Abstract
We study problems in interfacial fluid dynamics which do not have well-posed initial value problems. We prove existence of solutions for these problems by considering instead boundary value problems, where boundary data is specified at two different times. We develop a general framework, for problems on the real line and for problems which are spatially periodic. A variety of boundary conditions are considered, including Dirichlet, Neumann and mixed conditions. The framework is applied to two specific problems from interfacial fluid dynamics: a family of generalizations of the Boussinesq equations developed by Bona, Chen and Saut, and the vortex sheet.
Acknowledgements
This work has been completed as part of the first author's doctoral dissertation. The authors gratefully acknowledge support from the National Science Foundation through grants DMS-0926378, DMS-1008387 and DMS-1016267.