A mathematical justification for Korteweg-de Vries equation and Boussinesq equation of water surface waves
type: Item
instance of: scholarly article
title: A mathematical justification for Korteweg-de Vries equation and Boussinesq equation of water surface waves
zbMATH Open document ID: 0622.76021
author: Tadayoshi Kano
author: Takaaki Nishida
published in: Osaka Journal of Mathematics
publication date: 1986-01-01T00:00:00Z
review text: The Korteweg-de Vries equation and the Boussinesq equation have enjoyed the pride of place in almost all investigations of nonlinear fluid motion. Inspite of their importance neither of these equations have been rigourously derived starting from the equations governing the motion of inviscid irrotational fluids. The present authors have attempted to fill this void in the case of two dimensions. Clearly there are two parameters in the problem (1) the ratio of mean water depth to the wave length and (2) the ratio of the amplitude to the mean depth. There are two independent limiting cases to consider (1) shallow water approximation and (2) long wave approximations. Using a conformal mapping of the domain occupied by the fluid and using a scale of Banach spaces of analytic functions, they show that for the Cauchy problem, a short time solution exists which can be approximated uniformly by the corresponding solutions of the Boussinesq and KdV equations. This paper marks an important contribution to rigorous fluid dynamics.
Mathematics Subject Classification ID: 76B25
Mathematics Subject Classification ID: 35Q99
Mathematics Subject Classification ID: 76B15
zbMATH DE Number: 4010034
zbMATH Keywords: scaling
zbMATH Keywords: integro-differential equations
zbMATH Keywords: Korteweg-de Vries equation
zbMATH Keywords: Boussinesq equation
zbMATH Keywords: nonlinear fluid motion
zbMATH Keywords: inviscid irrotational fluids
zbMATH Keywords: shallow water approximation
zbMATH Keywords: long wave approximations
zbMATH Keywords: conformal mapping
zbMATH Keywords: Banach spaces of analytic functions
zbMATH Keywords: Cauchy problem
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4010034