数学物理学报(英文版) ›› 2009, Vol. 29 ›› Issue (6): 1721-1736.doi: 10.1016/S0252-9602(10)60013-6

• 论文 • 上一篇    下一篇

TIME ASYMPTOTIC BEHAVIOR OF THE BIPOLAR NAVIER-STOKES-POISSON SYSTEM

 Hai-Liang Li, Tong Yang, Chen Zou   

  1. Department of Mathematics, Capital Normal University, Beijing 100048, China; Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong; Department of Mechanics and Aerospace Engineering, COE, Peking University, Beijing 100871, China
  • 收稿日期:2009-11-03 出版日期:2009-11-20 发布日期:2009-11-20
  • 基金资助:

    The research of the first author was partially supported by the NNSFC No. 10871134, the NCET support of the Ministry of Education of China, the Huo Ying Dong Fund No.111033, the Chuang Xin Ren Cai Project of Beijing Municipal Commission of Education $\#$PHR201006107, and the Institute of Mathematics and Interdisciplinary Science at CNU. The research of the second author was supported by the General Research
    Fund of Hong Kong (CityU 103109), and the National Natural Science Foundation of China, 10871082

TIME ASYMPTOTIC BEHAVIOR OF THE BIPOLAR NAVIER-STOKES-POISSON SYSTEM

 Hai-Liang Li, Tong Yang, Chen Zou   

  1. Department of Mathematics, Capital Normal University, Beijing 100048, China; Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong; Department of Mechanics and Aerospace Engineering, COE, Peking University, Beijing 100871, China
  • Received:2009-11-03 Online:2009-11-20 Published:2009-11-20
  • Supported by:

    The research of the first author was partially supported by the NNSFC No. 10871134, the NCET support of the Ministry of Education of China, the Huo Ying Dong Fund No.111033, the Chuang Xin Ren Cai Project of Beijing Municipal Commission of Education $\#$PHR201006107, and the Institute of Mathematics and Interdisciplinary Science at CNU. The research of the second author was supported by the General Research
    Fund of Hong Kong (CityU 103109), and the National Natural Science Foundation of China, 10871082

摘要:

The bipolar Navier-Stokes-Poisson system (BNSP) has been used to simulate the transport of charged particles (ions and electrons for instance) under the influence of electrostatic force governed by the self-consistent Poisson equation. The optimal L2 time convergence rate for the global classical solution is obtained for a small initial perturbation of the constant equilibrium state. It is shown that due to the electric field, the difference of the charge densities tend to the equilibrium states at the optimal rate (1+t )-3/4 in L2-norm, while the individual momentum of the charged particles converges at the optimal rate (1+t )-1/4 which is slower than the rate (1+t )-3/4 for the compressible Navier-Stokes equations (NS). In addition, a new phenomenon on the charge transport is observed regarding the interplay between the two carriers that
almost counteracts the influence of the electric field so that  the total density and  momentum of the two carriers converges at a faster rate (1+t )-3/4+ε for any small constant ε > 0. The above estimates reveal the essential difference between the unipolar and the bipolar Navier-Stokes-Poisson systems.

关键词: bipolar Navier-Stokes-Poisson system, optimal time convergence rate, spectrum analysis, total momentum

Abstract:

The bipolar Navier-Stokes-Poisson system (BNSP) has been used to simulate the transport of charged particles (ions and electrons for instance) under the influence of electrostatic force governed by the self-consistent Poisson equation. The optimal L2 time convergence rate for the global classical solution is obtained for a small initial perturbation of the constant equilibrium state. It is shown that due to the electric field, the difference of the charge densities tend to the equilibrium states at the optimal rate (1+t )-3/4 in L2-norm, while the individual momentum of the charged particles converges at the optimal rate (1+t )-1/4 which is slower than the rate (1+t )-3/4 for the compressible Navier-Stokes equations (NS). In addition, a new phenomenon on the charge transport is observed regarding the interplay between the two carriers that
almost counteracts the influence of the electric field so that  the total density and  momentum of the two carriers converges at a faster rate (1+t )-3/4+ε for any small constant ε > 0. The above estimates reveal the essential difference between the unipolar and the bipolar Navier-Stokes-Poisson systems.

Key words: bipolar Navier-Stokes-Poisson system, optimal time convergence rate, spectrum analysis, total momentum

中图分类号: 

  • 35B40