A Mathematical Justification of the IsobeKakinuma Model for Water Waves with and without Bottom Topography
Abstract
We consider the IsobeKakinuma model for water waves in both cases of the flat and the variable bottoms. The IsobeKakinuma model is a system of EulerLagrange equations for an approximate Lagrangian which is derived from Luke's Lagrangian for water waves by approximating the velocity potential in the Lagrangian appropriately. The IsobeKakinuma model consists of (N+1) second order and a first order partial differential equations, where N is a nonnegative integer. We justify rigorously the IsobeKakinuma model as a higher order shallow water approximation in the strongly nonlinear regime by giving an error estimate between the solutions of the IsobeKakinuma model and of the full water wave problem in terms of the small nondimensional parameter δ , which is the ratio of the mean depth to the typical wavelength. It turns out that the error is of order O(δ ^{4N+2}) in the case of the flat bottom and of order O(δ ^{4[N/2]+2}) in the case of variable bottoms.
 Publication:

Journal of Mathematical Fluid Mechanics
 Pub Date:
 December 2018
 DOI:
 10.1007/s000210180398x
 arXiv:
 arXiv:1803.09236
 Bibcode:
 2018JMFM...20.1985I
 Keywords:

 Mathematics  Analysis of PDEs
 EPrint:
 doi:10.1007/s000210180398x