Hard sphere dynamics and the Enskog equation

Viktor I. Gerasimenko; Igor V. Gapyak

doi:10.3934/krm.2012.5.459

Article Contents

2012, Volume 5, Issue 3: 459-484. Doi: 10.3934/krm.2012.5.459

This issue Previous Article A quadratic Fourier representation of the Boltzmann collision operator with an application to the stability problem Next Article Optimization of a model Fokker-Planck equation

Hard sphere dynamics and the Enskog equation

Viktor I. Gerasimenko^1, and
Igor V. Gapyak^2,

1.
Institute of Mathematics of NAS of Ukraine, 3, Tereshchenkivs'ka Str., 01601 Kyiv-4
2.
Taras Shevchenko National University of Kyiv, Department of Mechanics and Mathematics, 2, Academician Glushkov Av.03187, Kyiv

Received: July 31, 2011

Revised: December 31, 2011

Published: August 2012

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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 which are the generating operators of a nonperturbative solution of the Cauchy problem of the BBGKY hierarchy the nonlinear kinetic Enskog equation is derived. 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 an equivalent. For the initial-value problem of the generalized Enskog equation the existence theorem is proved in the space of integrable functions.

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Mathematics Subject Classification: Primary: 35Q20; Secondary: 47J35.

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References

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