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M. Oliver,
Justification of the shallow water limit for a rigid lid
flow with bottom topography,
Theoretical and Computational Fluid Dynamics 9
(1997), 311-324.
Abstract:
The so-called lake equations arise as the shallow water limit of the
rigid lid equations - three dimensional Euler equations with a rigid
lid upper boundary condition - in a horizontally periodic basin with
bottom topography. We prove an a priori estimate in the
Sobolev space Hm for m>=3 which shows that a
solution to the rigid lid equations can be approximated by a solution
of the lake equations for an interval of time which can be estimated
in terms of the initial deviation from a columnar configuration and
the magnitude of the initial data in Hm, the
gradient of the bottom topography in
Hm+1, and the aspect ratio of the basin.
In particular, any solution to the lake equations remains close to
some solution of the rigid lid equations for an interval of time that
can be made arbitrarily large by choosing the aspect ratio of the
basin small.
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