The convergence of Hamiltonian structure in the shallow water approximation

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 three-dimensional 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 Dirichlet-Neumann problem for the Laplacian in a thin domain.