# 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 and Related Models, 2014, 7 (3) : 551-590. doi: 10.3934/krm.2014.7.551
##### References:
 [1] R. Adams, Sobolev Spaces, Pure and Applied Mathematics, Academic Press, New York-London, 1975. [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), 43-100. doi: 10.1081/PDE-100002246. [3] R. Alexandre and C. Villani, On the Landau approximation in plasma physics, Ann. Inst. H. Poincare Anal. Non Lineaire, 21 (2004), 61-95. doi: 10.1016/S0294-1449(03)00030-1. [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), 465-478. [5] P. Degond and M. Lemou, Dispersion relations for the linearized Fokker-Planck equation, Arch. Ration. Mech. Anal., 138 (1997), 137-167. doi: 10.1007/s002050050038. [6] R.-J. Duan and S.-Q. Liu, The Vlasov-Poisson-Boltzmann system without angular cutoff, Comm. Math. Phys., 324 (2013), 1-45. doi: 10.1007/s00220-013-1807-x. [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), 189-236. doi: 10.1007/s00220-007-0366-4. [8] R.-J. Duan and R. M. Strain, Optimal time decay of the Vlasov-Poisson-Boltzmann system in $\mathbb{R}^3$, Arch. Ration. Mech. Anal., 199 (2011), 291-328. doi: 10.1007/s00205-010-0318-6. [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), 1497-1546. doi: 10.1002/cpa.20381. [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.-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), 979-1028. doi: 10.1142/S0218202513500012. [12] R.-J. Duan, T. Yang and H.-J. Zhao, Global solutions to the Vlasov-Poisson-Landau system, preprint, arXiv:1112.3261, 2011. [13] Y. Guo, The Vlasov-Poisson-Boltzmann system near Maxwellians, Comm. Pure Appl. Math., 55 (2002), 1104-1135. doi: 10.1002/cpa.10040. [14] Y. Guo, The Landau equation in a periodic box, Comm. Math. Phys., 231 (2002), 391-434. doi: 10.1007/s00220-002-0729-9. [15] Y. Guo, The Boltzmann equation in the whole space, Indiana Univ. Math. J., 53 (2004), 1081-1094. doi: 10.1512/iumj.2004.53.2574. [16] Y. Guo, The Vlasov-Poisson-Laudau system in a periodic box, J. Amer. Math. Soc., 25 (2012), 759-812. doi: 10.1090/S0894-0347-2011-00722-4. [17] Y. Guo and Y.-J. Wang, Decay of dissipative equation and negative sobolev spaces, Comm. Partial Differential Equations, 37 (2012), 2165-2208. doi: 10.1080/03605302.2012.696296. [18] C. He and Y.-J. Lei, Besov spaces and one-species Vlasov-Poisson-Landau system in the whole space, preprint, 2014. [19] F. Hilton, Collisional transport in plasma, in Handbook of Plasma Physics, Volume I: Basic Plasma Physics I (eds. M. N. Rosenbluth and R. Z. Sagdeev), North-Holland Publishing Company, 1983, pp. 147. [20] N. A. Krall and A. W. Trivelpiece, Principles of Plasma Physics, McGraw-Hill, 1973. [21] M. S. Elias, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. [22] R. M. Strain, Optimal time decay of the non cut-off Boltzmann equation in the whole space, Kinetic and Related Models, 5 (2012), 583-613. doi: 10.3934/krm.2012.5.583. [23] 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. [24] R. M. Strain and Y. Guo, Almost exponential decay near Maxwellian, Comm. Partial Differential Equations, 31 (2006), 417-429. doi: 10.1080/03605300500361545. [25] R. M. Strain and Y. Guo, Exponential decay for soft potentials near Maxwellian, Arch. Ration. Mech. Anal., 187 (2008), 287-339. doi: 10.1007/s00205-007-0067-3. [26] R. M. Strain and K.-Y. Zhu, The Vlasov-Poisson-Landau system in $\mathbbR^3_x$, Arch. Ration. Mech. Anal., 210 (2013), 615-671. doi: 10.1007/s00205-013-0658-0. [27] C. Villani, On the Cauchy problem for Landau equation: Sequential stability, global existence, Adv. Diff. Eq., 1 (1996), 793-816. [28] C. Villani, A review of mathematical topics in collisional kinetic theory, in Handbook of mathematical fluid dynamics, I, North-Holland, Amsterdam, 2002, 71-305. doi: 10.1016/S1874-5792(02)80004-0. [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), 3281-3323. doi: 10.1137/120879129. [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), 1196-1232. doi: 10.1016/j.jde.2013.05.005. [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), 515-540. doi: 10.1007/s11425-013-4712-z.

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##### References:
 [1] R. Adams, Sobolev Spaces, Pure and Applied Mathematics, Academic Press, New York-London, 1975. [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), 43-100. doi: 10.1081/PDE-100002246. [3] R. Alexandre and C. Villani, On the Landau approximation in plasma physics, Ann. Inst. H. Poincare Anal. Non Lineaire, 21 (2004), 61-95. doi: 10.1016/S0294-1449(03)00030-1. [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), 465-478. [5] P. Degond and M. Lemou, Dispersion relations for the linearized Fokker-Planck equation, Arch. Ration. Mech. Anal., 138 (1997), 137-167. doi: 10.1007/s002050050038. [6] R.-J. Duan and S.-Q. Liu, The Vlasov-Poisson-Boltzmann system without angular cutoff, Comm. Math. Phys., 324 (2013), 1-45. doi: 10.1007/s00220-013-1807-x. [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), 189-236. doi: 10.1007/s00220-007-0366-4. [8] R.-J. Duan and R. M. Strain, Optimal time decay of the Vlasov-Poisson-Boltzmann system in $\mathbb{R}^3$, Arch. Ration. Mech. Anal., 199 (2011), 291-328. doi: 10.1007/s00205-010-0318-6. [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), 1497-1546. doi: 10.1002/cpa.20381. [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.-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), 979-1028. doi: 10.1142/S0218202513500012. [12] R.-J. Duan, T. Yang and H.-J. Zhao, Global solutions to the Vlasov-Poisson-Landau system, preprint, arXiv:1112.3261, 2011. [13] Y. Guo, The Vlasov-Poisson-Boltzmann system near Maxwellians, Comm. Pure Appl. Math., 55 (2002), 1104-1135. doi: 10.1002/cpa.10040. [14] Y. Guo, The Landau equation in a periodic box, Comm. Math. Phys., 231 (2002), 391-434. doi: 10.1007/s00220-002-0729-9. [15] Y. Guo, The Boltzmann equation in the whole space, Indiana Univ. Math. J., 53 (2004), 1081-1094. doi: 10.1512/iumj.2004.53.2574. [16] Y. Guo, The Vlasov-Poisson-Laudau system in a periodic box, J. Amer. Math. Soc., 25 (2012), 759-812. doi: 10.1090/S0894-0347-2011-00722-4. [17] Y. Guo and Y.-J. Wang, Decay of dissipative equation and negative sobolev spaces, Comm. Partial Differential Equations, 37 (2012), 2165-2208. doi: 10.1080/03605302.2012.696296. [18] C. He and Y.-J. Lei, Besov spaces and one-species Vlasov-Poisson-Landau system in the whole space, preprint, 2014. [19] F. Hilton, Collisional transport in plasma, in Handbook of Plasma Physics, Volume I: Basic Plasma Physics I (eds. M. N. Rosenbluth and R. Z. Sagdeev), North-Holland Publishing Company, 1983, pp. 147. [20] N. A. Krall and A. W. Trivelpiece, Principles of Plasma Physics, McGraw-Hill, 1973. [21] M. S. Elias, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. [22] R. M. Strain, Optimal time decay of the non cut-off Boltzmann equation in the whole space, Kinetic and Related Models, 5 (2012), 583-613. doi: 10.3934/krm.2012.5.583. [23] 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. [24] R. M. Strain and Y. Guo, Almost exponential decay near Maxwellian, Comm. Partial Differential Equations, 31 (2006), 417-429. doi: 10.1080/03605300500361545. [25] R. M. Strain and Y. Guo, Exponential decay for soft potentials near Maxwellian, Arch. Ration. Mech. Anal., 187 (2008), 287-339. doi: 10.1007/s00205-007-0067-3. [26] R. M. Strain and K.-Y. Zhu, The Vlasov-Poisson-Landau system in $\mathbbR^3_x$, Arch. Ration. Mech. Anal., 210 (2013), 615-671. doi: 10.1007/s00205-013-0658-0. [27] C. Villani, On the Cauchy problem for Landau equation: Sequential stability, global existence, Adv. Diff. Eq., 1 (1996), 793-816. [28] C. Villani, A review of mathematical topics in collisional kinetic theory, in Handbook of mathematical fluid dynamics, I, North-Holland, Amsterdam, 2002, 71-305. doi: 10.1016/S1874-5792(02)80004-0. [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), 3281-3323. doi: 10.1137/120879129. [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), 1196-1232. doi: 10.1016/j.jde.2013.05.005. [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), 515-540. doi: 10.1007/s11425-013-4712-z.
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