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doi: 10.3934/dcds.2020292

Uniform stability estimate for the Vlasov-Poisson-Boltzmann system

Hao Wang ,

School of Science, Wuhan Institute of Technology, Wuhan, 430072, China, School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China

*

Received  February 2020 Revised  May 2020 Published  August 2020

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.

Keywords: Vlasov-Poisson-Boltzmann system, compensating function, energy method, uniform stability estimate.
Mathematics Subject Classification: Primary:76P05, 82C40;Secondary:82D05.
Citation: Hao Wang. Uniform stability estimate for the Vlasov-Poisson-Boltzmann system. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2020292
References:
[1] R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, Vol. 65. Academic Press, New York-London, 1975.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

R. T. Glassey, The Cauchy Problem in Kinetic Theory, Philadelphia, PA, SIAM, 1996. doi: 10.1137/1.9781611971477.  Google Scholar

[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.  Google Scholar

[12]

Y. Guo, The Vlasov-Poisson-Boltzmann system near Maxwellians, Comm. Pure Appl. Math., 55 (2002), 1104-1135.  doi: 10.1002/cpa.10040.  Google Scholar

[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.  Google Scholar

[14]

Y. Guo, Boltzmann diffusive limit beyond the Navier-Stokes approximation, Comm. Pure Appl. Math., 59 (2006), 626-687.  doi: 10.1002/cpa.20121.  Google Scholar

[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.  Google Scholar

[16]

S. Kawashima, The Boltzmann equation and thirteen moments, Japan J. Appl. Math., 7 (1990), 301-320.  doi: 10.1007/BF03167846.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1] R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, Vol. 65. Academic Press, New York-London, 1975.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

R. T. Glassey, The Cauchy Problem in Kinetic Theory, Philadelphia, PA, SIAM, 1996. doi: 10.1137/1.9781611971477.  Google Scholar

[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.  Google Scholar

[12]

Y. Guo, The Vlasov-Poisson-Boltzmann system near Maxwellians, Comm. Pure Appl. Math., 55 (2002), 1104-1135.  doi: 10.1002/cpa.10040.  Google Scholar

[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.  Google Scholar

[14]

Y. Guo, Boltzmann diffusive limit beyond the Navier-Stokes approximation, Comm. Pure Appl. Math., 59 (2006), 626-687.  doi: 10.1002/cpa.20121.  Google Scholar

[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.  Google Scholar

[16]

S. Kawashima, The Boltzmann equation and thirteen moments, Japan J. Appl. Math., 7 (1990), 301-320.  doi: 10.1007/BF03167846.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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