September  2014, 7(3): 551-590. doi: 10.3934/krm.2014.7.551

One-species Vlasov-Poisson-Landau system near Maxwellians in the whole space

1. 

School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

Received  May 2014 Revised  May 2014 Published  July 2014

The classical one-species Vlasov-Poisson-Landau system describes dynamics of electrons interacting with its self-consistent electrostatic field as well as its grazing collisions modeled by the famous Landau (Fokker-Planck) collision kernel. We show in this manuscript that the Cauchy problem for the one-species Vlasov-Poisson-Landau system which includes the Coulomb potential admits a unique global solution near a given global Maxwellian in the whole space $\mathbb{R}^3_x$ provided that the initial perturbation satisfies certain regularity and smallness conditions. Compared with that of [12], which, to the best of our knowledge, is the only result concerning the one-species Vlasov-Poisson-Landau system available up to now, we do not ask the initial perturbation to satisfy the neutral condition and the minimal regularity assumption we imposed on the initial perturbation is also weaker.
Citation: Yuanjie Lei, Linjie Xiong, Huijiang Zhao. One-species Vlasov-Poisson-Landau system near Maxwellians in the whole space. Kinetic & Related Models, 2014, 7 (3) : 551-590. doi: 10.3934/krm.2014.7.551
References:
[1]

R. Adams, Sobolev Spaces,, Pure and Applied Mathematics, (1975). Google Scholar

[2]

A. Arnold, P. Markowich, G. Toscani and A. Unterreiter, On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations,, Comm. Partial Differential Equations, 26 (2001), 43. doi: 10.1081/PDE-100002246. Google Scholar

[3]

R. Alexandre and C. Villani, On the Landau approximation in plasma physics,, Ann. Inst. H. Poincare Anal. Non Lineaire, 21 (2004), 61. doi: 10.1016/S0294-1449(03)00030-1. Google Scholar

[4]

A. A. Arsenev and O. E. Buryak, On the connection between a solution of the Boltzmann equation and a solution of the Landau-Fokker-Planck equation,, Math. USSR. Sbornik, 69 (1991), 465. Google Scholar

[5]

P. Degond and M. Lemou, Dispersion relations for the linearized Fokker-Planck equation,, Arch. Ration. Mech. Anal., 138 (1997), 137. doi: 10.1007/s002050050038. Google Scholar

[6]

R.-J. Duan and S.-Q. Liu, The Vlasov-Poisson-Boltzmann system without angular cutoff,, Comm. Math. Phys., 324 (2013), 1. doi: 10.1007/s00220-013-1807-x. Google Scholar

[7]

R.-J. Duan, S. Ukai, T. Yang and H.-J. Zhao, Optimal decay estimates on the linearized Boltzmann equation with time dependent force and their applications,, Comm. Math. Phys., 277 (2008), 189. doi: 10.1007/s00220-007-0366-4. Google Scholar

[8]

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

[9]

R.-J. Duan and R. M. Strain, Optimal large-time behavior of the Vlasov-Maxwell-Boltzmann system in the whole space,, Comm. Pure. Appl. Math., 64 (2011), 1497. doi: 10.1002/cpa.20381. 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.-J. Duan, T. Yang and H.-J. Zhao, The Vlasov-Poisson-Boltzmann system for soft potentials,, Mathematical Models and Methods in Applied Sciences, 23 (2013), 979. doi: 10.1142/S0218202513500012. Google Scholar

[12]

R.-J. Duan, T. Yang and H.-J. Zhao, Global solutions to the Vlasov-Poisson-Landau system,, preprint, (2011). Google Scholar

[13]

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

[14]

Y. Guo, The Landau equation in a periodic box,, Comm. Math. Phys., 231 (2002), 391. doi: 10.1007/s00220-002-0729-9. Google Scholar

[15]

Y. Guo, The Boltzmann equation in the whole space,, Indiana Univ. Math. J., 53 (2004), 1081. doi: 10.1512/iumj.2004.53.2574. Google Scholar

[16]

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

[17]

Y. Guo and Y.-J. Wang, Decay of dissipative equation and negative sobolev spaces,, Comm. Partial Differential Equations, 37 (2012), 2165. doi: 10.1080/03605302.2012.696296. Google Scholar

[18]

C. He and Y.-J. Lei, Besov spaces and one-species Vlasov-Poisson-Landau system in the whole space,, preprint, (2014). Google Scholar

[19]

F. Hilton, Collisional transport in plasma,, in Handbook of Plasma Physics, (1983). Google Scholar

[20]

N. A. Krall and A. W. Trivelpiece, Principles of Plasma Physics,, McGraw-Hill, (1973). Google Scholar

[21]

M. S. Elias, Singular Integrals and Differentiability Properties of Functions,, Princeton Mathematical Series, (1970). Google Scholar

[22]

R. M. Strain, Optimal time decay of the non cut-off Boltzmann equation in the whole space,, Kinetic and Related Models, 5 (2012), 583. doi: 10.3934/krm.2012.5.583. Google Scholar

[23]

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

[24]

R. M. Strain and Y. Guo, Almost exponential decay near Maxwellian,, Comm. Partial Differential Equations, 31 (2006), 417. doi: 10.1080/03605300500361545. Google Scholar

[25]

R. M. Strain and Y. Guo, Exponential decay for soft potentials near Maxwellian,, Arch. Ration. Mech. Anal., 187 (2008), 287. doi: 10.1007/s00205-007-0067-3. Google Scholar

[26]

R. M. Strain and K.-Y. Zhu, The Vlasov-Poisson-Landau system in $\mathbbR^3_x$,, Arch. Ration. Mech. Anal., 210 (2013), 615. doi: 10.1007/s00205-013-0658-0. Google Scholar

[27]

C. Villani, On the Cauchy problem for Landau equation: Sequential stability, global existence,, Adv. Diff. Eq., 1 (1996), 793. Google Scholar

[28]

C. Villani, A review of mathematical topics in collisional kinetic theory,, in Handbook of mathematical fluid dynamics, (2002), 71. doi: 10.1016/S1874-5792(02)80004-0. Google Scholar

[29]

Y.-J. Wang, Golobal solution and time decay of the Vlasov-Poisson-Landau System in $\mathbbR^3_x$,, SIAM J. Math. Anal., 44 (2012), 3281. doi: 10.1137/120879129. Google Scholar

[30]

Q.-H. Xiao, L.-J. Xiong and H.-J. Zhao, The Vlasov-Posson-Boltzmann system with angular cutoff for soft potential,, J. Differential Equations, 255 (2013), 1196. doi: 10.1016/j.jde.2013.05.005. Google Scholar

[31]

Q.-H. Xiao, L.-J. Xiong and H.-J. Zhao, The Vlasov-Poisson-Boltzmann system for non-cutoff hard potentials,, Science China Mathematics, 57 (2014), 515. doi: 10.1007/s11425-013-4712-z. Google Scholar

show all references

References:
[1]

R. Adams, Sobolev Spaces,, Pure and Applied Mathematics, (1975). Google Scholar

[2]

A. Arnold, P. Markowich, G. Toscani and A. Unterreiter, On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker-Planck type equations,, Comm. Partial Differential Equations, 26 (2001), 43. doi: 10.1081/PDE-100002246. Google Scholar

[3]

R. Alexandre and C. Villani, On the Landau approximation in plasma physics,, Ann. Inst. H. Poincare Anal. Non Lineaire, 21 (2004), 61. doi: 10.1016/S0294-1449(03)00030-1. Google Scholar

[4]

A. A. Arsenev and O. E. Buryak, On the connection between a solution of the Boltzmann equation and a solution of the Landau-Fokker-Planck equation,, Math. USSR. Sbornik, 69 (1991), 465. Google Scholar

[5]

P. Degond and M. Lemou, Dispersion relations for the linearized Fokker-Planck equation,, Arch. Ration. Mech. Anal., 138 (1997), 137. doi: 10.1007/s002050050038. Google Scholar

[6]

R.-J. Duan and S.-Q. Liu, The Vlasov-Poisson-Boltzmann system without angular cutoff,, Comm. Math. Phys., 324 (2013), 1. doi: 10.1007/s00220-013-1807-x. Google Scholar

[7]

R.-J. Duan, S. Ukai, T. Yang and H.-J. Zhao, Optimal decay estimates on the linearized Boltzmann equation with time dependent force and their applications,, Comm. Math. Phys., 277 (2008), 189. doi: 10.1007/s00220-007-0366-4. Google Scholar

[8]

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

[9]

R.-J. Duan and R. M. Strain, Optimal large-time behavior of the Vlasov-Maxwell-Boltzmann system in the whole space,, Comm. Pure. Appl. Math., 64 (2011), 1497. doi: 10.1002/cpa.20381. 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.-J. Duan, T. Yang and H.-J. Zhao, The Vlasov-Poisson-Boltzmann system for soft potentials,, Mathematical Models and Methods in Applied Sciences, 23 (2013), 979. doi: 10.1142/S0218202513500012. Google Scholar

[12]

R.-J. Duan, T. Yang and H.-J. Zhao, Global solutions to the Vlasov-Poisson-Landau system,, preprint, (2011). Google Scholar

[13]

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

[14]

Y. Guo, The Landau equation in a periodic box,, Comm. Math. Phys., 231 (2002), 391. doi: 10.1007/s00220-002-0729-9. Google Scholar

[15]

Y. Guo, The Boltzmann equation in the whole space,, Indiana Univ. Math. J., 53 (2004), 1081. doi: 10.1512/iumj.2004.53.2574. Google Scholar

[16]

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

[17]

Y. Guo and Y.-J. Wang, Decay of dissipative equation and negative sobolev spaces,, Comm. Partial Differential Equations, 37 (2012), 2165. doi: 10.1080/03605302.2012.696296. Google Scholar

[18]

C. He and Y.-J. Lei, Besov spaces and one-species Vlasov-Poisson-Landau system in the whole space,, preprint, (2014). Google Scholar

[19]

F. Hilton, Collisional transport in plasma,, in Handbook of Plasma Physics, (1983). Google Scholar

[20]

N. A. Krall and A. W. Trivelpiece, Principles of Plasma Physics,, McGraw-Hill, (1973). Google Scholar

[21]

M. S. Elias, Singular Integrals and Differentiability Properties of Functions,, Princeton Mathematical Series, (1970). Google Scholar

[22]

R. M. Strain, Optimal time decay of the non cut-off Boltzmann equation in the whole space,, Kinetic and Related Models, 5 (2012), 583. doi: 10.3934/krm.2012.5.583. Google Scholar

[23]

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

[24]

R. M. Strain and Y. Guo, Almost exponential decay near Maxwellian,, Comm. Partial Differential Equations, 31 (2006), 417. doi: 10.1080/03605300500361545. Google Scholar

[25]

R. M. Strain and Y. Guo, Exponential decay for soft potentials near Maxwellian,, Arch. Ration. Mech. Anal., 187 (2008), 287. doi: 10.1007/s00205-007-0067-3. Google Scholar

[26]

R. M. Strain and K.-Y. Zhu, The Vlasov-Poisson-Landau system in $\mathbbR^3_x$,, Arch. Ration. Mech. Anal., 210 (2013), 615. doi: 10.1007/s00205-013-0658-0. Google Scholar

[27]

C. Villani, On the Cauchy problem for Landau equation: Sequential stability, global existence,, Adv. Diff. Eq., 1 (1996), 793. Google Scholar

[28]

C. Villani, A review of mathematical topics in collisional kinetic theory,, in Handbook of mathematical fluid dynamics, (2002), 71. doi: 10.1016/S1874-5792(02)80004-0. Google Scholar

[29]

Y.-J. Wang, Golobal solution and time decay of the Vlasov-Poisson-Landau System in $\mathbbR^3_x$,, SIAM J. Math. Anal., 44 (2012), 3281. doi: 10.1137/120879129. Google Scholar

[30]

Q.-H. Xiao, L.-J. Xiong and H.-J. Zhao, The Vlasov-Posson-Boltzmann system with angular cutoff for soft potential,, J. Differential Equations, 255 (2013), 1196. doi: 10.1016/j.jde.2013.05.005. Google Scholar

[31]

Q.-H. Xiao, L.-J. Xiong and H.-J. Zhao, The Vlasov-Poisson-Boltzmann system for non-cutoff hard potentials,, Science China Mathematics, 57 (2014), 515. doi: 10.1007/s11425-013-4712-z. Google Scholar

[1]

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

[2]

Yemin Chen. Smoothness of classical solutions to the Vlasov-Poisson-Landau system. Kinetic & Related Models, 2008, 1 (3) : 369-386. doi: 10.3934/krm.2008.1.369

[3]

Tariel Sanikidze, A.F. Tedeev. On the temporal decay estimates for the degenerate parabolic system. Communications on Pure & Applied Analysis, 2013, 12 (4) : 1755-1768. doi: 10.3934/cpaa.2013.12.1755

[4]

Blanca Ayuso, José A. Carrillo, Chi-Wang Shu. Discontinuous Galerkin methods for the one-dimensional Vlasov-Poisson system. Kinetic & Related Models, 2011, 4 (4) : 955-989. doi: 10.3934/krm.2011.4.955

[5]

Irene M. Gamba, Maria Pia Gualdani, Christof Sparber. A note on the time decay of solutions for the linearized Wigner-Poisson system. Kinetic & Related Models, 2009, 2 (1) : 181-189. doi: 10.3934/krm.2009.2.181

[6]

Paul Deuring, Stanislav Kračmar, Šárka Nečasová. A linearized system describing stationary incompressible viscous flow around rotating and translating bodies: Improved decay estimates of the velocity and its gradient. Conference Publications, 2011, 2011 (Special) : 351-361. doi: 10.3934/proc.2011.2011.351

[7]

Yemin Chen. Smoothness of classical solutions to the Vlasov-Maxwell-Landau system near Maxwellians. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 889-910. doi: 10.3934/dcds.2008.20.889

[8]

Xiaopeng Zhao. Space-time decay estimates of solutions to liquid crystal system in $\mathbb{R}^3$. Communications on Pure & Applied Analysis, 2019, 18 (1) : 1-13. doi: 10.3934/cpaa.2019001

[9]

Silvia Caprino, Guido Cavallaro, Carlo Marchioro. Time evolution of a Vlasov-Poisson plasma with magnetic confinement. Kinetic & Related Models, 2012, 5 (4) : 729-742. doi: 10.3934/krm.2012.5.729

[10]

Wenjia Jing, Panagiotis E. Souganidis, Hung V. Tran. Large time average of reachable sets and Applications to Homogenization of interfaces moving with oscillatory spatio-temporal velocity. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : 915-939. doi: 10.3934/dcdss.2018055

[11]

Robert T. Glassey, Walter A. Strauss. Perturbation of essential spectra of evolution operators and the Vlasov-Poisson-Boltzmann system. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 457-472. doi: 10.3934/dcds.1999.5.457

[12]

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

[13]

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

[14]

Gianluca Crippa, Silvia Ligabue, Chiara Saffirio. Lagrangian solutions to the Vlasov-Poisson system with a point charge. Kinetic & Related Models, 2018, 11 (6) : 1277-1299. doi: 10.3934/krm.2018050

[15]

Zili Chen, Xiuting Li, Xianwen Zhang. The two dimensional Vlasov-Poisson system with steady spatial asymptotics. Kinetic & Related Models, 2017, 10 (4) : 977-1009. doi: 10.3934/krm.2017039

[16]

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

[17]

Jean Dolbeault. An introduction to kinetic equations: the Vlasov-Poisson system and the Boltzmann equation. Discrete & Continuous Dynamical Systems - A, 2002, 8 (2) : 361-380. doi: 10.3934/dcds.2002.8.361

[18]

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

[19]

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

[20]

Cong He, Hongjun Yu. Large time behavior of the solution to the Landau Equation with specular reflective boundary condition. Kinetic & Related Models, 2013, 6 (3) : 601-623. doi: 10.3934/krm.2013.6.601

2018 Impact Factor: 1.38

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]