American Institute of Mathematical Sciences

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June  2016, 9(2): 393-405. doi: 10.3934/krm.2016.9.393

Global solutions to the relativistic Vlasov-Poisson-Fokker-Planck system

Lan Luo 1, and  Hongjun Yu 2,

1. 

School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China
2. 

School of Mathematical Sciences, South China Normal University, Guangzhou 510631

Received  July 2015 Revised  September 2015 Published  March 2016

Global solutions to the relativistic Vlasov-Poisson-Fokker-Planck system near the relativistic Maxwellian are constructed based on an approach by combining the compensating function and energy method. In addition, an exponential rate in time of the solution to its equilibrium is obtained.
Keywords: compensating function, global solution, exponential decay, energy method., The relativistic Vlasov-Poisson-Fokker-Planck system.
Mathematics Subject Classification: Primary: 82D10; Secondary: 35Q8.
Citation: Lan Luo, Hongjun Yu. Global solutions to the relativistic Vlasov-Poisson-Fokker-Planck system. Kinetic & Related Models, 2016, 9 (2) : 393-405. doi: 10.3934/krm.2016.9.393
References:
[1]

M. Bostan and Th. Goudon, Low field regime for the relativistic Vlasov-Maxwell-Fokker-Planck system: the one and one half dimensional case,, Kinet. Relat. Models, 1 (2008), 139.  doi: 10.3934/krm.2008.1.139.  Google Scholar

[2]

F. Bouchut, Existence and uniqueness of a global smooth solution for the Vlasov-Poisson-Fokker-Planck system in three dimensions,, J. Func. Anal., 111 (1993), 239.  doi: 10.1006/jfan.1993.1011.  Google Scholar

[3]

A. Carpio, Long-time behaviour for solutions of the Vlasov-Poisson-Fokker-Planck equation,, Math. Methods Appl. Sci., 21 (1998), 985.  doi: 10.1002/(SICI)1099-1476(19980725)21:11<985::AID-MMA919>3.0.CO;2-B.  Google Scholar

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J.-A. Carrillo and J. Soler, On the initial value problem for the Vlasov-Poisson-Fokker-Planck system with initial data in $L^p$ spaces,, Math. Methods Appl. Sci., 18 (1995), 825.  doi: 10.1002/mma.1670181006.  Google Scholar

[5]

J.-A. Carrillo, J. Soler and J. L. Vazquez, Asymptotic behaviour and self-similarity for the three-dimensional Vlasov-Poisson-Fokker-Planck system,, J. Func. Anal., 141 (1996), 99.  doi: 10.1006/jfan.1996.0123.  Google Scholar

[6]

M. Chae, The global classical solution of the Vlasov-Maxwell-Fokker-Planck system near Maxwellian,, Math. Models Methods Appl. Sci., 21 (2011), 1007.  doi: 10.1142/S0218202511005222.  Google Scholar

[7]

R. DiPerna and P.-L. Lions, Global weak solutions of kinetic equations,, Rend. Sem. Mat. Univ. Politec. Torino, 46 (1988), 259.   Google Scholar

[8]

R.-J. Duan and S.-Q. Liu, The Vlasov-Poisson-Boltzmann system without angular cutoff,, Comm. Math. Phys., 324 (2013), 1.   Google Scholar

[9]

R.-J. Duan and R.-M. Strain, Optimal time decay of the Vlasov-Poisson-Boltzmann system in $R^3$,, Arch. Ration. Mech. Anal., 199 (2011), 291.  doi: 10.1007/s00205-010-0318-6.  Google Scholar

[10]

R.-J. Duan, T. Yang and H.-J. Zhao, The Vlasov-Poisson-Boltzmann system in the whole space: The hard potential case,, J. Differential Equations, 252 (2012), 6356.  doi: 10.1016/j.jde.2012.03.012.  Google Scholar

[11]

R. T. Glassey, The Cauchy Problem in Kinetic Theory,, Society for Industrial and Applied Mathematics (SIAM), (1996).  doi: 10.1137/1.9781611971477.  Google Scholar

[12]

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

[13]

L. Hsiao, F.-C. Li and S. Wang, Combined quasineutral and inviscid limit of the Vlasov-Poisson-Fokker-Planck system,, Commun. Pure Appl. Anal., 7 (2008), 579.  doi: 10.3934/cpaa.2008.7.579.  Google Scholar

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[15]

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

[16]

R. Lai, On the one and one-half dimensional relativistic Vlasov-Maxwell-Fokker-Planck system with non-vanishing viscosity,, Math. Meth. Appl. Sci., 21 (1998), 1287.  doi: 10.1002/(SICI)1099-1476(19980925)21:14<1287::AID-MMA996>3.0.CO;2-G.  Google Scholar

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

[18]

B. O'Dwyer and H.-D. Victory Jr., On classical solutions of the Vlasov-Poisson-Fokker-Planck system,, Indiana Univ. Math. J., 39 (1990), 105.  doi: 10.1512/iumj.1990.39.39009.  Google Scholar

[19]

G. Rein and J. Weckler, Generic global classical solutions of the Vlasov-Fokker-Planck-Poisson system in three dimensions,, J. Differential Equations, 99 (1992), 59.  doi: 10.1016/0022-0396(92)90135-A.  Google Scholar

[20]

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

[21]

H.-D. Victory Jr., On the existence of global weak solutions for Vlasov-Poisson-Fokker-Planck systems,, J. Math. Anal. Appl., 160 (1991), 525.  doi: 10.1016/0022-247X(91)90324-S.  Google Scholar

[22]

C. Villani, Hypocoercivity,, Memoirs Amer. Math. Soc., 202 (2009).  doi: 10.1090/S0065-9266-09-00567-5.  Google Scholar

[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.  doi: 10.1016/j.jde.2009.11.027.  Google Scholar

[24]

T. Yang and H.-J. Yu, Global classical solutions for the Vlasov-Maxwell-Fokker-Planck system,, SIAM J. Math. Anal., 42 (2010), 459.  doi: 10.1137/090755796.  Google Scholar

[25]

T. Yang and H.-J. Zhao, Global existence of classical solutions to the Vlasov-Poisson-Boltzmann system,, Comm. Math. Phys., 268 (2006), 569.  doi: 10.1007/s00220-006-0103-4.  Google Scholar

show all references

References:
[1]

M. Bostan and Th. Goudon, Low field regime for the relativistic Vlasov-Maxwell-Fokker-Planck system: the one and one half dimensional case,, Kinet. Relat. Models, 1 (2008), 139.  doi: 10.3934/krm.2008.1.139.  Google Scholar

[2]

F. Bouchut, Existence and uniqueness of a global smooth solution for the Vlasov-Poisson-Fokker-Planck system in three dimensions,, J. Func. Anal., 111 (1993), 239.  doi: 10.1006/jfan.1993.1011.  Google Scholar

[3]

A. Carpio, Long-time behaviour for solutions of the Vlasov-Poisson-Fokker-Planck equation,, Math. Methods Appl. Sci., 21 (1998), 985.  doi: 10.1002/(SICI)1099-1476(19980725)21:11<985::AID-MMA919>3.0.CO;2-B.  Google Scholar

[4]

J.-A. Carrillo and J. Soler, On the initial value problem for the Vlasov-Poisson-Fokker-Planck system with initial data in $L^p$ spaces,, Math. Methods Appl. Sci., 18 (1995), 825.  doi: 10.1002/mma.1670181006.  Google Scholar

[5]

J.-A. Carrillo, J. Soler and J. L. Vazquez, Asymptotic behaviour and self-similarity for the three-dimensional Vlasov-Poisson-Fokker-Planck system,, J. Func. Anal., 141 (1996), 99.  doi: 10.1006/jfan.1996.0123.  Google Scholar

[6]

M. Chae, The global classical solution of the Vlasov-Maxwell-Fokker-Planck system near Maxwellian,, Math. Models Methods Appl. Sci., 21 (2011), 1007.  doi: 10.1142/S0218202511005222.  Google Scholar

[7]

R. DiPerna and P.-L. Lions, Global weak solutions of kinetic equations,, Rend. Sem. Mat. Univ. Politec. Torino, 46 (1988), 259.   Google Scholar

[8]

R.-J. Duan and S.-Q. Liu, The Vlasov-Poisson-Boltzmann system without angular cutoff,, Comm. Math. Phys., 324 (2013), 1.   Google Scholar

[9]

R.-J. Duan and R.-M. Strain, Optimal time decay of the Vlasov-Poisson-Boltzmann system in $R^3$,, Arch. Ration. Mech. Anal., 199 (2011), 291.  doi: 10.1007/s00205-010-0318-6.  Google Scholar

[10]

R.-J. Duan, T. Yang and H.-J. Zhao, The Vlasov-Poisson-Boltzmann system in the whole space: The hard potential case,, J. Differential Equations, 252 (2012), 6356.  doi: 10.1016/j.jde.2012.03.012.  Google Scholar

[11]

R. T. Glassey, The Cauchy Problem in Kinetic Theory,, Society for Industrial and Applied Mathematics (SIAM), (1996).  doi: 10.1137/1.9781611971477.  Google Scholar

[12]

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

[13]

L. Hsiao, F.-C. Li and S. Wang, Combined quasineutral and inviscid limit of the Vlasov-Poisson-Fokker-Planck system,, Commun. Pure Appl. Anal., 7 (2008), 579.  doi: 10.3934/cpaa.2008.7.579.  Google Scholar

[14]

H.-J Hwang and J. Jang, On the Vlasov-Poisson-Fokker-Planck equation near Maxwellian,, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 681.  doi: 10.3934/dcdsb.2013.18.681.  Google Scholar

[15]

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

[16]

R. Lai, On the one and one-half dimensional relativistic Vlasov-Maxwell-Fokker-Planck system with non-vanishing viscosity,, Math. Meth. Appl. Sci., 21 (1998), 1287.  doi: 10.1002/(SICI)1099-1476(19980925)21:14<1287::AID-MMA996>3.0.CO;2-G.  Google Scholar

[17]

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

[18]

B. O'Dwyer and H.-D. Victory Jr., On classical solutions of the Vlasov-Poisson-Fokker-Planck system,, Indiana Univ. Math. J., 39 (1990), 105.  doi: 10.1512/iumj.1990.39.39009.  Google Scholar

[19]

G. Rein and J. Weckler, Generic global classical solutions of the Vlasov-Fokker-Planck-Poisson system in three dimensions,, J. Differential Equations, 99 (1992), 59.  doi: 10.1016/0022-0396(92)90135-A.  Google Scholar

[20]

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

[21]

H.-D. Victory Jr., On the existence of global weak solutions for Vlasov-Poisson-Fokker-Planck systems,, J. Math. Anal. Appl., 160 (1991), 525.  doi: 10.1016/0022-247X(91)90324-S.  Google Scholar

[22]

C. Villani, Hypocoercivity,, Memoirs Amer. Math. Soc., 202 (2009).  doi: 10.1090/S0065-9266-09-00567-5.  Google Scholar

[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.  doi: 10.1016/j.jde.2009.11.027.  Google Scholar

[24]

T. Yang and H.-J. Yu, Global classical solutions for the Vlasov-Maxwell-Fokker-Planck system,, SIAM J. Math. Anal., 42 (2010), 459.  doi: 10.1137/090755796.  Google Scholar

[25]

T. Yang and H.-J. Zhao, Global existence of classical solutions to the Vlasov-Poisson-Boltzmann system,, Comm. Math. Phys., 268 (2006), 569.  doi: 10.1007/s00220-006-0103-4.  Google Scholar

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