Stability of the one-species VlasovPoissonBoltzmann system

Renjun Duan Tong Yang

Analysis of PDEs mathscidoc:1912.43953

SIAM Journal on Mathematical Analysis, 41, (6), 2353-2387, 2010.1
In this paper, we are concerned with the one-species VlasovPoissonBoltzmann system with a nonconstant background density in full space. There exists a stationary solution when the background density is a small perturbation of a positive constant state. We prove the nonlinear stability of solutions to the Cauchy problem near the stationary state in some Sobolev space without any time derivatives. This result is nontrivial even when the background density is a constant state. In the proof, the macroscopic balance laws are essentially used to deal with the a priori estimates on both the microscopic and macroscopic parts of the solution. Moreover, some interactive energy functionals are introduced to overcome difficulty stemming from the absence of time derivatives in the energy functional.
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@inproceedings{renjun2010stability,
  title={Stability of the one-species VlasovPoissonBoltzmann system},
  author={Renjun Duan, and Tong Yang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224210858668249517},
  booktitle={SIAM Journal on Mathematical Analysis},
  volume={41},
  number={6},
  pages={2353-2387},
  year={2010},
}
Renjun Duan, and Tong Yang. Stability of the one-species VlasovPoissonBoltzmann system. 2010. Vol. 41. In SIAM Journal on Mathematical Analysis. pp.2353-2387. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224210858668249517.
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