2025/04/24 by Ovidiu-Neculai Avadanei, Avadanei, Ovidiu-Neculai
Earth and Planetary Sciences · Mathematics · #35L81 #35P05 #35Q35 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Ocean Waves and Remote Sensing #Primary: 35Q75 #Secondary: 35L10
paper · pdf · doi:10.48550/arxiv.2504.17932
openalex publication_date 2025/04/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.