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Singular solutions of a Lane-Emden system
Uniform stability estimate for the Vlasov-Poisson-Boltzmann system
School of Science, Wuhan Institute of Technology, Wuhan, 430072, China, School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China |
This paper is concerned with the uniform stability estimate to the Cauchy problem of the Vlasov-Poisson-Boltzmann system. Our analysis is based on compensating function introduced by Kawashima and the standard energy method.
References:
| [1] |
R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, Vol. 65. Academic Press, New York-London, 1975.
|
| [2] |
C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Springer, New York, 1994.
doi: 10.1007/978-1-4419-8524-8. |
| [3] |
L. Desvillettes and J. Dolbeault,
On long time asymptotics of the Vlasov-Poisson-Boltzmann equation, Comm. Partial Differ. Eqs., 16 (1991), 451-489.
doi: 10.1080/03605309108820765. |
| [4] |
R. Duan and R. M. Strain,
Optimal time decay of the Vlasov-Poisson-Boltzmann system in $\mathbb{R}^ 3$., Arch. Ration. Mech. Anal., 199 (2011), 291-328.
doi: 10.1007/s00205-010-0318-6. |
| [5] |
R. Duan and T. Yang,
Stability of the one-species Vlasov-Poisson-Boltzmann system, SIAM J. Math. Anal., 41 (2010), 2353-2387.
doi: 10.1137/090745775. |
| [6] |
R. Duan, T. Yang and H. Zhao,
The Vlasov-Poisson-Boltzmann system in the whole space: The hard potential case, J. Diff. Eqs., 252 (2012), 6356-6386.
doi: 10.1016/j.jde.2012.03.012. |
| [7] |
R. Duan, T. Yang and C. Zhu,
Boltzmann equation with external force and Vlasov-Poisson-Boltzmann system in infinite vacuum, Discrete Contin. Dyn. Syst., 16 (2006), 253-277.
doi: 10.3934/dcds.2006.16.253. |
| [8] |
R. Duan and S. Liu,
The Vlasov-Poisson-Boltzmann system without angular cutoff, Comm. Math. Phys., 324 (2013), 1-45.
doi: 10.1007/s00220-013-1807-x. |
| [9] |
R. Duan, T. Yang and H. Zhao,
The Vlasov-Poisson-Boltzmann system for soft potentials, Math. Models Meth. Appl. Sci., 23 (2013), 979-1028.
doi: 10.1142/S0218202513500012. |
| [10] |
R. T. Glassey, The Cauchy Problem in Kinetic Theory, Philadelphia, PA, SIAM, 1996.
doi: 10.1137/1.9781611971477. |
| [11] |
R. T. Glassey and W. A. Strauss,
Perturbation of essential spectra of evolution operators and the Vlasov-Poisson-Boltzmann system, Discrete Contin. Dynam. Systems-A, 5 (1999), 457-472.
doi: 10.3934/dcds.1999.5.457. |
| [12] |
Y. Guo,
The Vlasov-Poisson-Boltzmann system near Maxwellians, Comm. Pure Appl. Math., 55 (2002), 1104-1135.
doi: 10.1002/cpa.10040. |
| [13] |
Y. Guo,
The Boltzmann equation in the whole space, Indiana Univ. Math. J., 53 (2004), 1081-1094.
doi: 10.1512/iumj.2004.53.2574. |
| [14] |
Y. Guo,
Boltzmann diffusive limit beyond the Navier-Stokes approximation, Comm. Pure Appl. Math., 59 (2006), 626-687.
doi: 10.1002/cpa.20121. |
| [15] |
Y. Guo and J. Jang,
Global Hilbert expansion for the Vlasov-Poisson-Boltzmann system, Comm. Math. Phys., 299 (2010), 469-501.
doi: 10.1007/s00220-010-1089-5. |
| [16] |
S. Kawashima,
The Boltzmann equation and thirteen moments, Japan J. Appl. Math., 7 (1990), 301-320.
doi: 10.1007/BF03167846. |
| [17] |
P.-L. Lions, On kinetic equations, In Proceedings of International Congress of Mathematician, Vol. Ⅰ, Ⅱ (Kyoto, 1990), 1173–1185, Math. Soc. Japan, Tokyo, 1991. |
| [18] |
S. Mischler,
On the initial boundary value problem for the Vlasov-Poisson-Boltzmann system, Comm. Math. Phys., 210 (2000), 447-466.
doi: 10.1007/s002200050787. |
| [19] |
X. Wang and H. Shi,
Decay and stability of solutions to the Fokker-Planck-Boltzmann equation in $\mathbb{R}^3$, Appl. Anal., 97 (2018), 1933-1959.
doi: 10.1080/00036811.2017.1344225. |
| [20] |
T. Yang, H. Yu and H. Zhao,
Cauchy problem for the Vlasov-Poisson-Boltzmann system, Arch. Rat. Mech. Anal., 182 (2006), 415-470.
doi: 10.1007/s00205-006-0009-5. |
| [21] |
T. Yang and H. Yu,
Optimal convergence rates of classical so lutions for Vlasov-Poisson-Boltzmann system, Comm. Math. Phys., 301 (2011), 319-355.
doi: 10.1007/s00220-010-1142-4. |
| [22] |
T. Yang and H. Zhao,
Global existence of classical solutions to the Vlasov-Poisson-Boltzmann system, Comm. Math. Phys., 268 (2006), 569-605.
doi: 10.1007/s00220-006-0103-4. |
show all references
References:
| [1] |
R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, Vol. 65. Academic Press, New York-London, 1975.
|
| [2] |
C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases, Springer, New York, 1994.
doi: 10.1007/978-1-4419-8524-8. |
| [3] |
L. Desvillettes and J. Dolbeault,
On long time asymptotics of the Vlasov-Poisson-Boltzmann equation, Comm. Partial Differ. Eqs., 16 (1991), 451-489.
doi: 10.1080/03605309108820765. |
| [4] |
R. Duan and R. M. Strain,
Optimal time decay of the Vlasov-Poisson-Boltzmann system in $\mathbb{R}^ 3$., Arch. Ration. Mech. Anal., 199 (2011), 291-328.
doi: 10.1007/s00205-010-0318-6. |
| [5] |
R. Duan and T. Yang,
Stability of the one-species Vlasov-Poisson-Boltzmann system, SIAM J. Math. Anal., 41 (2010), 2353-2387.
doi: 10.1137/090745775. |
| [6] |
R. Duan, T. Yang and H. Zhao,
The Vlasov-Poisson-Boltzmann system in the whole space: The hard potential case, J. Diff. Eqs., 252 (2012), 6356-6386.
doi: 10.1016/j.jde.2012.03.012. |
| [7] |
R. Duan, T. Yang and C. Zhu,
Boltzmann equation with external force and Vlasov-Poisson-Boltzmann system in infinite vacuum, Discrete Contin. Dyn. Syst., 16 (2006), 253-277.
doi: 10.3934/dcds.2006.16.253. |
| [8] |
R. Duan and S. Liu,
The Vlasov-Poisson-Boltzmann system without angular cutoff, Comm. Math. Phys., 324 (2013), 1-45.
doi: 10.1007/s00220-013-1807-x. |
| [9] |
R. Duan, T. Yang and H. Zhao,
The Vlasov-Poisson-Boltzmann system for soft potentials, Math. Models Meth. Appl. Sci., 23 (2013), 979-1028.
doi: 10.1142/S0218202513500012. |
| [10] |
R. T. Glassey, The Cauchy Problem in Kinetic Theory, Philadelphia, PA, SIAM, 1996.
doi: 10.1137/1.9781611971477. |
| [11] |
R. T. Glassey and W. A. Strauss,
Perturbation of essential spectra of evolution operators and the Vlasov-Poisson-Boltzmann system, Discrete Contin. Dynam. Systems-A, 5 (1999), 457-472.
doi: 10.3934/dcds.1999.5.457. |
| [12] |
Y. Guo,
The Vlasov-Poisson-Boltzmann system near Maxwellians, Comm. Pure Appl. Math., 55 (2002), 1104-1135.
doi: 10.1002/cpa.10040. |
| [13] |
Y. Guo,
The Boltzmann equation in the whole space, Indiana Univ. Math. J., 53 (2004), 1081-1094.
doi: 10.1512/iumj.2004.53.2574. |
| [14] |
Y. Guo,
Boltzmann diffusive limit beyond the Navier-Stokes approximation, Comm. Pure Appl. Math., 59 (2006), 626-687.
doi: 10.1002/cpa.20121. |
| [15] |
Y. Guo and J. Jang,
Global Hilbert expansion for the Vlasov-Poisson-Boltzmann system, Comm. Math. Phys., 299 (2010), 469-501.
doi: 10.1007/s00220-010-1089-5. |
| [16] |
S. Kawashima,
The Boltzmann equation and thirteen moments, Japan J. Appl. Math., 7 (1990), 301-320.
doi: 10.1007/BF03167846. |
| [17] |
P.-L. Lions, On kinetic equations, In Proceedings of International Congress of Mathematician, Vol. Ⅰ, Ⅱ (Kyoto, 1990), 1173–1185, Math. Soc. Japan, Tokyo, 1991. |
| [18] |
S. Mischler,
On the initial boundary value problem for the Vlasov-Poisson-Boltzmann system, Comm. Math. Phys., 210 (2000), 447-466.
doi: 10.1007/s002200050787. |
| [19] |
X. Wang and H. Shi,
Decay and stability of solutions to the Fokker-Planck-Boltzmann equation in $\mathbb{R}^3$, Appl. Anal., 97 (2018), 1933-1959.
doi: 10.1080/00036811.2017.1344225. |
| [20] |
T. Yang, H. Yu and H. Zhao,
Cauchy problem for the Vlasov-Poisson-Boltzmann system, Arch. Rat. Mech. Anal., 182 (2006), 415-470.
doi: 10.1007/s00205-006-0009-5. |
| [21] |
T. Yang and H. Yu,
Optimal convergence rates of classical so lutions for Vlasov-Poisson-Boltzmann system, Comm. Math. Phys., 301 (2011), 319-355.
doi: 10.1007/s00220-010-1142-4. |
| [22] |
T. Yang and H. Zhao,
Global existence of classical solutions to the Vlasov-Poisson-Boltzmann system, Comm. Math. Phys., 268 (2006), 569-605.
doi: 10.1007/s00220-006-0103-4. |
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