We consider the initial value problem to the Isobe-Kakinuma model for water waves.
As was shown by J. C. Luke, the water wave problem has a variational structure.
By approximating the velocity potential in Luke's Lagrangian, we obtain an approximate
Lagrangian for water waves. The Isobe-Kakinuma model is a corresponding Euler-Lagrange
equation for the approximate Lagrangian.
In this talk, we first explain a structure of the Isobe-Kakinuma model and then
justify the model rigorously as a higher order shallow water approximation by giving
an error estimate between the solutions of the model and of the full water wave problem.
It is revealed that the Isobe-Kakinuma model is a much more precise model than the
well-known Green-Naghdi equations.