Methods and Applications of Analysis

Volume 29 (2022)

Number 3

Local well-posedness of a Hamiltonian regularisation of the Saint-Venant system with uneven bottom

Pages: 295 – 302

DOI: https://dx.doi.org/10.4310/MAA.2022.v29.n3.a4

Authors

Billel Guelmame (LJAD, Inria & CNRS, Université Côte d’Azur, France)

Didier Clamond (LJAD, CNRS, Université Côte d’Azur, France)

Stéphane Junca (LJAD, Inria & CNRS, Université Côte d’Azur, France)

Abstract

We prove in this note the local (in time) well-posedness of a broad class of $2 \times 2$ symmetrisable hyperbolic system involving additional non-local terms. The latest result implies the local well-posedness of the non dispersive regularisation of the Saint-Venant system with uneven bottom introduced by Clamond et al. [2]. We also prove that, as long as the first derivatives are bounded, singularities cannot appear.

Keywords

dispersionless shallow water equations, nonlinear hyperbolic systems, Hamiltonian regularisation, energy conservation

2010 Mathematics Subject Classification

35B65, 35L65, 35Q35, 37K05, 76B15

Received 8 March 2022

Accepted 6 September 2022

Published 21 March 2023