Euler–Poisson Systems as Action-Minimizing Paths in the Wasserstein Space

Abstract

This paper uses a variational approach to establish existence of solutions (σt, vt) for the 1-d Euler–Poisson system by minimizing an action. We assume that the initial and terminal points σ0, σT are prescribed in \({\mathcal {P}_2(\mathbb {R})}\) , the set of Borel probability measures on the real line, of finite second-order moments. We show existence of a unique minimizer of the action when the time interval [0,T] satisfies T < π. These solutions conserve the Hamiltonian and they yield a path t → σt in \({\mathcal {P}_2(\mathbb {R})}\) . When σt = δy(t) is a Dirac mass, the Euler–Poisson system reduces to \({\ddot {y} + y=0}\) . The kinetic version of the Euler–Poisson, that is the Vlasov–Poisson system was studied in Ambrosio and Gangbo (Comm Pure Appl Math, to appear) as a Hamiltonian system.