Counterexamples to Strichartz estimates and gallery waves for the irrotational compressible Euler equation in a vacuum setting
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2025
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| author | Avadanei, Ovidiu-Neculai |
| author_facet | Avadanei, Ovidiu-Neculai |
| contents | We consider the free boundary problem for the irrotational compressible Euler equation in a physical vacuum setting. By using the irrotationality condition in the Eulerian formulation of Ifrim and Tataru, we derive a formulation of the problem in terms of the velocity potential function, which turns out to be an acoustic wave equation that is widely used in solar seismology. This paper is a first step towards understanding what Strichartz estimates are achievable for the aforementioned equation. Our object of study is the corresponding linearized problem in a model case, in which our domain is represented by the upper half-space. For this, we investigate the geodesics corresponding to the resulting acoustic metric, which have multiple periodic reflections next to the boundary. Inspired by their dynamics, we define a class of whispering gallery type modes associated to our problem, and prove Strichartz estimates for them. By using a construction akin to a wave packet, we also prove that one necessarily has a loss of derivatives in the Strichartz estimates for the acoustic wave equation satisfied by the potential function. In particular, this suggests that the low regularity well-posedness result obtained by Ifrim and Tataru might be optimal, at least in a certain frequency regime. To the best of our knowledge, these are the first results of this kind for the irrotational compressible Euler equations in a physical vacuum. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_17932 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Counterexamples to Strichartz estimates and gallery waves for the irrotational compressible Euler equation in a vacuum setting Avadanei, Ovidiu-Neculai Analysis of PDEs Primary: 35Q75, Secondary: 35L10, 35Q35, 35P05, 35L81 We consider the free boundary problem for the irrotational compressible Euler equation in a physical vacuum setting. By using the irrotationality condition in the Eulerian formulation of Ifrim and Tataru, we derive a formulation of the problem in terms of the velocity potential function, which turns out to be an acoustic wave equation that is widely used in solar seismology. This paper is a first step towards understanding what Strichartz estimates are achievable for the aforementioned equation. Our object of study is the corresponding linearized problem in a model case, in which our domain is represented by the upper half-space. For this, we investigate the geodesics corresponding to the resulting acoustic metric, which have multiple periodic reflections next to the boundary. Inspired by their dynamics, we define a class of whispering gallery type modes associated to our problem, and prove Strichartz estimates for them. By using a construction akin to a wave packet, we also prove that one necessarily has a loss of derivatives in the Strichartz estimates for the acoustic wave equation satisfied by the potential function. In particular, this suggests that the low regularity well-posedness result obtained by Ifrim and Tataru might be optimal, at least in a certain frequency regime. To the best of our knowledge, these are the first results of this kind for the irrotational compressible Euler equations in a physical vacuum. |
| title | Counterexamples to Strichartz estimates and gallery waves for the irrotational compressible Euler equation in a vacuum setting |
| topic | Analysis of PDEs Primary: 35Q75, Secondary: 35L10, 35Q35, 35P05, 35L81 |
| url | https://arxiv.org/abs/2504.17932 |