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

Abstract

This paper uses a variational approach to establish existence of solutions (σ _t , v _t ) for the 1-d Euler–Poisson system by minimizing an action. We assume that the initial and terminal points σ₀, σ _T 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 → σ _t in ${\mathcal {P}_2(\mathbb {R})}$ . When σ _t  = δ_y(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.