eu THE CONVERGENCE OF THE SERIES OF THE SOLUTION OF THE CAUCHY PROBLEM FOR THE BBGKY HIERARCHY OF EQUATIONS IN MANY-KIND PARTICLE SYSTEMS

The non-equilibrium state of many-kind particle systems can be described by a solution of the Cauchy problem for the BBGKY hierarchy of equations. A solution of the Cauchy problem for the BBGKY hierarchy of equations can be represented as the iteration series or the functional series [1, 2, 3, 4]. We can describe the cluster nature of the evolution of many-kind particle systems by the solution in the cumulant representation [3, 4].


Introduction
The non-equilibrium state of many-kind particle systems can be described by a solution of the Cauchy problem for the BBGKY hierarchy of equations.A solution of the Cauchy problem for the BBGKY hierarchy of equations can be represented as the iteration series or the functional series [1,2,3,4].We can describe the cluster nature of the evolution of many-kind particle systems by the solution in the cumulant representation [3,4].

Formulation of the Problem
Consider a many-kind particle system.Denote by 2σ i > 0 and m i > 0 the i-th particle diameter and mass respectively.Particles are characterized by their phase coordinates (q i , υ i ) ≡ x i .Let particles interact via the pair short-range hard core interaction potential Φ(q).For the configurations of this particle system, we have the inequalities The interaction potential Φ(q) satisfies the stability condition: The interaction potential Φ(q) also satisfies the condition: Consider the Banach space E ξ, β of sequences of bounded measurable functions f n (x −n 2 , . . ., x n 1 ) equal to zero on the set of forbidden configurations with the norm , where ξ, β > 0.
A solution of the Cauchy problem for the BBGKY hierarchy of equations is represented as the expansion in the Banach space of sequences of integrable functions [4]: where is the sum over all ordered partitions of the partially ordered set X Y into |P | nonempty pairwise disjoint partially ordered subsets and the set Y lies in one of the subsets X i .

The Convergence of the Series of the Solution of the Cauchy Problem for the BBGKY Hierarchy of Equations
We prove that the series (2) converges uniformly on Y on every compact set for a finite integration region in the space E ξ,β .
The following theorem holds.
Theorem 1.For the pair short-range hard core interaction potential Φ(q) and for the initial data F (0) ∈ E ξ, β , the series (2) converges uniformly on Y on every compact set in the space E ξ, β if Proof.The integrand of the solution (2) has the form for n 1 = n 2 = 0: Let us estimate the integrand (3) by using the stability condition and the condition (1) for the interaction potential: We denote an interaction region by Ω n 1 +n 2 , an uninteraction region by G in configuration variables.
The integral ( 2) is equal to zero in the region G .Indeed, not a single point of x −(n 2 +s 2 ) , . . ., x −(s 2 +1) , x s 1 +1 , . . ., x s 1 +n 1 not interacting with each other and with the cluster Y in the region G, we obtain such an equality: Thus, the integral (2) is equal to zero beyond the interaction region Ω n 1 +n 2 .Let us estimate the integral (2), using the inequality (4): where V n 1 +n 2 (t) is the interaction region volume.The estimate of the interaction region volume has the following form [2]: where |l s 1 +s 2 (2t)| is the length of the interval l s 1 +s 2 (2t) containing (s 1 + s 2 )particles at time 2t, R is the range of the interaction, D ≡ sup is the force acting upon the i-th particle by its nearest neighbours.Therefore, .
We denote Thus, We transform the expression A in the form:
We denote d = max D, 2 Rβ .Then holds the inequality (for any t): Indeed, if d = 2 Rβ , denoting R Rβ , this inequality also holds.From this inequality follows the inequality: Taking into account the inequality (6), we obtain