### 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 f_{0}(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.

Language | English |
---|---|

Pages | 4345–4379 |

Number of pages | 35 |

Journal | SIAM Journal on Mathematical Analysis (SIMA) |

Volume | 44 |

Issue number | 6 |

Early online date | 18 Dec 2012 |

DOIs | |

Status | Published - 2012 |

### Fingerprint

### Cite this

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

Research output: Contribution to journal › Article

*SIAM Journal on Mathematical Analysis (SIMA)*, vol. 44, no. 6, pp. 4345–4379. DOI: 10.1137/120865598

}

TY - JOUR

T1 - A semigroup approach to the justification of kinetic theory

AU - Matthies,Karsten

AU - Theil,Florian

PY - 2012

Y1 - 2012

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84871591363&partnerID=8YFLogxK

UR - http://dx.doi.org/10.1137/120865598

U2 - 10.1137/120865598

DO - 10.1137/120865598

M3 - Article

VL - 44

SP - 4345

EP - 4379

JO - SIAM Journal on Mathematical Analysis (SIMA)

T2 - SIAM Journal on Mathematical Analysis (SIMA)

JF - SIAM Journal on Mathematical Analysis (SIMA)

SN - 0036-1410

IS - 6

ER -