On Regularized Solution for BBGKY Hierarchy of One-Dimensional Infinite System

Abstract

We construct a regularized cumulant (semi-invariant) representation of a solution of the initial value problem for the BBGKY hierarchy for a one-dimensional infinite system of hard spheres interacting via a short-range potential. An existence theorem is proved for the initial data from the space of sequences of bounded functions. In this paper, we propose a regularization method for the solution of the BBGKY hierarchy in the cumulant representation. Due to this method, the structure of the solution expansions guarantees the mutual compensation of the divergent integrals in every term of the series. We establish convergence conditions for the series of the solution and prove an existence theorem of a local in time weak solution of the BBGKY hierarchy for the initial data from the space of sequences of functions which are bounded with respect to the configuration variables and exponentially decreasing with respect to the momentum variables.