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Zeitschrift für Analysis und ihre Anwendungen

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Volume 32, Issue 1, 2013, pp. 19–36
DOI: 10.4171/ZAA/1472

Published online: 2013-01-27

Theoretical Study of an Abstract Bubble Vibration Model

Yohan Penel[1], Stéphane Dellacherie[2] and Olivier Lafitte[3]

(1) Université Pierre et Marie Curie, Paris, France
(2) CEA Saclay, Gif-Sur-Yvette, France
(3) Université de Paris XIII/CNRS, Villetaneuse, France

We present the theoretical study of a hyperbolic-elliptic system of equations called the Abstract Bubble Vibration (Abv) model. This simpli ed system is derived 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 interval of existence for any smooth initial condition. In particular, we obtain a global-in-time existence result for small parameters. We then focus on 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.

Keywords: Elliptic-hyperbolic coupling, short time existence, uniqueness

Penel Yohan, Dellacherie Stéphane, Lafitte Olivier: Theoretical Study of an Abstract Bubble Vibration Model. Z. Anal. Anwend. 32 (2013), 19-36. doi: 10.4171/ZAA/1472