On the Existence of Weak Local in Time Solutions in the Form of a Cumulant Expansion for a Chain of Bogolyubov’s Equations of a One-Dimensional Symmetric Particle System

Abstract

We consider a Cauchy problem for a chain of Bogolyubov equations of an infinite onedimensional symmetric particle system, where the particles interact with each other by a finite-range pair potential with a hard core. We consider it in the space of sequences of bounded measurable functions. Based on the proposed method of a joint interval for estimates of the volume of the interaction domain and on the derived estimate itself we find a representation of a weak local with respect to time solution in the form of a cumulant expansion. We prove that the considered weak local with respect to time solution is an equilibrium solution if the initial data are equilibrium distribution functions.