Microscopic solutions of the Boltzmann-Enskog equation in the series representation

Mario Pulvirenti; Sergio Simonella; Anton Trushechkin

doi:10.3934/krm.2018036

Article Contents

2018, Volume 11, Issue 4: 911-931. Doi: 10.3934/krm.2018036

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Microscopic solutions of the Boltzmann-Enskog equation in the series representation

1.
International Research Center M & MOCS, Università dell'Aquila, Palazzo Caetani, Cisterna di Latina, (LT) 04012, Italy
2.
CNRS and UMPA (UMR CNRS 5669), École Normale Supérieure de Lyon, 46 allée dItalie, 69364 Lyon Cedex 07, France
3.
Steklov Mathematical Institute of Russian Academy of Sciences, Gubkina Street 8, Moscow 119991, Russia
4.
National Research Nuclear University MEPhI, Kashirskoe Highway 31, Moscow 115409, Russia
5.
National University of Science and Technology MISIS, Leninsky Avenue 2, Moscow 119049, Russia

Received: September 2017

Revised: January 2018

Published: April 2018

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Abstract

The Boltzmann-Enskog equation for a hard sphere gas is known to have so called microscopic solutions, i.e., solutions of the form of time-evolving empirical measures of a finite number of hard spheres. However, the precise mathematical meaning of these solutions should be discussed, since the formal substitution of empirical measures into the equation is not well-defined. Here we give a rigorous mathematical meaning to the microscopic solutions to the Boltzmann-Enskog equation by means of a suitable series representation.

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Mathematics Subject Classification: Primary: 82C05, 82C40; Secondary: 35Q20.

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Figure 1. $\mathbf {r_5} = \{1, 1, 2, 3, 2\}$

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Figure 2.

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Figure 3.

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Figure 4.

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Figure 5. Initial configuration $\delta(\zeta_1-z_1)\delta(\zeta_2-z_2)\delta(\zeta_3-z_1)\delta(\zeta_4-z_1)\cdots$

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Figure 6. Sum of compositions of trees

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Figure 7.

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