Existence of global solutions to the 1D Abstract Bubble Vibration model

Abstract

The Abstract Bubble Vibration model ($\mathrm{ABV}$) is a system of two PDEs consisting of a transport equation and a Poisson equation. It has been derived in order to provide a better understanding of hyperbolic-elliptic couplings which are involved in low Mach number models. While a local existence theorem has already been proven in any dimension for the ($\mathrm{ABV}$) model, we get interested in this paper in the one-dimensional case, where we prove the existence of global-in-time solutions no matter how smooth the data. We also provide explicit expressions of these solutions thanks to the method of characteristics that we apply to the transport equation taking advantage of the coupling with the Poisson equation. We then illustrate numerically these results using two different schemes depending on the smoothness of data.