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December  2017, 10(4): 1089-1125. doi: 10.3934/krm.2017043

Spectrum structure and optimal decay rate of the relativistic Vlasov-Poisson-Landau system

1. 

Department of Mathematics, Capital Normal University, China

2. 

School of Mathematical Sciences, South China Normal University, China

3. 

Department of Mathematics and Information Sciences, Guangxi University, China

* Corresponding author: Mingying Zhong

Received  October 2016 Revised  January 2017 Published  March 2017

Fund Project: The first author is supported by the NNSFC grants No. 11225102,11231006,11461161007 and 11671384, and by the Key Project of Beijing Municipal Education Commission no. CIT and TCD2014 0323. The second author is supported by the NNSFC grants No. 11371151. The third author is supported by National Natural Science Foundation of China No. 11671100 and 11301094, Project supported by Guangxi Natural Science Foundation No. 2014GXNSFBA118020.

It is interesting to analyze the mutual influence of relativistic effect and electrostatic potential force on the qualitative behaviors of charge particles simulated by the one-species relativistic Vlasov-Poisson-Landau (rVPL) system with the physical Coulombic interaction. In this paper, we first study the spectrum structure on the linearized rVPL system and obtain the optimal time decay rates of the solutions to the linearized system, and then we construct global strong solutions to the nonlinear system around a global relativistic Maxwellian. Finally we make use of time decay rates of the solutions to the linearized system and uniform energy estimates to establish the time decay of the global solution to the original Cauchy problem for the rVPL system to the absolute Maxwellian at the optimal convergence rate $(1+t)^{-3/4}$. This time rate is faster than the optimal rate $(1+t)^{-1/4}$ of classical Vlasov-Poisson-Boltzmann [2,10] and Vlasov-Poisson-Landau system [7,8,17] and this fast time decay rate is caused by the combined influence of relativistic effect and electrostatic potential force.

Citation: Hai-Liang Li, Hongjun Yu, Mingying Zhong. Spectrum structure and optimal decay rate of the relativistic Vlasov-Poisson-Landau system. Kinetic and Related Models, 2017, 10 (4) : 1089-1125. doi: 10.3934/krm.2017043
References:
[1]

R.-J. Duan, Global smooth dynamics of a fully ionized plasma with long-range collisions, Ann. I. H. Poincaré, 31 (2014), 751-778.  doi: 10.1016/j.anihpc.2013.07.004.

[2]

R.-J. Duan and R. 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.

[3]

R. S. Ellis and M. A. Pinsky, The first and second fluid approximations to the linearized Boltzmann equation, J. Math. pure et appl., 54 (1975), 125-156. 

[4]

Y. Guo, The Vlasov-Poisson-Landau system in a periodic box, J. Amer. Math. Soc., 25 (2012), 759-812.  doi: 10.1090/S0894-0347-2011-00722-4.

[5]

F. L. Hinton, Collisional transport in plasma. In Handbook of Plasma Physics, Volume Ⅰ: Basic Plasma Physics Ⅰ Rosenbluth M. N. and Sagdeev R. Z. (eds. ), Amsterdam: North-Holland Publishing Company, 1983, pp. 147.

[6]

T. Kato, Perturbation Theory of Linear Operator Springer, New York, 1996.

[7]

Y. Lei and H. Zhao, Negative Sobolev Spaces and the Two-species Vlasov-Maxwell-Landau System in the Whole Space, J. Funct. Anal., 267 (2014), 3710-3757.  doi: 10.1016/j.jfa.2014.09.011.

[8]

Y. LeiL. Xiong and H. Zhao, One-species Vlasov-Poisson-Landau system near Maxwellians in the whole space, Kinet. Relat. Models, 7 (2014), 551-590.  doi: 10.3934/krm.2014.7.551.

[9]

M. Lemou, Linearized quantum and relativistic Fokker-Planck-Landau equations, Math. Methods Appl. Sci., 23 (2000), 1093-1119.  doi: 10.1002/1099-1476(200008)23:12<1093::AID-MMA153>3.0.CO;2-8.

[10]

H. -L. Li, T. Yang and M. Zhong, Spectrum analysis for the Vlasov-Poisson-Boltzmann system, preprint, arXiv: 1402.3633v1, [math. AP].

[11]

H.-L. LiT. Yang and M. Zhong, Spectrum analysis and optimal decay rates of the bipolar Vlasov-Poisson-Boltzmann equations, Indiana Univ. Math. J., 65 (2016), 665-725.  doi: 10.1512/iumj.2016.65.5730.

[12]

H.-L. LiT. YangJ. Sun and M. Zhong, Large time behavior of solutions to Vlasov-Poisson-Landau (Fokker-Planck) equations (in Chinese), Sci Sin Math, 46 (2016), 981-1004. 

[13]

S.-Q. Liu and H.-J. Zhao, Optimal large-time decay of the relativistic Landau-Maxwell system, J. Differential Equations, 256 (2014), 832-857.  doi: 10.1016/j.jde.2013.10.004.

[14]

T.-P. LiuT. Yang and S.-H. Yu, Energy method for the Boltzmann equation, Physica D, 188 (2004), 178-192.  doi: 10.1016/j.physd.2003.07.011.

[15]

L. Luo and H.-J. Yu, Spectrum analysis of the linearized relativistic Landau equation, J. Stat. Phys., 163 (2016), 914-935.  doi: 10.1007/s10955-016-1501-4.

[16]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[17]

R. M. Strain, The Vlasov-Poisson-Landau System in $\mathbb{R}^3_x$, Arch. Rational Mech. Anal., 210 (2013), 615-671.  doi: 10.1007/s00205-013-0658-0.

[18]

R. M. Strain and Y. Guo, Stability of the relativistic Maxwellian in a collisional plasma, Comm. Math. Phys., 251 (2004), 263-320.  doi: 10.1007/s00220-004-1151-2.

[19]

R. M. Strain and Y. Guo, Almost exponential decay near Maxwellian, Comm. P.D.E., 31 (2006), 417-429.  doi: 10.1080/03605300500361545.

[20]

S. Ukai and T. Yang, Mathematical Theory of Boltzmann Equation Lecture Notes Series-No. 8, Hong Kong: Liu Bie Ju Center for Mathematical Sciences, City University of Hong Kong, 2006.

[21]

Y. Wang, Global solution and time decay of the Vlasov-Poisson-Landau system in $\mathbb{R}^3$, SIAM J. Math. Anal., 44 (2012), 3281-3323.  doi: 10.1137/120879129.

[22]

T. Yang and H.-J. Yu, Spectrum analysis of some kinetic equations, Arch. Ration. Mech. Anal., 222 (2016), 731-768.  doi: 10.1007/s00205-016-1010-2.

[23]

T. Yang and H.-J. Yu, Hypocoercivity of the relativistic Boltzmann and Landau equations in the whole space, J. Differential Equations, 248 (2010), 1518-1560.  doi: 10.1016/j.jde.2009.11.027.

[24]

T. Yang and H.-J. Yu, Global solutions to the relativistic Landau-Maxwell system in the whole space, J. Math. Pures Appl., 97 (2012), 602-634.  doi: 10.1016/j.matpur.2011.09.006.

[25]

T. YangH.-J. Yu and H.-J. Zhao, Cauchy problem for the Vlasov-Poisson-Boltzmann system, Arch. Ration. Mech. Anal., 182 (2006), 415-470.  doi: 10.1007/s00205-006-0009-5.

[26]

T. Yang and H.-J. 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.-J. Duan, Global smooth dynamics of a fully ionized plasma with long-range collisions, Ann. I. H. Poincaré, 31 (2014), 751-778.  doi: 10.1016/j.anihpc.2013.07.004.

[2]

R.-J. Duan and R. 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.

[3]

R. S. Ellis and M. A. Pinsky, The first and second fluid approximations to the linearized Boltzmann equation, J. Math. pure et appl., 54 (1975), 125-156. 

[4]

Y. Guo, The Vlasov-Poisson-Landau system in a periodic box, J. Amer. Math. Soc., 25 (2012), 759-812.  doi: 10.1090/S0894-0347-2011-00722-4.

[5]

F. L. Hinton, Collisional transport in plasma. In Handbook of Plasma Physics, Volume Ⅰ: Basic Plasma Physics Ⅰ Rosenbluth M. N. and Sagdeev R. Z. (eds. ), Amsterdam: North-Holland Publishing Company, 1983, pp. 147.

[6]

T. Kato, Perturbation Theory of Linear Operator Springer, New York, 1996.

[7]

Y. Lei and H. Zhao, Negative Sobolev Spaces and the Two-species Vlasov-Maxwell-Landau System in the Whole Space, J. Funct. Anal., 267 (2014), 3710-3757.  doi: 10.1016/j.jfa.2014.09.011.

[8]

Y. LeiL. Xiong and H. Zhao, One-species Vlasov-Poisson-Landau system near Maxwellians in the whole space, Kinet. Relat. Models, 7 (2014), 551-590.  doi: 10.3934/krm.2014.7.551.

[9]

M. Lemou, Linearized quantum and relativistic Fokker-Planck-Landau equations, Math. Methods Appl. Sci., 23 (2000), 1093-1119.  doi: 10.1002/1099-1476(200008)23:12<1093::AID-MMA153>3.0.CO;2-8.

[10]

H. -L. Li, T. Yang and M. Zhong, Spectrum analysis for the Vlasov-Poisson-Boltzmann system, preprint, arXiv: 1402.3633v1, [math. AP].

[11]

H.-L. LiT. Yang and M. Zhong, Spectrum analysis and optimal decay rates of the bipolar Vlasov-Poisson-Boltzmann equations, Indiana Univ. Math. J., 65 (2016), 665-725.  doi: 10.1512/iumj.2016.65.5730.

[12]

H.-L. LiT. YangJ. Sun and M. Zhong, Large time behavior of solutions to Vlasov-Poisson-Landau (Fokker-Planck) equations (in Chinese), Sci Sin Math, 46 (2016), 981-1004. 

[13]

S.-Q. Liu and H.-J. Zhao, Optimal large-time decay of the relativistic Landau-Maxwell system, J. Differential Equations, 256 (2014), 832-857.  doi: 10.1016/j.jde.2013.10.004.

[14]

T.-P. LiuT. Yang and S.-H. Yu, Energy method for the Boltzmann equation, Physica D, 188 (2004), 178-192.  doi: 10.1016/j.physd.2003.07.011.

[15]

L. Luo and H.-J. Yu, Spectrum analysis of the linearized relativistic Landau equation, J. Stat. Phys., 163 (2016), 914-935.  doi: 10.1007/s10955-016-1501-4.

[16]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[17]

R. M. Strain, The Vlasov-Poisson-Landau System in $\mathbb{R}^3_x$, Arch. Rational Mech. Anal., 210 (2013), 615-671.  doi: 10.1007/s00205-013-0658-0.

[18]

R. M. Strain and Y. Guo, Stability of the relativistic Maxwellian in a collisional plasma, Comm. Math. Phys., 251 (2004), 263-320.  doi: 10.1007/s00220-004-1151-2.

[19]

R. M. Strain and Y. Guo, Almost exponential decay near Maxwellian, Comm. P.D.E., 31 (2006), 417-429.  doi: 10.1080/03605300500361545.

[20]

S. Ukai and T. Yang, Mathematical Theory of Boltzmann Equation Lecture Notes Series-No. 8, Hong Kong: Liu Bie Ju Center for Mathematical Sciences, City University of Hong Kong, 2006.

[21]

Y. Wang, Global solution and time decay of the Vlasov-Poisson-Landau system in $\mathbb{R}^3$, SIAM J. Math. Anal., 44 (2012), 3281-3323.  doi: 10.1137/120879129.

[22]

T. Yang and H.-J. Yu, Spectrum analysis of some kinetic equations, Arch. Ration. Mech. Anal., 222 (2016), 731-768.  doi: 10.1007/s00205-016-1010-2.

[23]

T. Yang and H.-J. Yu, Hypocoercivity of the relativistic Boltzmann and Landau equations in the whole space, J. Differential Equations, 248 (2010), 1518-1560.  doi: 10.1016/j.jde.2009.11.027.

[24]

T. Yang and H.-J. Yu, Global solutions to the relativistic Landau-Maxwell system in the whole space, J. Math. Pures Appl., 97 (2012), 602-634.  doi: 10.1016/j.matpur.2011.09.006.

[25]

T. YangH.-J. Yu and H.-J. Zhao, Cauchy problem for the Vlasov-Poisson-Boltzmann system, Arch. Ration. Mech. Anal., 182 (2006), 415-470.  doi: 10.1007/s00205-006-0009-5.

[26]

T. Yang and H.-J. 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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