Theoretical study of an abstract bubble vibration model

We present the theoretical study of a hyperbolic-elliptic system of equations called Abstract Bubble Vibration (Abv) model. This simplified system is derived under non-physical assumptions from a model describing a diphasic low Mach number flow. It is thus aimed at providing mathematical properties of the coupling between the hyperbolic transport equation and the elliptic Poisson equation. We prove an existence and uniqueness result including the approximation of the time of existence for any smooth initial condition. In particular, we obtain a global-in-time existence result for small initial data. We then pay attention to properties of solutions (depending of their smoothness) such as maximum principle or evenness. In particular, an explicit formula of the mean value of solutions is given. (authors)