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

The Abstract Bubble Vibration model (Abv) is a set 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 a more general low Mach number system modelling bubbles. While a local existence theorem has already been proven in any dimension for the Abv model, we get interested in this paper in the one-dimensional case, where we prove the existence of global solutions no matter how smooth the data. We also provide explicit expressions of the solutions by means of the method of characteristics that we apply to the transport equation despite the coupling with the Poisson equation. We then compute these solutions by means of a numerical scheme based on a discrete version of the method of characteristics.

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