L1 and BV-type stability of the Boltzmann equation with external forces

Journal of Differential Equations

Volumn 227, Issue 1, 2006, Pages 1-28

  Duan, Renjun ^a   Yang, Tong ^a   Zhu, Changjiang ^b  

^a CITY UNIVERSITY OF HONG KONG   (Hong Kong)

^b CENTRAL CHINA NORMAL UNIVERSITY   (China)

Author keywords

Bi characteristics; Boltzmann equation; Lyapunov functional; Stability

Indexed keywords

EID: 33646598396     PISSN: 00220396     EISSN: 10902732     Source Type: Journal    
DOI: 10.1016/j.jde.2006.01.010     Document Type: Article

References (28)

1. 1 for the spatially homogeneous Boltzmann equation. Arch. Ration. Mech. Anal. 103 (1988) 151-168
2. Asano K. Local solutions to the initial and initial boundary value problem for the Boltzmann equation with an external force. J. Math. Kyoto Univ. 24 2 (1984) 225-238
3. Bellomo N., Palczewski A., and Toscani G. Mathematical Topics in Nonlinear Kinetic Theory (1988), World Scientific, Singapore
4. Bellomo N., Lachowicz M., Palczewski A., and Toscani G. On the initial value problem for the Boltzmann equation with a force term. Transport Theory Statist. Phys. 18 1 (1989) 87-102
5. Bouchut F., Golse F., and Pulvirenti M. Kinetic Equations and Asymptotic Theory. In: Perthame B., and Desvillettes L. (Eds). Ser. Appl. Math. vol. 4 (2000), Gauthier-Villars, Paris
6. Cercignani C. The Boltzmann Equation and Its Applications (1988), Springer, New York
7. Cercignani C., Illner R., and Pulvirenti M. The Mathematical Theory of Dilute Gases (1994), Springer, New York
8. M. Chae, S.-Y. Ha, New Lyapunov functionals of the Vlasov-Poisson system, SIAM J. Math. Anal., in press
9. Desvillettes L., and Villani C. On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation. Invent. Math. 159 (2005) 245-316
10. DiPerna R., and Lions P.-L. On the Cauchy problem for the Boltzmann equation: Global existence and weak stability results. Ann. of Math. 130 (1990) 1189-1214
11. Duan R.J., Yang T., and Zhu C.J. Global existence to the Boltzmann equation with external forces in infinite vacuum. J. Math. Phys. 46 (2005) 053307
12. R.J. Duan, T. Yang, C.J. Zhu, Boltzmann equation with external force and Vlasov-Poisson-Boltzmann system in infinite vacuum, 2005, Discrete Contin. Dyn. Syst., in press
13. Glikson A. On the existence of general solutions of the initial-value problem for the nonlinear Boltzmann equation with a cut-off. Arch. Ration. Mech. Anal. 45 (1972) 35-46
14. Glikson A. On solution of the nonlinear Boltzmann equation with a cut off in an unbounded domain. Arch. Ration. Mech. Anal. 47 (1972) 389-394
15. Guo Y. The Vlasov-Maxwell-Boltzmann system near Maxwellians. Invent. Math. 153 (2003) 593-630
16. Guo Y. The Vlasov-Poisson-Boltzmann system near Maxwellians. Comm. Pure Appl. Math. 55 9 (2002) 1104-1135
17. Guo Y. The Vlasov-Poisson-Boltzmann system near vacuum. Comm. Math. Phys. 218 2 (2001) 293-313
18. 1 stability of the Boltzmann equation for the hard sphere model. Arch. Ration. Mech. Anal. 171 (2004) 279-296
19. Ha S.-Y. Nonlinear functionals of the Boltzmann equation and uniform stability estimates. J. Differential Equations 215 (2005) 178-205
20. Illner R., and Shinbrot M. Global existence for a rare gas in an infinite vacuum. Comm. Math. Phys. 95 (1984) 217-226
21. Liu T.-P., Yang T., and Yu S.-H. Energy method for the Boltzmann equation. Phys. D 188 3-4 (2004) 178-192
22. Nishida T., and Imai K. Global solutions to the initial-value problem for the nonlinear Boltzmann. Publ. Res. Inst. Math. Sci. 12 (1976) 229-239
23. 3: Asymptotic behavior of solutions. J. Statist. Phys. 50 (1988) 611-632
24. Toscani G., and Bellomo N. 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
25. Ukai S. Solutions of Boltzmann equation. Patterns and Waves-Qualitative Analysis of Nonlinear Differential Equations. Stud. Math. Appl. vol. 18 (1986) 37-96
26. Ukai S., and Asano K. On the Cauchy problem of the Boltzmann equation with a soft potential. Publ. Res. Inst. Math. Sci. 18 (1982) 477-519
27. Ukai S., Yang T., and Zhao H.-J. Global solutions to the Boltzmann equation with external forces. Anal. Appl. (Singap.) 3 2 (2005) 157-193
28. Villani C. A Review of Mathematical Topics in Collisional Kinetic Theory. In: Friedlander S., and Serre D. (Eds). Handbook of Fluid Mechanics (2001)