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February  2019, 12(1): 159-176. doi: 10.3934/krm.2019007

## Elastic limit and vanishing external force for granular systems

 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

Received  February 2017 Revised  February 2018 Published  July 2018

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

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).

Citation: Fei Meng, Xiao-Ping Yang. Elastic limit and vanishing external force for granular systems. Kinetic & Related Models, 2019, 12 (1) : 159-176. doi: 10.3934/krm.2019007
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