Theoretical Study of an Abstract Bubble Vibration Model

  • Yohan Penel

    Université Pierre et Marie Curie, Paris, France
  • Stéphane Dellacherie

    CEA Saclay, Gif-Sur-Yvette, France
  • Olivier Lafitte

    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 simplified 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.

Cite this article

Yohan Penel, Stéphane Dellacherie, Olivier Lafitte, Theoretical Study of an Abstract Bubble Vibration Model. Z. Anal. Anwend. 32 (2013), no. 1, pp. 19–36

DOI 10.4171/ZAA/1472