A semigroup approach to the justification of kinetic theory

Karsten Matthies, Florian Theil

Research output: Contribution to journalArticle

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Abstract

This paper develops a method to rigorously show the validity of continuum description for the deterministic dynamics of many interacting particles with random initial data. We consider a hard sphere flow where particles are removed after the first collision. A fixed number of particles is drawn randomly according to an initial density f0(u; v) depending on d-dimensional position u and velocity v. In the Boltzmann Grad scaling, we derive the validity of a Boltzmann
equation without gain term for arbitrary long times, when we assume finiteness of moments up to order two and initial data that are L in space. We characterize the many particle flow by collision trees which encode possible collisions. The convergence of the many-particle dynamics to the Boltzmann dynamics is achieved via the convergence of associated probability measures on collision
trees. These probability measures satisfy nonlinear Kolmogorov equations, which are shown to be well-posed by semigroup methods.

LanguageEnglish
Pages4345–4379
Number of pages35
JournalSIAM Journal on Mathematical Analysis (SIMA)
Volume44
Issue number6
Early online date18 Dec 2012
DOIs
StatusPublished - 2012

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Kinetic theory
Kinetic Theory
Justification
Semigroup
Collision
Ludwig Boltzmann
Boltzmann equation
Trees (mathematics)
Probability Measure
Nonlinear equations
Kolmogorov Equation
Hard Spheres
Finiteness
Boltzmann Equation
Nonlinear Equations
Continuum
Scaling
Moment
Arbitrary
Term

Cite this

A semigroup approach to the justification of kinetic theory. / Matthies, Karsten; Theil, Florian.

In: SIAM Journal on Mathematical Analysis (SIMA), Vol. 44, No. 6, 2012, p. 4345–4379.

Research output: Contribution to journalArticle

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