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Global nonlinear stability of rarefaction waves for compressible Navier-Stokes equations with temperature and density dependent transport coefficients
The relativistic Vlasov-Maxwell-Boltzmann system for short range interaction
1. | Department of Mathematics, Jinan University, Guangzhou 510632, China |
2. | Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071 |
References:
[1] |
S. Calogero, The Newtonian limit of the relativistic Boltzmann equation, J. Math. Phys., 45 (2004), 4042-4052.
doi: 10.1063/1.1793328. |
[2] |
C. Cercignani and G. M. Kremer, The Relativistic Boltzmann Equation: Theory and Applications, Progress in Mathematical Physics, 22, Birkhäuser Verlag, Basel, 2002.
doi: 10.1007/978-3-0348-8165-4. |
[3] |
S. R. de Groot, W. A. van Leeuwen and Ch. G. van Weert, Relativistic Kinetic Theory. Principles and Applications, North-Holland Publishing Co., Amsterdam-New York, 1980. |
[4] |
R. J. Duan, Global smooth dynamics of a fully ionized plasma with long-range collisions, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 31 (2014), 751-778.
doi: 10.1016/j.anihpc.2013.07.004. |
[5] |
R. J. Duan, Y. J. Lei, T. Yang and H. J. Zhao, The Vlasov-Maxwell-Boltzmann system near Maxwellians in the whole space with very soft potentials, arXiv:1411.5150v1. |
[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. Q. Liu, T. Yang and H. J. Zhao, Stability of the nonrelativistic Vlasov- Maxwell-Boltzmann system for angular non-cutoff potentials, Kinetic and Related Models, 6 (2013), 159-204. |
[8] |
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. |
[9] |
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. |
[10] |
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. |
[11] |
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. |
[12] |
M. Dudyński and M. L. Ekiel-J.ezewska, Global existence proof for relativistic Boltzmann equation, J. Statist.Phys., 66 (1992), 991-1001.
doi: 10.1007/BF01055712. |
[13] |
M. Escobedo, S. Mischler and M. A. Valle, Homogeneous Boltzmann Equation in Quantum Relativistic Kinetic Theory, Electronic Journal of Differential Equations, Monograph, vol. 4, Southwest Texas State University, San Marcos, TX, 2003. |
[14] |
R. T. Glassey, The Cauchy Problem in Kinetic Theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996.
doi: 10.1137/1.9781611971477. |
[15] |
R. T. Glassey, Global solutions to the Cauchy problem for the relativistic Boltzmann equation with near-vacuum data, Comm. Math. Phys., 264 (2006), 705-724.
doi: 10.1007/s00220-006-1522-y. |
[16] |
R. T. Glassey and W. A. Strauss, On the derivatives of the collision map of relativistic particles, Transport Theory Statist. Phys., 20 (1991), 55-68.
doi: 10.1080/00411459108204708. |
[17] |
R. T. Glassey and W. A. Strauss, Asymptotic stability of the relativistic Maxwellian, Publ. Res. Inst. Math. Sci., 29 (1993), 301-347.
doi: 10.2977/prims/1195167275. |
[18] |
R. T. Glassey and W. A. Strauss, Asymptotic stability of the relativistic Maxwellian via fourteen moments, Transport Theory Statist. Phys., 24 (1995), 657-678.
doi: 10.1080/00411459508206020. |
[19] |
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. |
[20] |
Y. Guo, The Vlasov-Maxwell-Boltzmann system near Maxwellians, Invent. Math., 153 (2003), 593-630.
doi: 10.1007/s00222-003-0301-z. |
[21] |
Y. Guo and R. M. Strain, Momentum regularity and stability of the relativistic Vlasov- Maxwell-Boltzmann system, Comm. Math. Phys., 310 (2012), 649-673.
doi: 10.1007/s00220-012-1417-z. |
[22] |
S. Y. Ha, Y. D. Kim, H. Lee and S. E. Noh, Asymptotic completeness for relativistic kinetic equations with short-range interaction forces, Methods Appl. Anal., 14 (2007), 251-262.
doi: 10.4310/MAA.2007.v14.n3.a3. |
[23] |
L. Hsiao and H. J. Yu, Asymptotic stability of the relativistic Maxwellian, Math. Methods Appl. Sci., 29 (2006), 1481-1499.
doi: 10.1002/mma.736. |
[24] |
L. Hsiao and H. J. Yu, Global classical solutions to the initial value problem for the relativistic Landau equation, J. Differential Equations, 228 (2006), 641-660.
doi: 10.1016/j.jde.2005.10.022. |
[25] |
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. |
[26] |
N. A. Krall and A. W. Trivelpiece, Principles of Plasma Physics, McGraw-Hill, 1973. |
[27] |
Y. J. Lei and H.-J. Zhao, The Vlasov-Maxwell-Boltzmann system with a uniform ionic background near Maxwellians, J. Differential Equations, 260 (2016), 2830-2897.
doi: 10.1016/j.jde.2015.10.021. |
[28] |
S. Q. Liu and H. J. Zhao, Optimal large-time decay of the relativistic Landau-Maxwell system, J. Differential Equations, 256 (2014), 832-857.
doi: 10.1016/j.jde.2013.10.004. |
[29] |
A. Majda and A. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, 2002. |
[30] |
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. |
[31] |
R. M. Strain, Some Applications of an Energy Method in Collisional Kinetic Theory, Phd Thesis, Brown University, Providence RI, 2005. |
[32] |
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. |
[33] |
R. M. Strain and Y. Guo, Almost exponential decay near Maxwellian, Comm. Partial Differential Equations, 31 (2006), 417-429.
doi: 10.1080/03605300500361545. |
[34] |
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. |
[35] |
R. M. Strain, Global Newtonian limit for the relativistic Boltzmann equation near vacuum, SIAM J. Math. Anal., 42 (2010), 1568-1601.
doi: 10.1137/090762695. |
[36] |
R. M. Strain, Asymptotic stability of the relativistic Boltzmann equation for the soft-potentials, Comm. Math. Phys., 300 (2010), 529-597.
doi: 10.1007/s00220-010-1129-1. |
[37] |
R. M. Strain, Coordinates in the relativistic Boltzmann theory, Kinetic and Related Models, 4 (2011), 345-359.
doi: 10.3934/krm.2011.4.345. |
[38] |
R. M. Strain and K. Y. Zhu, Large-time decay of the soft potential relativistic Boltzmann equation in $\mathbb R^3$, Kinetic and Related Models, 5 (2012), 383-415.
doi: 10.3934/krm.2012.5.383. |
[39] |
C. Villani, A review of mathematical topics in collisional kinetic theory, North-Holland, Amsterdam, Handbook of mathematical fluid dynamics, I (2002), 71-305.
doi: 10.1016/S1874-5792(02)80004-0. |
[40] |
L. S. Wang, Q. H. Xiao, L. J. Xiong and H. J. Zhao, The Vlasov-Poisson-Boltzmann system near Maxwellians for long-range interactions, Acta Mathematica Scientia, 2016 (in press). |
[41] |
Q. H. Xiao, Large-time behavior of the two-species relativistic Landau-Maxwell system in $\mathbb R^3_x$, J. Differential Equations, 259 (2015), 3520-3558.
doi: 10.1016/j.jde.2015.04.031. |
[42] |
Q. H. Xiao, L. J. Xiong and H. J. Zhao, The Vlasov-Poisson-Boltzmann system with angular cutoff for soft potentials, J. Differential Equations, 255 (2013), 1196-1232.
doi: 10.1016/j.jde.2013.05.005. |
[43] |
Q. H. Xiao, L. J. Xiong and H. J. Zhao, The Vlasov-Poisson-Boltzmann system for non-cutoff hard potentials, Sci. China Math., 57 (2014), 515-540.
doi: 10.1007/s11425-013-4712-z. |
[44] |
Q. H. Xiao, L. J. Xiong and H. J. Zhao, The Vlasov-Poisson-Boltzmann system for the whole range of cutoff soft potentials, arXiv:1403.2584v1. |
[45] |
T. Yang and H. J. Yu, Hypocoercivity of the relativistic Boltzmann and Landau equa- tions in the whole space, J. Differential Equations, 248 (2010), 1518-1560.
doi: 10.1016/j.jde.2009.11.027. |
show all references
References:
[1] |
S. Calogero, The Newtonian limit of the relativistic Boltzmann equation, J. Math. Phys., 45 (2004), 4042-4052.
doi: 10.1063/1.1793328. |
[2] |
C. Cercignani and G. M. Kremer, The Relativistic Boltzmann Equation: Theory and Applications, Progress in Mathematical Physics, 22, Birkhäuser Verlag, Basel, 2002.
doi: 10.1007/978-3-0348-8165-4. |
[3] |
S. R. de Groot, W. A. van Leeuwen and Ch. G. van Weert, Relativistic Kinetic Theory. Principles and Applications, North-Holland Publishing Co., Amsterdam-New York, 1980. |
[4] |
R. J. Duan, Global smooth dynamics of a fully ionized plasma with long-range collisions, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 31 (2014), 751-778.
doi: 10.1016/j.anihpc.2013.07.004. |
[5] |
R. J. Duan, Y. J. Lei, T. Yang and H. J. Zhao, The Vlasov-Maxwell-Boltzmann system near Maxwellians in the whole space with very soft potentials, arXiv:1411.5150v1. |
[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. Q. Liu, T. Yang and H. J. Zhao, Stability of the nonrelativistic Vlasov- Maxwell-Boltzmann system for angular non-cutoff potentials, Kinetic and Related Models, 6 (2013), 159-204. |
[8] |
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. |
[9] |
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. |
[10] |
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. |
[11] |
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. |
[12] |
M. Dudyński and M. L. Ekiel-J.ezewska, Global existence proof for relativistic Boltzmann equation, J. Statist.Phys., 66 (1992), 991-1001.
doi: 10.1007/BF01055712. |
[13] |
M. Escobedo, S. Mischler and M. A. Valle, Homogeneous Boltzmann Equation in Quantum Relativistic Kinetic Theory, Electronic Journal of Differential Equations, Monograph, vol. 4, Southwest Texas State University, San Marcos, TX, 2003. |
[14] |
R. T. Glassey, The Cauchy Problem in Kinetic Theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996.
doi: 10.1137/1.9781611971477. |
[15] |
R. T. Glassey, Global solutions to the Cauchy problem for the relativistic Boltzmann equation with near-vacuum data, Comm. Math. Phys., 264 (2006), 705-724.
doi: 10.1007/s00220-006-1522-y. |
[16] |
R. T. Glassey and W. A. Strauss, On the derivatives of the collision map of relativistic particles, Transport Theory Statist. Phys., 20 (1991), 55-68.
doi: 10.1080/00411459108204708. |
[17] |
R. T. Glassey and W. A. Strauss, Asymptotic stability of the relativistic Maxwellian, Publ. Res. Inst. Math. Sci., 29 (1993), 301-347.
doi: 10.2977/prims/1195167275. |
[18] |
R. T. Glassey and W. A. Strauss, Asymptotic stability of the relativistic Maxwellian via fourteen moments, Transport Theory Statist. Phys., 24 (1995), 657-678.
doi: 10.1080/00411459508206020. |
[19] |
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. |
[20] |
Y. Guo, The Vlasov-Maxwell-Boltzmann system near Maxwellians, Invent. Math., 153 (2003), 593-630.
doi: 10.1007/s00222-003-0301-z. |
[21] |
Y. Guo and R. M. Strain, Momentum regularity and stability of the relativistic Vlasov- Maxwell-Boltzmann system, Comm. Math. Phys., 310 (2012), 649-673.
doi: 10.1007/s00220-012-1417-z. |
[22] |
S. Y. Ha, Y. D. Kim, H. Lee and S. E. Noh, Asymptotic completeness for relativistic kinetic equations with short-range interaction forces, Methods Appl. Anal., 14 (2007), 251-262.
doi: 10.4310/MAA.2007.v14.n3.a3. |
[23] |
L. Hsiao and H. J. Yu, Asymptotic stability of the relativistic Maxwellian, Math. Methods Appl. Sci., 29 (2006), 1481-1499.
doi: 10.1002/mma.736. |
[24] |
L. Hsiao and H. J. Yu, Global classical solutions to the initial value problem for the relativistic Landau equation, J. Differential Equations, 228 (2006), 641-660.
doi: 10.1016/j.jde.2005.10.022. |
[25] |
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. |
[26] |
N. A. Krall and A. W. Trivelpiece, Principles of Plasma Physics, McGraw-Hill, 1973. |
[27] |
Y. J. Lei and H.-J. Zhao, The Vlasov-Maxwell-Boltzmann system with a uniform ionic background near Maxwellians, J. Differential Equations, 260 (2016), 2830-2897.
doi: 10.1016/j.jde.2015.10.021. |
[28] |
S. Q. Liu and H. J. Zhao, Optimal large-time decay of the relativistic Landau-Maxwell system, J. Differential Equations, 256 (2014), 832-857.
doi: 10.1016/j.jde.2013.10.004. |
[29] |
A. Majda and A. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, 2002. |
[30] |
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. |
[31] |
R. M. Strain, Some Applications of an Energy Method in Collisional Kinetic Theory, Phd Thesis, Brown University, Providence RI, 2005. |
[32] |
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. |
[33] |
R. M. Strain and Y. Guo, Almost exponential decay near Maxwellian, Comm. Partial Differential Equations, 31 (2006), 417-429.
doi: 10.1080/03605300500361545. |
[34] |
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. |
[35] |
R. M. Strain, Global Newtonian limit for the relativistic Boltzmann equation near vacuum, SIAM J. Math. Anal., 42 (2010), 1568-1601.
doi: 10.1137/090762695. |
[36] |
R. M. Strain, Asymptotic stability of the relativistic Boltzmann equation for the soft-potentials, Comm. Math. Phys., 300 (2010), 529-597.
doi: 10.1007/s00220-010-1129-1. |
[37] |
R. M. Strain, Coordinates in the relativistic Boltzmann theory, Kinetic and Related Models, 4 (2011), 345-359.
doi: 10.3934/krm.2011.4.345. |
[38] |
R. M. Strain and K. Y. Zhu, Large-time decay of the soft potential relativistic Boltzmann equation in $\mathbb R^3$, Kinetic and Related Models, 5 (2012), 383-415.
doi: 10.3934/krm.2012.5.383. |
[39] |
C. Villani, A review of mathematical topics in collisional kinetic theory, North-Holland, Amsterdam, Handbook of mathematical fluid dynamics, I (2002), 71-305.
doi: 10.1016/S1874-5792(02)80004-0. |
[40] |
L. S. Wang, Q. H. Xiao, L. J. Xiong and H. J. Zhao, The Vlasov-Poisson-Boltzmann system near Maxwellians for long-range interactions, Acta Mathematica Scientia, 2016 (in press). |
[41] |
Q. H. Xiao, Large-time behavior of the two-species relativistic Landau-Maxwell system in $\mathbb R^3_x$, J. Differential Equations, 259 (2015), 3520-3558.
doi: 10.1016/j.jde.2015.04.031. |
[42] |
Q. H. Xiao, L. J. Xiong and H. J. Zhao, The Vlasov-Poisson-Boltzmann system with angular cutoff for soft potentials, J. Differential Equations, 255 (2013), 1196-1232.
doi: 10.1016/j.jde.2013.05.005. |
[43] |
Q. H. Xiao, L. J. Xiong and H. J. Zhao, The Vlasov-Poisson-Boltzmann system for non-cutoff hard potentials, Sci. China Math., 57 (2014), 515-540.
doi: 10.1007/s11425-013-4712-z. |
[44] |
Q. H. Xiao, L. J. Xiong and H. J. Zhao, The Vlasov-Poisson-Boltzmann system for the whole range of cutoff soft potentials, arXiv:1403.2584v1. |
[45] |
T. Yang and H. J. Yu, Hypocoercivity of the relativistic Boltzmann and Landau equa- tions in the whole space, J. Differential Equations, 248 (2010), 1518-1560.
doi: 10.1016/j.jde.2009.11.027. |
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