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 is Galilean invariant, linearly non-dispersive conserves formally an H1-like energy. In this paper, we extend regularisation in two directions. First, consider the more general barotropic Euler system, equations being very special case. Second, introduce class regularisations, showing thus that of (2018) not unique. Considering high-frequency approximation regularisation, obtain new two-component Hunter–Saxton system. We prove both systems – are locally (in time) well-posed, and, if singularities appear finite time, they necessary first derivatives.