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Vladikavkazskii Matematicheskii Zhurnal, 2022, Volume 24, Number 3, Pages 21–36
DOI: https://doi.org/10.46698/x4972-4013-9236-n (Mi vmj822) 		 
Existence of global classical solutions for the Saint-Venant equations

R. Aziba, S. Georgievb, A. Kheloufia, K. Mebarkia

a Bejaia University, Laboratory of Applied Mathematics, Faculty of Exact Sciences, Bejaia 06000, Algeria
b Sofia University �St. Kliment Ohridski�, 15 Tzar Osvoboditel Blvd., Sofia 1504, Bulgaria
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DOI: https://doi.org/10.46698/x4972-4013-9236-n
Abstract: Nowadays, investigations of the existence of global classical solutions for non linear evolution equations is a topic of active mathematical research. In this article, we are concerned with a classical system of shallow water equations which describes long surface waves in a fluid of variable depth. This system was proposed in 1871 by Adhémar Jean-Claude Barré de Saint-Venant. Namely, we investigate an initial value problem for the one dimensional Saint-Venant equations. We are especially interested in question of what sufficient conditions the initial data and the topography of the bottom must verify in order that the considered system has global classical solutions. In order to prove our main results we use a new topological approach based on the fixed point abstract theory of the sum of two operators in Banach spaces. This basic and new idea yields global existence theorems for many of the interesting equations of mathematical physics.
Key words: Saint-Venant equations, classical solution, fixed point, initial value problem.
Funding agency
The authors R. Azib, A. Kheloufi and K. Mebarki acknowledge support of �Direction Générale de la Recherche Scientifique et du Développement Technologique (DGRSDT)�, MESRS, Algeria.
Received: 14.09.2021
Bibliographic databases:
Document Type: Article
UDC: 517.95
MSC: 35Q35, 35A09, 35E15
Language: English
Citation: R. Azib, S. Georgiev, A. Kheloufi, K. Mebarki, “Existence of global classical solutions for the Saint-Venant equations”, Vladikavkaz. Mat. Zh., 24:3 (2022), 21–36
Citation in format AMSBIB
\Bibitem{AziGeoKhe22}
\by R.~Azib, S.~Georgiev, A.~Kheloufi, K.~Mebarki
\paper Existence of global classical solutions for the Saint-Venant equations
\jour Vladikavkaz. Mat. Zh.
\yr 2022
\vol 24
\issue 3
\pages 21--36
\mathnet{http://mi.mathnet.ru/vmj822}
\crossref{https://doi.org/10.46698/x4972-4013-9236-n}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=4489388}
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