# American Institute of Mathematical Sciences

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

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##### 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
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