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Diffusion asymptotics of a kinetic model for gaseous mixtures
Stability of the nonrelativistic Vlasov-Maxwell-Boltzmann system for angular non-cutoff potentials
1. | Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong |
2. | Department of Mathematics, Jinan Unviersity, Guangdong, China |
3. | Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong |
4. | School of Mathematics and Statistics, Wuhan University, China |
References:
[1] |
R. Alexandre, L. Desvillettes, C. Villani and B. Wennberg, Entropy dissipation and long-range interactions, Arch. Ration. Mech. Anal., 152 (2000), 327-355.
doi: 10.1007/s002050000083. |
[2] |
R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, Regularizing effect and local existence for non-cutoff Boltzmann equation, Arch. Ration. Mech. Anal., 198 (2010), 39-123.
doi: 10.1007/s00205-010-0290-1. |
[3] |
R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, Global existence and full regularity of the Boltzmann equation without angular cutoff, Comm. Math. Phys., 304 (2011), 513-581.
doi: 10.1007/s00220-011-1242-9. |
[4] |
R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, The Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential, J. Funct. Anal., 263 (2012), 915-1010.
doi: 10.1016/j.jfa.2011.10.007. |
[5] |
R. Alexandre and C. Villani, On the Landau approximation in plasma physics, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 61-95. |
[6] |
L. Desvillettes and C. Villani, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation, Invent. Math., 159 (2005), 245-316.
doi: 10.1007/s00222-004-0389-9. |
[7] |
R.-J. Duan, Global smooth flows for the compressible Euler-Maxwell system: Relaxation case, J. Hyperbolic Differ. Equ., 8 (2011), 375-413.
doi: 10.1142/S0219891611002421. |
[8] |
R.-J. Duan, Global smooth dynamics of a fully ionized plasma with long-range collisions, preprint, 2012. |
[9] |
R.-J. Duan and S.-Q. Liu, The Vlasov-Poisson-Boltzmann system without angular cutoff, preprint, 2012. |
[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, to appear in Math. Models Methods Appl. Sci. (2012), arXiv:1112.1453v1. |
[12] |
R.-J. Duan, T. Yang and H.-J. Zhao, Global solutions to the Vlasov-Poisson-Landau system, preprint 2011, arXiv:1112.3261v1. |
[13] |
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), 291-328.
doi: 10.1007/s00205-010-0318-6. |
[14] |
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. |
[15] |
H. Grad, On the kinetic theory of rarefied gases, Comm. Pure Appl. Math., 2 (1949), 331-407.
doi: 10.1002/cpa.3160020403. |
[16] |
H. Grad, Principles of the kinetic theory of gases, in "Handbuch der Physik," Springer-Berlin, XII (1958), 205-294. |
[17] |
P. T. Gressman and R. M. Strain, Global classical solutions of the Boltzmann equation without angular cut-off, J. Amer. Math. Soc., 24 (2011), 771-847.
doi: 10.1090/S0894-0347-2011-00697-8. |
[18] |
P. T. Gressman and R. M. Strain, Sharp anisotropic estimates for the Boltzmann collision operator and its entropy production, Adv. Math., 227 (2011), 2349-2384.
doi: 10.1016/j.aim.2011.05.005. |
[19] |
Y. Guo, The Landau equation in a periodic box, Comm. Math. Phys., 231 (2002), 391-434.
doi: 10.1007/s00220-002-0729-9. |
[20] |
Y. Guo, The Vlasov-Poisson-Landau system in a periodic box, J. Amer. Math. Soc., 25 (2012), 759-812.
doi: 10.1090/S0894-0347-2011-00722-4. |
[21] |
Y. Guo, The Vlasov-Maxwell-Boltzmann system near Maxwellians, Invent. Math., 153 (2003), 593-630.
doi: 10.1007/s00222-003-0301-z. |
[22] |
T. Hosono and S. Kawashima, Decay property of regularity-loss type and application to some nonlinear hyperbolic-elliptic system, Math. Models Methods Appl. Sci., 16 (2006), 1839-1859.
doi: 10.1142/S021820250600173X. |
[23] |
N. A. Krall and A. W. Trivelpiece, "Principles of Plasma Physics," McGraw-Hill, 1973. |
[24] |
P.-L. Lions, On Boltzmann and Landau equations, Philos. Trans. Roy. Soc. London Ser. A, 346 (1994), 191-204.
doi: 10.1098/rsta.1994.0018. |
[25] |
S.-Q. Liu and H.-J. Yu, Optimal time decay of the Landau equation with potential force, preprint, 2011. |
[26] |
T.-P. Liu and S.-H. Yu, Boltzmann equation: Micro-macro decompositions and positivity of shock profiles, Comm. Math. Phys., 246 (2004), 133-179.
doi: 10.1007/s00220-003-1030-2. |
[27] |
T.-P. Liu and S.-H. Yu, The Green's function and large-time behavior of solutions for the one-dimensional Boltzmann equation, Comm. Pure Appl. Math., 57 (2004), 1543-1608.
doi: 10.1002/cpa.20011. |
[28] |
C. Mouhot, Explicit coercivity estimates for the linearized Boltzmann and Landau operators, Comm. Partial Differential Equations, 31 (2006), 1321-1348.
doi: 10.1080/03605300600635004. |
[29] |
R. M. Strain, The Vlasov-Maxwell-Boltzmann system in the whole space, Comm. Math. Phys., 268 (2006), 543-567.
doi: 10.1007/s00220-006-0109-y. |
[30] |
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. |
[31] |
R. M. Strain and Y. Guo, Almost exponential decay near Maxwellian, Comm. Partial Differential Equations, 31 (2006), 417-429.
doi: 10.1080/03605300500361545. |
[32] |
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. |
[33] |
R. M. Strain and Y. Guo, Momentum regularity and stability of the relativistic Vlasov-Maxwell-Boltzmann system, Commun. Math. Phys., 10 (2012), 649-673.
doi: 10.1007/s00220-012-1417-z. |
[34] |
R. M. Strain and K. Zhu, The Vlasov-Poisson-Landau system in $\mathbb{R}^3$, preprint 2012, arXiv:1202.2471v1. |
[35] |
S. Ukai, On the existence of global solutions of mixed problem for non-linear Boltzmann equation, Proceedings of the Japan Academy, 50 (1974), 179-184.
doi: 10.3792/pja/1195519027. |
[36] |
C. Villani, A review of mathematical topics in collisional kinetic theory, North-Holland, Amsterdam, Handbook of mathematical fluid dynamics, I (2002), 71-305. |
[37] |
Y. Wang, Global solution and time decay of the Vlasov-Poisson-Landau system in $\mathbb{R}^3$, preprint 2012, arXiv:1205.6277. |
[38] |
M.-Q. Zhan, Local existence of solutions to the Landau-Maxwell system, Math. Methods Appl. Sci., 17 (1994), 613-641.
doi: 10.1002/mma.1670170804. |
show all references
References:
[1] |
R. Alexandre, L. Desvillettes, C. Villani and B. Wennberg, Entropy dissipation and long-range interactions, Arch. Ration. Mech. Anal., 152 (2000), 327-355.
doi: 10.1007/s002050000083. |
[2] |
R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, Regularizing effect and local existence for non-cutoff Boltzmann equation, Arch. Ration. Mech. Anal., 198 (2010), 39-123.
doi: 10.1007/s00205-010-0290-1. |
[3] |
R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, Global existence and full regularity of the Boltzmann equation without angular cutoff, Comm. Math. Phys., 304 (2011), 513-581.
doi: 10.1007/s00220-011-1242-9. |
[4] |
R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, The Boltzmann equation without angular cutoff in the whole space: I, Global existence for soft potential, J. Funct. Anal., 263 (2012), 915-1010.
doi: 10.1016/j.jfa.2011.10.007. |
[5] |
R. Alexandre and C. Villani, On the Landau approximation in plasma physics, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 61-95. |
[6] |
L. Desvillettes and C. Villani, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation, Invent. Math., 159 (2005), 245-316.
doi: 10.1007/s00222-004-0389-9. |
[7] |
R.-J. Duan, Global smooth flows for the compressible Euler-Maxwell system: Relaxation case, J. Hyperbolic Differ. Equ., 8 (2011), 375-413.
doi: 10.1142/S0219891611002421. |
[8] |
R.-J. Duan, Global smooth dynamics of a fully ionized plasma with long-range collisions, preprint, 2012. |
[9] |
R.-J. Duan and S.-Q. Liu, The Vlasov-Poisson-Boltzmann system without angular cutoff, preprint, 2012. |
[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, to appear in Math. Models Methods Appl. Sci. (2012), arXiv:1112.1453v1. |
[12] |
R.-J. Duan, T. Yang and H.-J. Zhao, Global solutions to the Vlasov-Poisson-Landau system, preprint 2011, arXiv:1112.3261v1. |
[13] |
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), 291-328.
doi: 10.1007/s00205-010-0318-6. |
[14] |
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. |
[15] |
H. Grad, On the kinetic theory of rarefied gases, Comm. Pure Appl. Math., 2 (1949), 331-407.
doi: 10.1002/cpa.3160020403. |
[16] |
H. Grad, Principles of the kinetic theory of gases, in "Handbuch der Physik," Springer-Berlin, XII (1958), 205-294. |
[17] |
P. T. Gressman and R. M. Strain, Global classical solutions of the Boltzmann equation without angular cut-off, J. Amer. Math. Soc., 24 (2011), 771-847.
doi: 10.1090/S0894-0347-2011-00697-8. |
[18] |
P. T. Gressman and R. M. Strain, Sharp anisotropic estimates for the Boltzmann collision operator and its entropy production, Adv. Math., 227 (2011), 2349-2384.
doi: 10.1016/j.aim.2011.05.005. |
[19] |
Y. Guo, The Landau equation in a periodic box, Comm. Math. Phys., 231 (2002), 391-434.
doi: 10.1007/s00220-002-0729-9. |
[20] |
Y. Guo, The Vlasov-Poisson-Landau system in a periodic box, J. Amer. Math. Soc., 25 (2012), 759-812.
doi: 10.1090/S0894-0347-2011-00722-4. |
[21] |
Y. Guo, The Vlasov-Maxwell-Boltzmann system near Maxwellians, Invent. Math., 153 (2003), 593-630.
doi: 10.1007/s00222-003-0301-z. |
[22] |
T. Hosono and S. Kawashima, Decay property of regularity-loss type and application to some nonlinear hyperbolic-elliptic system, Math. Models Methods Appl. Sci., 16 (2006), 1839-1859.
doi: 10.1142/S021820250600173X. |
[23] |
N. A. Krall and A. W. Trivelpiece, "Principles of Plasma Physics," McGraw-Hill, 1973. |
[24] |
P.-L. Lions, On Boltzmann and Landau equations, Philos. Trans. Roy. Soc. London Ser. A, 346 (1994), 191-204.
doi: 10.1098/rsta.1994.0018. |
[25] |
S.-Q. Liu and H.-J. Yu, Optimal time decay of the Landau equation with potential force, preprint, 2011. |
[26] |
T.-P. Liu and S.-H. Yu, Boltzmann equation: Micro-macro decompositions and positivity of shock profiles, Comm. Math. Phys., 246 (2004), 133-179.
doi: 10.1007/s00220-003-1030-2. |
[27] |
T.-P. Liu and S.-H. Yu, The Green's function and large-time behavior of solutions for the one-dimensional Boltzmann equation, Comm. Pure Appl. Math., 57 (2004), 1543-1608.
doi: 10.1002/cpa.20011. |
[28] |
C. Mouhot, Explicit coercivity estimates for the linearized Boltzmann and Landau operators, Comm. Partial Differential Equations, 31 (2006), 1321-1348.
doi: 10.1080/03605300600635004. |
[29] |
R. M. Strain, The Vlasov-Maxwell-Boltzmann system in the whole space, Comm. Math. Phys., 268 (2006), 543-567.
doi: 10.1007/s00220-006-0109-y. |
[30] |
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. |
[31] |
R. M. Strain and Y. Guo, Almost exponential decay near Maxwellian, Comm. Partial Differential Equations, 31 (2006), 417-429.
doi: 10.1080/03605300500361545. |
[32] |
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. |
[33] |
R. M. Strain and Y. Guo, Momentum regularity and stability of the relativistic Vlasov-Maxwell-Boltzmann system, Commun. Math. Phys., 10 (2012), 649-673.
doi: 10.1007/s00220-012-1417-z. |
[34] |
R. M. Strain and K. Zhu, The Vlasov-Poisson-Landau system in $\mathbb{R}^3$, preprint 2012, arXiv:1202.2471v1. |
[35] |
S. Ukai, On the existence of global solutions of mixed problem for non-linear Boltzmann equation, Proceedings of the Japan Academy, 50 (1974), 179-184.
doi: 10.3792/pja/1195519027. |
[36] |
C. Villani, A review of mathematical topics in collisional kinetic theory, North-Holland, Amsterdam, Handbook of mathematical fluid dynamics, I (2002), 71-305. |
[37] |
Y. Wang, Global solution and time decay of the Vlasov-Poisson-Landau system in $\mathbb{R}^3$, preprint 2012, arXiv:1205.6277. |
[38] |
M.-Q. Zhan, Local existence of solutions to the Landau-Maxwell system, Math. Methods Appl. Sci., 17 (1994), 613-641.
doi: 10.1002/mma.1670170804. |
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