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July/August 2017 The Cauchy problem on large time for a Boussinesq-Peregrine equation with large topography variations
Mesognon-Gireau Benoit
Adv. Differential Equations 22(7/8): 457-504 (July/August 2017).

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].

Funding Statement

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

Citation

Download Citation

Mesognon-Gireau Benoit. "The Cauchy problem on large time for a Boussinesq-Peregrine equation with large topography variations." Adv. Differential Equations 22 (7/8) 457 - 504, July/August 2017.

Information

Accepted: 1 October 2016; Published: July/August 2017
First available in Project Euclid: 4 May 2017

zbMATH: 1372.35231
MathSciNet: MR3646468

Subjects:
Primary: 35B25, 35Q53, 76B03, 76B15

Rights: Copyright © 2017 Khayyam Publishing, Inc.

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Vol.22 • No. 7/8 • July/August 2017
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