L1 stability for the Vlasov-Poisson-Boltzmann system around vacuum

Mathematical Models and Methods in Applied Sciences

Volumn 16, Issue 9, 2006, Pages 1505-1526

  Duan, Renjun ^a   Zhang, Mei ^b   Zhu, Changjiang ^b  

^a CITY UNIVERSITY OF HONG KONG   (Hong Kong)

^b CENTRAL CHINA NORMAL UNIVERSITY   (China)

Author keywords

Nonlinear functionals; Stability; Vlasov Poisson Boltzmann system

Indexed keywords

EID: 33747678674     PISSN: 02182025     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218202506001613     Document Type: Article

References (28)

1. 1 for the spatially homogeneous Boltzmann equation, Arch. Rational Mech. Anal. 103 (1988) 151-168.
2. C. Bardos and P. Degond, Global existence for the Vlasov-Poisson equation in three space variables with small initial data, AIHP, Anal. Nonlinear: 2 (1985) 101-118.
3. N. Bellomo, M. Lachowicz, A. Palczewski and G. Toscani, On the initial value problem for the Boltzmann equation with a force term, Transp. Th. Statis. Phys. 18 (1989) 87-102.
4. N. Bellomo, A. Palczewski and G. Toscani, Mathematical Topics in Nonlinear Kinetic Theory (World Scientific, 1988).
5. 1 stability estimates for n × n conservation laws, Arch. Rational Mech. Anal. 149 (1999) 1-22.
6. A. Carpio, E. Cebrian and F. J. Mustieles, Long time asymptotics for the semiconductor Vlasov-Poisson-Boltzmann equations, Math. Mod. Meth. Appl. Sci. 11 (2001) 1631-1655.
7. C. Cercignani, The Boltzmann Equation and Its Applications (Springer-Verlag, 1988).
8. C. Cercignani, R. Illner and M. Pulviventi, The Mathematical Theory of Dilute Gases (Springer-Verlag, 1994).
9. M. Chae and S.-Y. Ha, New Lyapunov functionals of the Vlasov-Poisson system, SIAM J. Math. Anal. 37 (2006) 1709-1731.
10. L. Desvillettes and J. Dolbeault, On long time asymptotics of the Vlasov-Poisson-Boltzmann equation, Commun. Partial Diff. Eqns. 16 (1995) 451-489.
11. 1 and BV-type stability of the Boltzmann equation with external forces, J. Diff. Eqns. 227 (2006) 1-28.
12. R. J. Duan, T. Yang and C. J. Zhu, Botlzmann equation with external force and Vlasov-Poisson-Boltzmann system in infinite vacuum, Discr. Cont. Dyn. Syst. 16 (2006) 253-277.
13. E. Frénod and E. Sonnendrücker, Long time behavior of the two-dimensional Vlasov equation with a strong external magnetic field, Math. Mod. Meth. Appl. Sci. 10 (2000) 539-553.
14. R. Glassey, The Cauchy Problem in Kinetic Theory (SIAM, 1996).
15. S. Graffi, A. Martinez and M. Pulvirenti, Mean-field approximation of quantum systems and classical limit, Math. Mod. Meth. Appl. Sci. 13 (2003) 59-73.
16. Y. Guo, The Vlasov-Poisson-Boltzmann system near Maxwellians, Commun. Pure Appl. Math. 55 (2002) 1104-1135.
17. Y. Guo, The Vlasov-Poisson-Boltzmann system near vacuum, Commun. Math. Phys. 218 (2001) 293-313.
18. 1 stability of the Boltzmann equation for the hard sphere model, Arch. Rational Mech. Anal. 171 (2004) 279-296.
19. S.-Y. Ha, Lyapunov functionals for the Enskog-Boltzmann equation, Indiana Univ. Math. J. 54 (2005) 997-1014.
20. S.-Y. Ha, Nonlinear functionals of the Boltzmann equation and uniform stability estimates, J. Diff. Eqns. 215 (2005) 178-205.
21. T.-P. Liu and T. Yang, Well-posedness theory for hyperbolic conservation laws, Commun. Pure Appl. Math. 52 (1999) 1553-1586.
22. S. Mischler, On the initial boundary value problem for the Vlasov-Poisson-Boltzmann system, Commun. Math. Phys. 210 (2000) 447-466.
23. 3: Asymptotic behavior of solutions, J. Statist. Phys. 50 (1988) 611-632.
24. F. Poupaud and J. Soler, Parabolic limit and stability of the Vlasov-Fokker-Planck system, Math. Mod. Meth. Appl. Sci. 10 (2000) 1027-1045.
25. G. Toscani and N. Bellomo, Global existence, uniqueness and stability of the nonlinear Boltzmann equation with almost general gas-particle interaction potential, Rend. Circ. Mat. Palermo(2) Suppl. 8 (1985) 419-433.
26. C. Villani, A review of mathematical topics in collisional kinetic theory, in Handbook of Mathematical Fluid Dynamics, eds. S. Friedlander and D. Serre (North-Holland, 2002), Vol. 1, pp. 71-305.
27. S. Wollman and E. Ozizmir, Numerical approximation of the Vlasov-Poisson-Fokker-Planck system in one dimension, J. Comput. Phys. 202 (2005) 602-644.
28. T. Yang, H. J. Yu and H. J. Zhao, Cauchy problem for the Vlasov-Poisson-Boltzmann system, to appear in Arch. Rational Mech. Anal.