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The paper deals with a rigorous description of the kinetic evolution of a hard sphere system in the low-density (BoltzmannGrad) scaling limit within the framework of marginal observables governed by the dual BBGKY (Bogolyubov-BornGreen-Kirkwood-Yvon) hierarchy. For initial states specified by means of a one-particle distribution function, the link between the Boltzmann-Grad asymptotic behavior of a nonperturbative solution of the Cauchy problem of the dual BBGKY hierarchy for marginal observables and a solution of the Boltzmann kinetic equation for hard sphere fluids is established. One of the advantages of such an approach to the derivation of the Boltzmann equation is an opportunity to describe the process of the propagation of initial correlations in scaling limits.

This paper discusses some of the latest advances in the mathematical understanding of the nature of kinetic equations that describe the collective behavior of many-particle systems with collisional dynamics.

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For a quadratic equation containing a small parameter regularly, and for the Duffing equation with small nonlinearity, their asymptotic solutions are constructed in the form of asymptotic Poincare series for a small parameter. The properties of the obtained asymptotic solutions when a small parameter tends to zero are analyzed. The sense of the theorem on the continuous dependence of the solution on the parameter for systems with regular perturbation is demonstrated. A Taylor series for the exact solution of the quadratic equation with small parameter is compared with its obtained asymptotic expansion. For the Duffing equation with small nonlinearity, we compare the graphs of the exact and asymptotic solutions under the same initial conditions.