Hamiltonian regularisation of the unidimensional barotropic Euler equations

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 H1-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.