Volume 3, Number 3, Summer 1995
THE CONVERGENCE OF HAMILTONIAN STRUCTURES
IN THE SHALLOW WATER APPROXIMATION
ZHONG GE, HANS PETER KRUSE, JERROLD E. MARSDEN AND CLINT SCOVEL
Abstract. It is shown that the Hamiltonian structure of
the shallow water equations is, in a precise sense, the limit
of the Hamiltonian structure for that of a threedimensional
ideal fluid with a free boundary problem as the fluid thickness
tends to zero. The procedure fits into an emerging general
scheme of convergence of Hamiltonian structures as parameters
tend to special values. The main technical difficulty in
the proof is how to deal with the condition of incompressibility.
This is treated using special estimates for the solution
of a mixed DirichletNeumann problem for the Laplacian in a
thin domain.
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