We use the requirement of smoothness of the flow as a selection principle that constrains the choice of scaling indices. An important consequence of our analysis is that for all the solutions we construct, the polytropic index $γ$ is strictly bigger than $\frac{4}{3}$, which is in sharp contrast to the known results in the isentropic case. |
| title | Supersonic Gravitational Collapse for Non-Isentropic Gaseous Stars |
| topic | Analysis of PDEs Mathematical Physics Dynamical Systems 35Q85, 35B44, 34C05 |
| url | https://arxiv.org/abs/2311.18795 |