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Existence of global solutions to the 1D abstract bubble vibration model. (English) Zbl 1289.35262
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.

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
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