Hubal H. Method of Cluster Expansions of Representation of Bogolyubov Equations' Solutions of Some Many-Particle Dynamical Systems.

The thesis is devoted to the proof of the existence of solutions represented by the non-equilibrium cluster expansion method of Bogolyubov equations for non-symmetrical one-dimensional many-particle dynamical systems. It is developed the methods of estimates of the interaction region volume and proved the existence of the weak global in time solution in cumulant representation of the initial value problem for the Bogolyubov chain of equations for infinite one-dimensional non-symmetrical system of particles interacting via the hard-core potential of finite range with initial data which are locally perturbed equilibrium distribution functions in the space of sequences of bounded functions. It is constructed new estimate and proved the existence of the global in time solution in the form of iteration series of the Cauchy problem for the Bogolyubov chain of equations of infinite one-dimensional non-symmetrical system of particles, interacting as hard spheres, for arbitrary initial data from this space.It is proved existence of the strong global in time solution in the cumulant representation of the Cauchy problem for the Bogolyubov chain of equations for one-dimensional non-symmetrical system of many particles, interacting as hard spheres, with initial data possessing the chaos property in the space of summable functions, and reduced such initial value problem to the corresponding initial value problem for the generalized kinetic equation.The generalized kinetic equation for non-symmetrical systems is obtained for the first time.