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

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

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.
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

[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

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

[1]

Ling Hsiao, Fucai Li, Shu Wang. Combined quasineutral and inviscid limit of the Vlasov-Poisson-Fokker-Planck system. Communications on Pure & Applied Analysis, 2008, 7 (3) : 579-589. doi: 10.3934/cpaa.2008.7.579

[2]

Kosuke Ono, Walter A. Strauss. Regular solutions of the Vlasov-Poisson-Fokker-Planck system. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 751-772. doi: 10.3934/dcds.2000.6.751

[3]

Maxime Hauray, Samir Salem. Propagation of chaos for the Vlasov-Poisson-Fokker-Planck system in 1D. Kinetic & Related Models, 2019, 12 (2) : 269-302. doi: 10.3934/krm.2019012

[4]

Hyung Ju Hwang, Juhi Jang. On the Vlasov-Poisson-Fokker-Planck equation near Maxwellian. Discrete & Continuous Dynamical Systems - B, 2013, 18 (3) : 681-691. doi: 10.3934/dcdsb.2013.18.681

[5]

Stephen Pankavich, Nicholas Michalowski. Global classical solutions for the "One and one-half'' dimensional relativistic Vlasov-Maxwell-Fokker-Planck system. Kinetic & Related Models, 2015, 8 (1) : 169-199. doi: 10.3934/krm.2015.8.169

[6]

Mihai Bostan, Thierry Goudon. Low field regime for the relativistic Vlasov-Maxwell-Fokker-Planck system; the one and one half dimensional case. Kinetic & Related Models, 2008, 1 (1) : 139-170. doi: 10.3934/krm.2008.1.139

[7]

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

[8]

José A. Carrillo, Renjun Duan, Ayman Moussa. Global classical solutions close to equilibrium to the Vlasov-Fokker-Planck-Euler system. Kinetic & Related Models, 2011, 4 (1) : 227-258. doi: 10.3934/krm.2011.4.227

[9]

Linjie Xiong, Tao Wang, Lusheng Wang. Global existence and decay of solutions to the Fokker-Planck-Boltzmann equation. Kinetic & Related Models, 2014, 7 (1) : 169-194. doi: 10.3934/krm.2014.7.169

[10]

Xiuting Li. The energy conservation for weak solutions to the relativistic Nordström-Vlasov system. Evolution Equations & Control Theory, 2016, 5 (1) : 135-145. doi: 10.3934/eect.2016.5.135

[11]

José Antonio Alcántara, Simone Calogero. On a relativistic Fokker-Planck equation in kinetic theory. Kinetic & Related Models, 2011, 4 (2) : 401-426. doi: 10.3934/krm.2011.4.401

[12]

Bingkang Huang, Lan Zhang. A global existence of classical solutions to the two-dimensional Vlasov-Fokker-Planck and magnetohydrodynamics equations with large initial data. Kinetic & Related Models, 2019, 12 (2) : 357-396. doi: 10.3934/krm.2019016

[13]

Simone Calogero, Stephen Pankavich. On the spatially homogeneous and isotropic Einstein-Vlasov-Fokker-Planck system with cosmological scalar field. Kinetic & Related Models, 2018, 11 (5) : 1063-1083. doi: 10.3934/krm.2018041

[14]

Jack Schaeffer. Global existence for the Vlasov-Poisson system with steady spatial asymptotic behavior. Kinetic & Related Models, 2012, 5 (1) : 129-153. doi: 10.3934/krm.2012.5.129

[15]

Meixia Xiao, Xianwen Zhang. On global solutions to the Vlasov-Poisson system with radiation damping. Kinetic & Related Models, 2018, 11 (5) : 1183-1209. doi: 10.3934/krm.2018046

[16]

Jonathan Zinsl. Exponential convergence to equilibrium in a Poisson-Nernst-Planck-type system with nonlinear diffusion. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2915-2930. doi: 10.3934/dcds.2016.36.2915

[17]

Hyung Ju Hwang, Jaewoo Jung, Juan J. L. Velázquez. On global existence of classical solutions for the Vlasov-Poisson system in convex bounded domains. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 723-737. doi: 10.3934/dcds.2013.33.723

[18]

Renjun Duan, Shuangqian Liu. Cauchy problem on the Vlasov-Fokker-Planck equation coupled with the compressible Euler equations through the friction force. Kinetic & Related Models, 2013, 6 (4) : 687-700. doi: 10.3934/krm.2013.6.687

[19]

Maxime Herda, Luis Miguel Rodrigues. Anisotropic Boltzmann-Gibbs dynamics of strongly magnetized Vlasov-Fokker-Planck equations. Kinetic & Related Models, 2019, 12 (3) : 593-636. doi: 10.3934/krm.2019024

[20]

Katherine Zhiyuan Zhang. Focusing solutions of the Vlasov-Poisson system. Kinetic & Related Models, 2019, 12 (6) : 1313-1327. doi: 10.3934/krm.2019051

2018 Impact Factor: 1.38

Metrics

  • PDF downloads (16)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]