Counterexamples to Strichartz estimates and gallery waves for the irrotational compressible Euler equation in a vacuum setting
Ovidiu-Neculai Avadanei
TL;DR
This paper studies the free boundary problem for the irrotational compressible Euler equations in a physical vacuum, reformulating the system via a velocity potential that satisfies an acoustic wave equation with respect to an acoustic metric. Focusing on a model in the upper half-space, it analyzes the linearized problem, constructs whispering gallery modes that reflect along the boundary, and proves Strichartz estimates for these modes while revealing a derivative loss that signals optimality for the low-regularity well-posedness threshold. The key contributions are the introduction of gallery waves as a mechanism for Strichartz failures in the irrotational compressible Euler setting and the demonstration that intrinsic scaling Strichartz estimates cannot hold, even in the simplest model, highlighting fundamental dispersion limitations in physical vacuum regimes. Together, these results provide the first rigorous counterexamples to Strichartz estimates in this context and illuminate how boundary geometry and acoustic metrics influence dispersive behavior with potential implications for nonlinear well-posedness and stability analyses.
Abstract
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.