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

doi:10.1016/S0252-9602(10)60013-6

Acta mathematica scientia,Series B ›› 2009, Vol. 29 ›› Issue (6): 1721-1736.doi: 10.1016/S0252-9602(10)60013-6

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

Hai-Liang Li, Tong Yang, Chen Zou

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

Abstract

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 L²time 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 )-^3/4in 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+εfor 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

CLC Number:

35B40

Hai-Liang Li, Tong Yang, Chen Zou. TIME ASYMPTOTIC BEHAVIOR OF THE BIPOLAR NAVIER-STOKES-POISSON SYSTEM[J].Acta mathematica scientia,Series B, 2009, 29(6): 1721-1736.

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

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