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Large-time decay of the soft potential relativistic Boltzmann equation in $\mathbb{R}^3_x$
1. | University of Pennsylvania, Department of Mathematics, David Rittenhouse Lab, 209 South 33rd Street, Philadelphia, PA 19104, United States |
2. | University of Pennsylvania, Department of Mathematics, David RittenhouseLab, 209 South 33rd Street, Philadelphia, PA 19104-6395, United States |
References:
[1] |
Håkan Andréasson, Regularity of the gain term and strong $L^1$ convergence to equilibrium for the relativistic Boltzmann equation, SIAM J. Math. Anal., 27 (1996), 1386-1405.
doi: 10.1137/0527076. |
[2] |
Russel E. Caflisch, The Boltzmann equation with a soft potential. I. Linear, spatially-homogeneous, Comm. Math. Phys., 74 (1980), 71-95. |
[3] |
Russel E. Caflisch, The Boltzmann equation with a soft potential. II. Nonlinear, spatially-periodic, Comm. Math. Phys., 74 (1980), 97-109. |
[4] |
Simone Calogero, The Newtonian limit of the relativistic Boltzmann equation, J. Math. Phys., 45 (2004), 4042-4052.
doi: 10.1063/1.1793328. |
[5] |
Carlo Cercignani and Gilberto Medeiros Kremer, "The Relativistic Boltzmann Equation: Theory and Applications," Progress in Mathematical Physics, 22, Birkhäuser Verlag, Basel, 2002. |
[6] |
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. |
[7] |
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. |
[8] |
R. J. DiPerna and P.-L. Lions, On the Cauchy problem for Boltzmann equations: Global existence and weak stability, Ann. of Math. (2), 130 (1989), 321-366.
doi: 10.2307/1971423. |
[9] |
Renjun Duan and Robert 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. |
[10] |
Renjun Duan and Robert M. Strain, Optimal large-time behavior of the Vlasov-Maxwell-Boltzmann system in the whole space, Commun. Pure Appl. Math., 64 (2011), 1497-1546. |
[11] |
Marek Dudyński, On the linearized relativistic Boltzmann equation. II. Existence of hydrodynamics, J. Statist. Phys., 57 (1989), 199-245.
doi: 10.1007/BF01023641. |
[12] |
Marek Dudyński and Maria L. Ekiel-Jeżewska, The relativistic Boltzmann equation-mathematical and physical aspects, J. Tech. Phys., 48 (2007), 39-47. |
[13] |
Marek Dudyński and Maria L. Ekiel-Jeżewska, On the linearized relativistic Boltzmann equation. I. Existence of solutions, Comm. Math. Phys., 115 (1988), 607-629.
doi: 10.1007/BF01224130. |
[14] |
Marek Dudyński and Maria L. Ekiel-Jeżewska, Global existence proof for relativistic Boltzmann equation, J. Statist. Phys., 66 (1992), 991-1001.
doi: 10.1007/BF01055712. |
[15] |
Marek Dudyński and Maria L. Ekiel-Jeżewska, Causality of the linearized relativistic Boltzmann equation, Phys. Rev. Lett., 55 (1985), 2831-2834.
doi: 10.1103/PhysRevLett.55.2831. |
[16] |
Marek Dudyński and Maria L. Ekiel-Jeżewska, Errata: "Causality of the linearized relativistic Boltzmann equation,'' Investigación Oper., 6 (1985), 2228. |
[17] |
Seung-Yeal Ha, Yong Duck Kim, Ho Lee and Se Eun Noh, Asymptotic completeness for relativistic kinetic equations with short-range interaction forces, Methods Appl. Anal., 14 (2007), 251-262. |
[18] |
Seung-Yeal Ha, Ho Lee, Xiongfeng Yang and Seok-Bae Yun, Uniform $L^2$-stability estimates for the relativistic Boltzmann equation, J. Hyperbolic Differ. Equ., 6 (2009), 295-312. |
[19] |
Ling Hsiao and Hongjun Yu, Asymptotic stability of the relativistic Maxwellian, Math. Methods Appl. Sci., 29 (2006), 1481-1499.
doi: 10.1002/mma.736. |
[20] |
Ling Hsiao and Hongjun Yu, Global classical solutions to the initial value problem for the relativistic Landau equation, J. Differential Equations, 228 (2006), 641-660. |
[21] |
Robert T. Glassey, "The Cauchy Problem in Kinetic Theory," Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996. |
[22] |
Robert 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. |
[23] |
Robert T. Glassey and Walter A. Strauss, On the derivatives of the collision map of relativistic particles, Transport Theory Statist. Phys., 20 (1991), 55-68.
doi: 10.1080/00411459108204708. |
[24] |
Robert T. Glassey and Walter A. Strauss, Asymptotic stability of the relativistic Maxwellian, Publ. Res. Inst. Math. Sci., 29 (1993), 301-347.
doi: 10.2977/prims/1195167275. |
[25] |
Robert T. Glassey and Walter A. Strauss, Asymptotic stability of the relativistic Maxwellian via fourteen moments, Transport Theory Statist. Phys., 24 (1995), 657-678.
doi: 10.1080/00411459508206020. |
[26] |
Philip T. Gressman and Robert M. Strain, Global classical solutions of the Boltzmann equation with long-range interactions, Proc. Nat. Acad. Sci. USA, 107 (2010), 5744-5749.
doi: 10.1073/pnas.1001185107. |
[27] |
Philip T. Gressman and Robert 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. |
[28] |
Philip T. Gressman and Robert 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. |
[29] |
Yan Guo, The Vlasov-Maxwell-Boltzmann system near Maxwellians, Invent. Math., 153 (2003), 593-630.
doi: 10.1007/s00222-003-0301-z. |
[30] |
Yan Guo, Classical solutions to the Boltzmann equation for molecules with an angular cutoff, Arch. Ration. Mech. Anal., 169 (2003), 305-353.
doi: 10.1007/s00205-003-0262-9. |
[31] |
Yan Guo, Decay and continuity of the Boltzmann equation in bounded domains, Arch. Ration. Mech. Anal., 197 (2010), 713-809.
doi: 10.1007/s00205-009-0285-y. |
[32] |
Yan Guo and Robert 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. |
[33] |
Yan Guo and Walter A. Strauss, Instability of periodic BGK equilibria, Comm. Pure Appl. Math., 48 (1995), 861-894.
doi: 10.1002/cpa.3160480803. |
[34] |
Zhenglu Jiang, On the Cauchy problem for the relativistic Boltzmann equation in a periodic box: Global existence, Transport Theory Statist. Phys., 28 (1999), 617-628.
doi: 10.1080/00411459908214520. |
[35] |
Zhenglu Jiang, On the relativistic Boltzmann equation, Acta Math. Sci. (English Ed.), 18 (1998), 348-360. |
[36] |
Shuichi Kawashima, The Boltzmann equation and thirteen moments, Japan J. Appl. Math., 7 (1990), 301-320.
doi: 10.1007/BF03167846. |
[37] |
P.-L. Lions, Compactness in Boltzmann's equation via Fourier integral operators and applications. I, II, III, J. Math. Kyoto Univ., 34 (1994), 391-427, 429-461, 539-584. |
[38] |
Tai-Ping Liu and Shih-Hsien Yu, Initial-boundary value problem for one-dimensional wave solutions of the Boltzmann equation, Comm. Pure Appl. Math., 60 (2007), 295-356.
doi: 10.1002/cpa.20172. |
[39] |
Tai-Ping Liu and Shih-Hsien 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. |
[40] |
Clément Mouhot and Cédric Villani, On Landau damping, Acta Math., 207 (2012), 29-201. |
[41] |
Jared Speck and Robert M. Strain, Hilbert expansion from the Boltzmann equation to relativistic fluids, Comm. Math. Phys., 304 (2011), 229-280.
doi: 10.1007/s00220-011-1207-z. |
[42] |
Robert M. Strain and Yan Guo, Stability of the relativistic Maxwellian in a collisional plasma, Comm. Math. Phys., 251 (2004), 263-320.
doi: 10.1007/s00220-004-1151-2. |
[43] |
Robert M. Strain and Yan Guo, Almost exponential decay near Maxwellian, Comm. Partial Differential Equations, 31 (2006), 417-429. |
[44] |
Robert M. Strain and Yan Guo, Exponential decay for soft potentials near Maxwellian, Arch. Ration. Mech. Anal., 187 (2008), 287-339.
doi: 10.1007/s00205-007-0067-3. |
[45] |
Robert M. Strain, Global Newtonian limit for the relativistic Boltzmann equation near vacuum, SIAM J. Math. Anal., 42 (2010), 1568-1601.
doi: 10.1137/090762695. |
[46] |
Robert 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. |
[47] |
Robert M. Strain, Coordinates in the relativistic Boltzmann theory, Kinetic and Related Models, 4 (2011), 345-359.
doi: 10.3934/krm.2011.4.345. |
[48] |
Robert M. Strain, Optimal time decay of the non cut-off Boltzmann equation in the whole space, preprint, arXiv:1011.5561v2, 2010. |
[49] |
Seiji Ukai and Kiyoshi Asano, On the Cauchy problem of the Boltzmann equation with a soft potential, Publ. Res. Inst. Math. Sci., 18 (1982), 57-99.
doi: 10.2977/prims/1195183569. |
[50] |
Ivan Vidav, Spectra of perturbed semigroups with applications to transport theory, J. Math. Anal. Appl., 30 (1970), 264-279.
doi: 10.1016/0022-247X(70)90160-5. |
[51] |
Bernt Wennberg, The geometry of binary collisions and generalized Radon transforms, Arch. Rational Mech. Anal., 139 (1997), 291-302.
doi: 10.1007/s002050050054. |
[52] |
Tong Yang and Hongjun Yu, Hypocoercivity of the relativistic Boltzmann and Landau equations in the whole space, J. Differential Equations, 248 (2010), 1518-1560. |
[53] |
Hongjun Yu, Smoothing effects for classical solutions of the relativistic Landau-Maxwell system, J. Differential Equations, 246 (2009), 3776-3817. |
show all references
References:
[1] |
Håkan Andréasson, Regularity of the gain term and strong $L^1$ convergence to equilibrium for the relativistic Boltzmann equation, SIAM J. Math. Anal., 27 (1996), 1386-1405.
doi: 10.1137/0527076. |
[2] |
Russel E. Caflisch, The Boltzmann equation with a soft potential. I. Linear, spatially-homogeneous, Comm. Math. Phys., 74 (1980), 71-95. |
[3] |
Russel E. Caflisch, The Boltzmann equation with a soft potential. II. Nonlinear, spatially-periodic, Comm. Math. Phys., 74 (1980), 97-109. |
[4] |
Simone Calogero, The Newtonian limit of the relativistic Boltzmann equation, J. Math. Phys., 45 (2004), 4042-4052.
doi: 10.1063/1.1793328. |
[5] |
Carlo Cercignani and Gilberto Medeiros Kremer, "The Relativistic Boltzmann Equation: Theory and Applications," Progress in Mathematical Physics, 22, Birkhäuser Verlag, Basel, 2002. |
[6] |
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. |
[7] |
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. |
[8] |
R. J. DiPerna and P.-L. Lions, On the Cauchy problem for Boltzmann equations: Global existence and weak stability, Ann. of Math. (2), 130 (1989), 321-366.
doi: 10.2307/1971423. |
[9] |
Renjun Duan and Robert 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. |
[10] |
Renjun Duan and Robert M. Strain, Optimal large-time behavior of the Vlasov-Maxwell-Boltzmann system in the whole space, Commun. Pure Appl. Math., 64 (2011), 1497-1546. |
[11] |
Marek Dudyński, On the linearized relativistic Boltzmann equation. II. Existence of hydrodynamics, J. Statist. Phys., 57 (1989), 199-245.
doi: 10.1007/BF01023641. |
[12] |
Marek Dudyński and Maria L. Ekiel-Jeżewska, The relativistic Boltzmann equation-mathematical and physical aspects, J. Tech. Phys., 48 (2007), 39-47. |
[13] |
Marek Dudyński and Maria L. Ekiel-Jeżewska, On the linearized relativistic Boltzmann equation. I. Existence of solutions, Comm. Math. Phys., 115 (1988), 607-629.
doi: 10.1007/BF01224130. |
[14] |
Marek Dudyński and Maria L. Ekiel-Jeżewska, Global existence proof for relativistic Boltzmann equation, J. Statist. Phys., 66 (1992), 991-1001.
doi: 10.1007/BF01055712. |
[15] |
Marek Dudyński and Maria L. Ekiel-Jeżewska, Causality of the linearized relativistic Boltzmann equation, Phys. Rev. Lett., 55 (1985), 2831-2834.
doi: 10.1103/PhysRevLett.55.2831. |
[16] |
Marek Dudyński and Maria L. Ekiel-Jeżewska, Errata: "Causality of the linearized relativistic Boltzmann equation,'' Investigación Oper., 6 (1985), 2228. |
[17] |
Seung-Yeal Ha, Yong Duck Kim, Ho Lee and Se Eun Noh, Asymptotic completeness for relativistic kinetic equations with short-range interaction forces, Methods Appl. Anal., 14 (2007), 251-262. |
[18] |
Seung-Yeal Ha, Ho Lee, Xiongfeng Yang and Seok-Bae Yun, Uniform $L^2$-stability estimates for the relativistic Boltzmann equation, J. Hyperbolic Differ. Equ., 6 (2009), 295-312. |
[19] |
Ling Hsiao and Hongjun Yu, Asymptotic stability of the relativistic Maxwellian, Math. Methods Appl. Sci., 29 (2006), 1481-1499.
doi: 10.1002/mma.736. |
[20] |
Ling Hsiao and Hongjun Yu, Global classical solutions to the initial value problem for the relativistic Landau equation, J. Differential Equations, 228 (2006), 641-660. |
[21] |
Robert T. Glassey, "The Cauchy Problem in Kinetic Theory," Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996. |
[22] |
Robert 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. |
[23] |
Robert T. Glassey and Walter A. Strauss, On the derivatives of the collision map of relativistic particles, Transport Theory Statist. Phys., 20 (1991), 55-68.
doi: 10.1080/00411459108204708. |
[24] |
Robert T. Glassey and Walter A. Strauss, Asymptotic stability of the relativistic Maxwellian, Publ. Res. Inst. Math. Sci., 29 (1993), 301-347.
doi: 10.2977/prims/1195167275. |
[25] |
Robert T. Glassey and Walter A. Strauss, Asymptotic stability of the relativistic Maxwellian via fourteen moments, Transport Theory Statist. Phys., 24 (1995), 657-678.
doi: 10.1080/00411459508206020. |
[26] |
Philip T. Gressman and Robert M. Strain, Global classical solutions of the Boltzmann equation with long-range interactions, Proc. Nat. Acad. Sci. USA, 107 (2010), 5744-5749.
doi: 10.1073/pnas.1001185107. |
[27] |
Philip T. Gressman and Robert 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. |
[28] |
Philip T. Gressman and Robert 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. |
[29] |
Yan Guo, The Vlasov-Maxwell-Boltzmann system near Maxwellians, Invent. Math., 153 (2003), 593-630.
doi: 10.1007/s00222-003-0301-z. |
[30] |
Yan Guo, Classical solutions to the Boltzmann equation for molecules with an angular cutoff, Arch. Ration. Mech. Anal., 169 (2003), 305-353.
doi: 10.1007/s00205-003-0262-9. |
[31] |
Yan Guo, Decay and continuity of the Boltzmann equation in bounded domains, Arch. Ration. Mech. Anal., 197 (2010), 713-809.
doi: 10.1007/s00205-009-0285-y. |
[32] |
Yan Guo and Robert 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. |
[33] |
Yan Guo and Walter A. Strauss, Instability of periodic BGK equilibria, Comm. Pure Appl. Math., 48 (1995), 861-894.
doi: 10.1002/cpa.3160480803. |
[34] |
Zhenglu Jiang, On the Cauchy problem for the relativistic Boltzmann equation in a periodic box: Global existence, Transport Theory Statist. Phys., 28 (1999), 617-628.
doi: 10.1080/00411459908214520. |
[35] |
Zhenglu Jiang, On the relativistic Boltzmann equation, Acta Math. Sci. (English Ed.), 18 (1998), 348-360. |
[36] |
Shuichi Kawashima, The Boltzmann equation and thirteen moments, Japan J. Appl. Math., 7 (1990), 301-320.
doi: 10.1007/BF03167846. |
[37] |
P.-L. Lions, Compactness in Boltzmann's equation via Fourier integral operators and applications. I, II, III, J. Math. Kyoto Univ., 34 (1994), 391-427, 429-461, 539-584. |
[38] |
Tai-Ping Liu and Shih-Hsien Yu, Initial-boundary value problem for one-dimensional wave solutions of the Boltzmann equation, Comm. Pure Appl. Math., 60 (2007), 295-356.
doi: 10.1002/cpa.20172. |
[39] |
Tai-Ping Liu and Shih-Hsien 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. |
[40] |
Clément Mouhot and Cédric Villani, On Landau damping, Acta Math., 207 (2012), 29-201. |
[41] |
Jared Speck and Robert M. Strain, Hilbert expansion from the Boltzmann equation to relativistic fluids, Comm. Math. Phys., 304 (2011), 229-280.
doi: 10.1007/s00220-011-1207-z. |
[42] |
Robert M. Strain and Yan Guo, Stability of the relativistic Maxwellian in a collisional plasma, Comm. Math. Phys., 251 (2004), 263-320.
doi: 10.1007/s00220-004-1151-2. |
[43] |
Robert M. Strain and Yan Guo, Almost exponential decay near Maxwellian, Comm. Partial Differential Equations, 31 (2006), 417-429. |
[44] |
Robert M. Strain and Yan Guo, Exponential decay for soft potentials near Maxwellian, Arch. Ration. Mech. Anal., 187 (2008), 287-339.
doi: 10.1007/s00205-007-0067-3. |
[45] |
Robert M. Strain, Global Newtonian limit for the relativistic Boltzmann equation near vacuum, SIAM J. Math. Anal., 42 (2010), 1568-1601.
doi: 10.1137/090762695. |
[46] |
Robert 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. |
[47] |
Robert M. Strain, Coordinates in the relativistic Boltzmann theory, Kinetic and Related Models, 4 (2011), 345-359.
doi: 10.3934/krm.2011.4.345. |
[48] |
Robert M. Strain, Optimal time decay of the non cut-off Boltzmann equation in the whole space, preprint, arXiv:1011.5561v2, 2010. |
[49] |
Seiji Ukai and Kiyoshi Asano, On the Cauchy problem of the Boltzmann equation with a soft potential, Publ. Res. Inst. Math. Sci., 18 (1982), 57-99.
doi: 10.2977/prims/1195183569. |
[50] |
Ivan Vidav, Spectra of perturbed semigroups with applications to transport theory, J. Math. Anal. Appl., 30 (1970), 264-279.
doi: 10.1016/0022-247X(70)90160-5. |
[51] |
Bernt Wennberg, The geometry of binary collisions and generalized Radon transforms, Arch. Rational Mech. Anal., 139 (1997), 291-302.
doi: 10.1007/s002050050054. |
[52] |
Tong Yang and Hongjun Yu, Hypocoercivity of the relativistic Boltzmann and Landau equations in the whole space, J. Differential Equations, 248 (2010), 1518-1560. |
[53] |
Hongjun Yu, Smoothing effects for classical solutions of the relativistic Landau-Maxwell system, J. Differential Equations, 246 (2009), 3776-3817. |
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