Advances in Differential Equations

The Cauchy problem on large time for a Boussinesq-Peregrine equation with large topography variations

Mesognon-Gireau Benoit

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Abstract

We prove, in this paper, a long time existence result for a modified Boussinesq-Peregrine equation in dimension $1$, describing the motion of Water Waves in shallow water, in the case of a non flat bottom. More precisely, the dimensionless equations depend strongly on three parameters $\epsilon,\mu,\beta$ measuring the amplitude of the waves, the shallowness and the amplitude of the bathymetric variations, respectively. For the Boussinesq-Peregrine model, one has small amplitude variations ($\epsilon = O(\mu)$). We first give a local existence result for the original Boussinesq Peregrine equation as derived by Boussinesq ([9], [8]) and Peregrine ([22]) in all dimensions. We then introduce a new model which has formally the same precision as the Boussinesq-Peregrine equation, and give a local existence result in all dimensions. We finally prove a local existence result on a time interval of size $\frac{1}{\epsilon}$ in dimension $1$ for this new equation, without any assumption on the smallness of the bathymetry $\beta$, which is an improvement of the long time existence result for the Boussinesq systems in the case of flat bottom ($\beta=0$) by [24].

Note

Author has been partially funded by the ANR project Dyficolti ANR-13-BS01-0003-01.

Article information

Source
Adv. Differential Equations Volume 22, Number 7/8 (2017), 457-504.

Dates
Accepted: October 2016
First available in Project Euclid: 4 May 2017

Permanent link to this document
https://projecteuclid.org/euclid.ade/1493863419

Zentralblatt MATH identifier
1372.35231

Subjects
Primary: 35B25: Singular perturbations 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10] 76B15: Water waves, gravity waves; dispersion and scattering, nonlinear interaction [See also 35Q30] 76B03: Existence, uniqueness, and regularity theory [See also 35Q35]

Citation

Benoit, Mesognon-Gireau. The Cauchy problem on large time for a Boussinesq-Peregrine equation with large topography variations. Adv. Differential Equations 22 (2017), no. 7/8, 457--504. https://projecteuclid.org/euclid.ade/1493863419


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