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A framework for hyperbolic approximation of kinetic equations using quadrature-based projection methods
One-species Vlasov-Poisson-Landau system near Maxwellians in the whole space
1. | School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China |
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. |
show all references
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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