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Global solutions to the relativistic Vlasov-Poisson-Fokker-Planck system
1. | School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, China |
2. | School of Mathematical Sciences, South China Normal University, Guangzhou 510631 |
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), 139-170.
doi: 10.3934/krm.2008.1.139. |
[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), 239-258.
doi: 10.1006/jfan.1993.1011. |
[3] |
A. Carpio, Long-time behaviour for solutions of the Vlasov-Poisson-Fokker-Planck equation, Math. Methods Appl. Sci., 21 (1998), 985-1014.
doi: 10.1002/(SICI)1099-1476(19980725)21:11<985::AID-MMA919>3.0.CO;2-B. |
[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), 825-839.
doi: 10.1002/mma.1670181006. |
[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), 99-132.
doi: 10.1006/jfan.1996.0123. |
[6] |
M. Chae, The global classical solution of the Vlasov-Maxwell-Fokker-Planck system near Maxwellian, Math. Models Methods Appl. Sci., 21 (2011), 1007-1025.
doi: 10.1142/S0218202511005222. |
[7] |
R. DiPerna and P.-L. Lions, Global weak solutions of kinetic equations, Rend. Sem. Mat. Univ. Politec. Torino, 46 (1988), 259-288. |
[8] |
R.-J. Duan and S.-Q. Liu, The Vlasov-Poisson-Boltzmann system without angular cutoff, Comm. Math. Phys., 324 (2013), 1-45. |
[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), 291-328.
doi: 10.1007/s00205-010-0318-6. |
[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), 6356-6386.
doi: 10.1016/j.jde.2012.03.012. |
[11] |
R. T. Glassey, The Cauchy Problem in Kinetic Theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996.
doi: 10.1137/1.9781611971477. |
[12] |
Y. Guo, The Vlasov-Poisson-Boltzmann system near Maxwellians, Comm. Pure Appl. Math., 55 (2002), 1104-1135.
doi: 10.1002/cpa.10040. |
[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), 579-589.
doi: 10.3934/cpaa.2008.7.579. |
[14] |
H.-J Hwang and J. Jang, On the Vlasov-Poisson-Fokker-Planck equation near Maxwellian, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 681-691.
doi: 10.3934/dcdsb.2013.18.681. |
[15] |
S. Kawashima, The Boltzmann equation and thirteen moments, Japan J. Appl. Math., 7 (1990), 301-320.
doi: 10.1007/BF03167846. |
[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), 1287-1296.
doi: 10.1002/(SICI)1099-1476(19980925)21:14<1287::AID-MMA996>3.0.CO;2-G. |
[17] |
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. |
[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), 105-156.
doi: 10.1512/iumj.1990.39.39009. |
[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), 59-77.
doi: 10.1016/0022-0396(92)90135-A. |
[20] |
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. |
[21] |
H.-D. Victory Jr., On the existence of global weak solutions for Vlasov-Poisson-Fokker-Planck systems, J. Math. Anal. Appl., 160 (1991), 525-555.
doi: 10.1016/0022-247X(91)90324-S. |
[22] |
C. Villani, Hypocoercivity, Memoirs Amer. Math. Soc., 202 (2009), iv+141 pp.
doi: 10.1090/S0065-9266-09-00567-5. |
[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 classical solutions for the Vlasov-Maxwell-Fokker-Planck system, SIAM J. Math. Anal., 42 (2010), 459-488.
doi: 10.1137/090755796. |
[25] |
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] |
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), 139-170.
doi: 10.3934/krm.2008.1.139. |
[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), 239-258.
doi: 10.1006/jfan.1993.1011. |
[3] |
A. Carpio, Long-time behaviour for solutions of the Vlasov-Poisson-Fokker-Planck equation, Math. Methods Appl. Sci., 21 (1998), 985-1014.
doi: 10.1002/(SICI)1099-1476(19980725)21:11<985::AID-MMA919>3.0.CO;2-B. |
[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), 825-839.
doi: 10.1002/mma.1670181006. |
[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), 99-132.
doi: 10.1006/jfan.1996.0123. |
[6] |
M. Chae, The global classical solution of the Vlasov-Maxwell-Fokker-Planck system near Maxwellian, Math. Models Methods Appl. Sci., 21 (2011), 1007-1025.
doi: 10.1142/S0218202511005222. |
[7] |
R. DiPerna and P.-L. Lions, Global weak solutions of kinetic equations, Rend. Sem. Mat. Univ. Politec. Torino, 46 (1988), 259-288. |
[8] |
R.-J. Duan and S.-Q. Liu, The Vlasov-Poisson-Boltzmann system without angular cutoff, Comm. Math. Phys., 324 (2013), 1-45. |
[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), 291-328.
doi: 10.1007/s00205-010-0318-6. |
[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), 6356-6386.
doi: 10.1016/j.jde.2012.03.012. |
[11] |
R. T. Glassey, The Cauchy Problem in Kinetic Theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996.
doi: 10.1137/1.9781611971477. |
[12] |
Y. Guo, The Vlasov-Poisson-Boltzmann system near Maxwellians, Comm. Pure Appl. Math., 55 (2002), 1104-1135.
doi: 10.1002/cpa.10040. |
[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), 579-589.
doi: 10.3934/cpaa.2008.7.579. |
[14] |
H.-J Hwang and J. Jang, On the Vlasov-Poisson-Fokker-Planck equation near Maxwellian, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 681-691.
doi: 10.3934/dcdsb.2013.18.681. |
[15] |
S. Kawashima, The Boltzmann equation and thirteen moments, Japan J. Appl. Math., 7 (1990), 301-320.
doi: 10.1007/BF03167846. |
[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), 1287-1296.
doi: 10.1002/(SICI)1099-1476(19980925)21:14<1287::AID-MMA996>3.0.CO;2-G. |
[17] |
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. |
[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), 105-156.
doi: 10.1512/iumj.1990.39.39009. |
[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), 59-77.
doi: 10.1016/0022-0396(92)90135-A. |
[20] |
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. |
[21] |
H.-D. Victory Jr., On the existence of global weak solutions for Vlasov-Poisson-Fokker-Planck systems, J. Math. Anal. Appl., 160 (1991), 525-555.
doi: 10.1016/0022-247X(91)90324-S. |
[22] |
C. Villani, Hypocoercivity, Memoirs Amer. Math. Soc., 202 (2009), iv+141 pp.
doi: 10.1090/S0065-9266-09-00567-5. |
[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 classical solutions for the Vlasov-Maxwell-Fokker-Planck system, SIAM J. Math. Anal., 42 (2010), 459-488.
doi: 10.1137/090755796. |
[25] |
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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