Hamiltonian regularisation of the unidimensional barotropic Euler equations. (April 2022)

• Record Type: Journal Article
• Authors: Guelmame, Billel; Clamond, Didier; Junca, Stéphane
• Is Part Of: Nonlinear analysis. Volume 64(2022)
• Publication Date: 2022-04
• Subjects: Barotropic flows -- Euler system -- Nonlinear hyperbolic systems -- Generalised Hunter–Saxton system -- Regularisation -- Energy conservation; Nonlinear functional analysis -- Periodicals; Analyse fonctionnelle non linéaire -- Périodiques; Nonlinear functional analysis; Periodicals; 515.7248
• DOI: 10.1016/j.nonrwa.2021.103455
• Languages: English
• ISSNs: 1468-1218

Abstract

Abstract: Recently, a Hamiltonian regularised shallow water (Saint-Venant) system has been introduced by Clamond and Dutykh (2018). This system is Galilean invariant, linearly non-dispersive and conserves formally an H 1 -like energy. In this paper, we extend this regularisation in two directions. First, we consider the more general barotropic Euler system, the shallow water equations being formally a very special case. Second, we introduce a class regularisations, showing thus that this Hamiltonian regularisation of Clamond and Dutykh (2018) is not unique. Considering the high-frequency approximation of this regularisation, we obtain a new two-component Hunter–Saxton system. We prove that both systems – the regularised barotropic Euler system and the two-component Hunter–Saxton system – are locally (in time) well-posed, and, if singularities appear in finite time, they are necessary in the first derivatives.