Elastic limit and vanishing external force for granular systems

Fei Meng; Xiao-Ping Yang

doi:10.3934/krm.2019007

Article Contents

2019, Volume 12, Issue 1: 159-176. Doi: 10.3934/krm.2019007

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Elastic limit and vanishing external force for granular systems

Fei Meng^1, , and
Xiao-Ping Yang^2,

1.
School of Science, Nanjing University of Posts and Telecommunications, Nanjing 210003, China
2.
Department of Mathematics, Nanjing University, Nanjing 210093, China

* Corresponding author: Fei Meng
* Corresponding author: Fei Meng

Received: January 31, 2017

Revised: January 31, 2018

Published: July 2018

This work is supported by the National Natural Science Foundation of China(Grant No. 11531005); the Scientific Research Foundation of NUPT(Grant No. NY218091).

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Abstract

We consider two popular models derived from the theory of granular gases. The first model is the inelastic Boltzmann equation with a diffusion term representing the heat bath, the second model is obtained by a self-similar transformation for the inelastic Boltzmann equation in the homogeneous cooling problem. We prove that the steady states of the two models converge to a Maxwellian equilibrium or a Dirac distribution in the elastic limit and the vanishing external force, respectively. Our results show that the limits of the steady states depend on the ratio of external energy and dissipated energy due to inelastic collision. These results provide a partial answer to a question proposed by Gamba, Panferov and Villani (Comm. Math. Phys. 246,503-541. 2004).

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Mathematics Subject Classification: Primary: 76P05, 76T25; Secondary: 74E20.

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References

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