On free surfaces in hydrostatic flows

Abstract

In the following note we investigate the occurrence of non-trivial free-surfaces for hydrostatic flows. We show that, for travelling-wave solutions of the full Euler equations where we invoke the hydrostatic approximation, no non-flat travelling wave flows occur unless the resulting flow is outside the domain of current local or global existence results for travelling water waves.

Keywords:

• hydrostatic limit
• free surface
• laminar flows

AMS Subject Classifications::

• 35Q31
• 35Q35
• 35R35

Acknowledgements

The author would like to thank the organizers of the programme ‘Nonlinear Water Waves’, Erwin Schrödinger Institute (Vienna), April–June 2011, where work on this article was undertaken. The support of the WWTF is gratefully acknowledged.