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

摘要

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 σ_0, σ_T are prescribed in P_2(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 P_2(R). When σ_t = δ_y(t) is a Dirac mass, the Euler–Poisson system reduces to ¨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.