Long-wavelength limit for the Green–Naghdi equations

This paper studies the long-wavelength limit for the one-dimensional Green–Naghdi (GN) equations, which are often used to describe the propagation of fully nonlinear waves in coastal oceanography. We prove that, under the long-wavelength, small-amplitude approximation, the formal Korteweg–de Vries (KdV) equation for the GN equations is mathematically valid in the time interval for which the KdV dynamics survive. The main idea in the proof is to apply the Gardner–Morikawa transform, the reductive perturbation method, and some error energy estimates. The main novelties of this paper are the construction of valid approximate solutions of the GN equations with respect to the small wave amplitude parameter and global uniform energy estimates for the error system.