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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Received 10 October 1996 and accepted 15 May 1997
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Oliver, M. Justification of the Shallow-Water Limit for a Rigid-Lid Flow with Bottom Topography . Theoret. Comput. Fluid Dynamics 9, 311–324 (1997). https://doi.org/10.1007/s001620050047
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DOI: https://doi.org/10.1007/s001620050047