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Applicable Analysis
An International Journal
Volume 92, 2013 - Issue 5
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Articles

Temporal boundary value problems in interfacial fluid dynamics

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Pages 922-948 | Received 01 Sep 2011, Accepted 11 Nov 2011, Published online: 11 Jan 2012
 

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.

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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.

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