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In the paper we discuss possible approaches to the problem of the rigorousderivation of quantum kinetic equations from underlying many-particle dynamics.For the description of a many-particle evolution we construct solutions of theCauchy problems of the BBGKY hierarchy and the dual BBGKY hierarchy in suitableBanach spaces. In the framework of the conventional approach to the descriptionof kinetic evolution the mean-field asymptotics of the quantum BBGKY hierarchysolution is constructed. We develop also alternative approaches. One method isbased on the construction of the solution asymptotics of the initial-valueproblem of the quantum dual BBGKY hierarchy. One more approach is based on thegeneralized quantum kinetic equation that is a consequence of the equivalenceof the Cauchy problems of such evolution equation and the BBGKY hierarchy withinitial data determined by the one-particle density operator.

Quantum Kinetic Equations and Evolution of Many-Particle Systems