Penel, Yohan Existence of global solutions to the 1D abstract bubble vibration model. (English) Zbl 1289.35262 Differ. Integral Equ. 26, No. 1-2, 59-80 (2013). This work deals with a 1D system of two coupled equations, a transport equation coupled to a Poisson equation. The model describes certain aspects of the dynamics of bubbles in a boundary domain. A global-in-time existence result is obtained under weak regularity conditions on the initial data (essentially bounded data). Moreover, a semi-analytical solution is also obtained when initial data is chosen corresponding to a single bubble. These mathematical results are then backed up with numerical solutions that can highlight main features of the bubble model. An instructive discussion of different numerical techniques that can be used are provided. Reviewer: Steinar Evje (Stavanger) Cited in 3 Documents MSC: 35Q35 PDEs in connection with fluid mechanics 35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness 35B65 Smoothness and regularity of solutions to PDEs 35A01 Existence problems for PDEs: global existence, local existence, non-existence 35L03 Initial value problems for first-order hyperbolic equations Keywords:abstract bubble vibration model; weak solution; bubble-kind solution; Poisson equation PDFBibTeX XMLCite \textit{Y. Penel}, Differ. Integral Equ. 26, No. 1--2, 59--80 (2013; Zbl 1289.35262)