Theoretical study of an abstract bubble vibration model. (English) Zbl 1262.35176
Summary: 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.
MSC:
| 35M30 | Mixed-type systems of PDEs |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
| 35A09 | Classical solutions to PDEs |
| 35Q35 | PDEs in connection with fluid mechanics |
Keywords:
short time existence; uniqueness; hyperbolic-elliptic system; diphasic low Mach number flow; explicit formula of the mean valueReferences:
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