
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
Abstract:
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 oneparticle 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
initialvalue 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 BoltzmannGrad
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.]

