Large Time Behavior of the Vlasov-Poisson-Boltzmann System

Abstract and Applied Analysis

Research Article

Li Yang¹  Shuilin Jin²  Li Li³ 

[1] Department of Foundation, Harbin Finance University, Harbin 150030, China;Department of Mathematics, Harbin Institute of Technology, Harbin 150080, China, hit.edu.cn;Natural Science Research Center, Harbin Institute of Technology, Harbin 150080, China, hit.edu.cn

Others  :  1297233
DOI  :  10.1155/2013/632903

 received in 2013-05-18, accepted in 2013-07-01,  发布年份 2013

【 摘 要 】

The motion of dilute charged particles can be modeled by Vlasov-Poisson-Boltzmann system. We study the large time stability of the VPB system. To be precise, we prove that when time goes to infinity, the solution of VPB system tends to global Maxwellian state in a rate Ot−∞, by using a method developed for Boltzmann equation without force in the work of Desvillettes and Villani (2005). The improvement of the present paper is the removal of condition on parameter λ as in the work ofLi (2008).

【 授权许可】

CC BY   
Copyright © 2013 Li Li et al. 2013

【 参考文献 】

• [1]S. Ukai. (1974). On the existence of global solutions of mixed problem for non-linear Boltzmann equation. Proceedings of the Japan Academy.50:179-184. DOI: 10.3792/pja/1195519027.
• [2]R. E. Caflisch. (1980). The Boltzmann equation with a soft potential. I. Linear, spatially-homogeneous. Communications in Mathematical Physics.74(1):71-95. DOI: 10.3792/pja/1195519027.
• [3]R. M. Strain, Y. Guo. (2008). Exponential decay for soft potentials near Maxwellian. Archive for Rational Mechanics and Analysis.187(2):287-339. DOI: 10.3792/pja/1195519027.
• [4]L. Desvillettes, C. Villani. (2005). On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation. Inventiones Mathematicae.159(2):245-316. DOI: 10.3792/pja/1195519027.
• [5]L. Li. (2008). On the trend to equilibrium for the Vlasov-Poisson-Boltzmann equation. Journal of Differential Equations.244(6):1467-1501. DOI: 10.3792/pja/1195519027.
• [6]Y. Guo. (2002). The Vlasov-Poisson-Boltzmann system near Maxwellians. Communications on Pure and Applied Mathematics.55(9):1104-1135. DOI: 10.3792/pja/1195519027.
• [7]R. M. Strain. (2006). The Vlasov-Maxwell-Boltzmann system in the whole space. Communications in Mathematical Physics.268(2):543-567. DOI: 10.3792/pja/1195519027.
• [8]T. Yang, H. J. Yu, H. J. Zhao. (2006). Cauchy problem for the Vlasov-Poisson-Boltzmann system. Archive for Rational Mechanics and Analysis.182(3):415-470. DOI: 10.3792/pja/1195519027.
• [9]T. Yang, H. J. Zhao. (2006). Global existence of classical solutions to the Vlasov-Poisson-Boltzmann system. Communications in Mathematical Physics.268(3):569-605. DOI: 10.3792/pja/1195519027.
• [10]C. Villani. (2003). Cercignani's conjecture is sometimes true and always almost true. Communications in Mathematical Physics.234(3):455-490. DOI: 10.3792/pja/1195519027.