Symmetry and Integrability of Equations of Mathematical Physics − 2011

I.V. Gapyak (National Taras Shevchenko University of Kyiv, Ukraine)
V.I. Gerasimenko (Institute of Mathematics of NAS of Ukraine, Kyiv, Ukraine)

On the nonlinear kinetic equations of infinitely many hard spheres


We develop a rigorous formalism for the description of the kinetic evolution of infinitely many hard spheres. On the basis of the kinetic cluster expansions of cumulants of groups of operators of finitely many hard spheres the nonlinear kinetic Enskog equation and its generalizations are justified.
It is established that for initial states which are specified in terms of one-particle distribution functions the description of the evolution by the Cauchy problem of the BBGKY hierarchy and by the Cauchy problem of the generalized Enskog kinetic equation together with a sequence of explicitly defined functionals of a solution of stated kinetic equation are equivalent. For the initial-value problem of the generalized Enskog equation the existence theorem is proved in the space of integrable functions.
The links of the specific Enskog equations for hard spheres with the generalized Enskog equation are discussed. In particular, we establish that the Boltzmann-Grad asymptotics of a solution of the generalized Enskog kinetic equation is governed by the Boltzmann equation and that the limit marginal functionals of the state are the products of a solution of the derived Boltzmann equation which means the propagation a chaos property in time.

The talk is based on the results published in [I.V. Gapyak, V.I. Gerasimenko. On Rigorous Derivation of the Enskog Kinetic Equation, 2011, arXiv:1107.5572, 28 p.]