Advances in Differential Equations

Comparisons between the BBM equation and a Boussinesq system

A. A. Alazman, J. P. Albert, J. L. Bona, M. Chen, and J. Wu

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Abstract

This project aims to cast light on a Boussinesq system of equations modelling two-way propagation of surface waves. Included in the study are existence results, comparisons between the Boussinesq equations and other wave models, and several numerical simulations. The existence theory is in fact a local well-posedness result that becomes global when the solution satisfies a practically reasonable constraint. The comparison result is concerned with initial velocities and wave profiles that correspond to unidirectional propagation. In this circumstance, it is shown that the solution of the Boussinesq system is very well approximated by an associated solution of the KdV or BBM equation over a long time scale of order $\frac{1}{\epsilon}$, where $\epsilon$ is the ratio of the maximum wave amplitude to the undisturbed depth of the liquid. This result confirms earlier numerical simulations and suggests further numerical experiments, some of which are reported here. Our results are related to recent results of Bona, Colin and Lannes [11] comparing Boussinesq systems of equations to the full two-dimensional Euler equations (see also the recent work of Schneider and Wayne [26] and Wright [30].

Article information

Source
Adv. Differential Equations Volume 11, Number 2 (2006), 121-166.

Dates
First available in Project Euclid: 18 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ade/1355867715

Mathematical Reviews number (MathSciNet)
MR2194497

Zentralblatt MATH identifier
1104.35039

Subjects
Primary: 35Q35: PDEs in connection with fluid mechanics
Secondary: 35B30: Dependence of solutions on initial and boundary data, parameters [See also 37Cxx] 35Q51: Soliton-like equations [See also 37K40] 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10] 76B03: Existence, uniqueness, and regularity theory [See also 35Q35] 76B15: Water waves, gravity waves; dispersion and scattering, nonlinear interaction [See also 35Q30]

Citation

Alazman, A. A.; Albert, J. P.; Bona, J. L.; Chen, M.; Wu, J. Comparisons between the BBM equation and a Boussinesq system. Adv. Differential Equations 11 (2006), no. 2, 121--166. https://projecteuclid.org/euclid.ade/1355867715.


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