Temporal boundary value problems in interfacial fluid dynamics

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.

Keywords:

• PDE
• boundary value problems
• Boussinesq equations
• vortex sheet
• fluid dynamics

AMS Subject Classifications::

• 35Q35
• 76B47
• 35J60

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.