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Asymptotic behavior of the solutions of the inhomogeneous Porous Medium Equation with critical vanishing density
Uniform $L^1$-stability of the relativistic Boltzmann equation near vacuum
1. | Department of Mathematical Sciences, Seoul National University, Seoul 151-747, South Korea |
2. | University of Pennsylvania, Department of Mathematics, David Rittenhouse Lab, 209 South 33rd Street, Philadelphia, PA 19104-6395, Uruguay |
References:
[1] |
R. Alonso and I. M. Gamba, Distributional and classical solutions to the Cauchy Boltzmann problem for soft potentials with integrable angular cross section, J. Stat. Phys., 137 (2009), 1147-1165.
doi: 10.1007/s10955-009-9873-3. |
[2] |
D. Bancel, Problème de Cauchy pour l'équation de Boltzmann en relativité générale, Ann. Inst. Henri Poincareé, XVIII 3 (1973), 263-284. |
[3] |
D. Bancel and Y. Choquet-Bruhat, Uniqureness and local stability for the Einstein-Maxwell-Boltzmann system, Comm. Math. Phys., 33 (1973), 83-96. |
[4] |
K. Bichteler, On the Cauchy problem of the relativistic Boltzmann equation, Comm. Math. Phys., 4 (1967), 352-364. |
[5] |
S. Calogero, The Newtonian limit of the relativistic Botlzmann equation, J. Math. Phys., 45 (2004), 4042-4052.
doi: 10.1063/1.1793328. |
[6] |
M. Dudyński and M. Ekiel Jezewska, On the linearized Relativistic Boltzmann equation, Comm. Math. Phys., 115 (1988), 607-629. |
[7] |
M. Dudyński and M. Ekiel Jezewska, Global existence proof for relativistic Boltzmann equation, J. Stat. Phys., 66 (1992), 991-1001. |
[8] |
M. Dudyński and M. Ekiel Jezewska, The relativistic Boltzmann equation-mathematical and physical aspects, J. Tech. Phys., 48 (2007), 39-47. |
[9] |
R. J. DiPerna and P.-L. Lions, On the Cauchy problem for Boltzmann equations: global existence and weak stability, Ann. of Math., 130 (1989), 321-366. |
[10] |
R. T. Glassey, Global solutioins 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. |
[11] |
R. T. Glassey and W. Strauss, Asymptotic stability of the relativistic Maxwellian via fourteen moments, Trans. Th. Stat. Phys., 24 (1995), 657-678. |
[12] |
R. T. Glassey and W. Strauss, Asymptotic stability of the relativistic Maxwellian, Publ. R.I.M.S. Kyoto Univ., 29 (1993), 301-347. |
[13] |
J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math., 18 (1965), 697-715. |
[14] |
S.-Y. Ha, Nonlinear functionals of the Boltzmann equation and uniform stability estimates, J. Differential Equations, 215 (2005), 178-205.
doi: 10.1016/j.jde.2004.07.022. |
[15] |
S.-Y Ha, $L_1$-stability of the Boltzmann equation for the hard-sphere model, Arch. Ration. Mech. Anal., 173 (2004), 279-296.
doi: 10.1007/s00205-004-0321-x. |
[16] |
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. |
[17] |
S.-Y. Ha and S.-B. Yun, Uniform $L^1$-stability estmate of the Bolzmann equation near a local Maxwellian, Physica D, 220 (2006), 79-97.
doi: 10.1016/j.physd.2006.06.011. |
[18] |
L. Hsiao and H. Yu, Asymptotic stability of the relativistic Maxwellian, Math. Methods Appl. Sci., 29 (2006), 1481-1499.
doi: 10.1002/mma.736. |
[19] |
R. Illner and M. Shinbrot, The Boltzmann equation, global existence for a rare gas in an infinite vacuum, Comm. Math. Phys., 95 (1984), 217-226. |
[20] |
S. Kaniel and M. Shinbrot, The Boltzmann equation 1. Uniqueness and local existence, Comm. Math. Phys., 58 (1978), 65-84. |
[21] |
A. Lichnerowich and R. Marrot, Propriés statistiques des ensembles de particules en relativité restreinte, F. R. Acad. Sci. Paris, 210 (1940), 759-761. |
[22] |
J. Polewczak, Classical Solution of the nonlinear Boltzmann equation in all $\mathbb{R}^3$ Asymptotic behavior of solutions, J. Stat. Phys., 50 (1988), 611-632. |
[23] |
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. |
[24] |
R. M. Strain, Coordinates in the relativistic Boltzmann theory, Kinetic and Related Models, 4 (2011), 345-359.
doi: 10.3934/krm.2011.4.345. |
[25] |
R. M. Strain and K. Zhu, Large-time decay of the soft potential relativistic Boltzmann equation in $\mathbb{R}_{x}^{3}$, Kinetic and Related Models, 5 (2012), 383-415.
doi: 3934/krm.2012.5.383. |
[26] |
G. Toscani, H-thoerem and asymptotic trend of the solution for a rarefied gas in a vacuum, Arch. Rational Mech. Anal., 100 (1987), 1-12. |
[27] |
T. Yang and H. Yu, Hypocoercivity of the relativistic Boltzmann and Landau equations in the whole space, J. Differential Equations, 248 (2010), 1518-1560.
doi: 10.1016/j.jde.2009.11.027. |
show all references
References:
[1] |
R. Alonso and I. M. Gamba, Distributional and classical solutions to the Cauchy Boltzmann problem for soft potentials with integrable angular cross section, J. Stat. Phys., 137 (2009), 1147-1165.
doi: 10.1007/s10955-009-9873-3. |
[2] |
D. Bancel, Problème de Cauchy pour l'équation de Boltzmann en relativité générale, Ann. Inst. Henri Poincareé, XVIII 3 (1973), 263-284. |
[3] |
D. Bancel and Y. Choquet-Bruhat, Uniqureness and local stability for the Einstein-Maxwell-Boltzmann system, Comm. Math. Phys., 33 (1973), 83-96. |
[4] |
K. Bichteler, On the Cauchy problem of the relativistic Boltzmann equation, Comm. Math. Phys., 4 (1967), 352-364. |
[5] |
S. Calogero, The Newtonian limit of the relativistic Botlzmann equation, J. Math. Phys., 45 (2004), 4042-4052.
doi: 10.1063/1.1793328. |
[6] |
M. Dudyński and M. Ekiel Jezewska, On the linearized Relativistic Boltzmann equation, Comm. Math. Phys., 115 (1988), 607-629. |
[7] |
M. Dudyński and M. Ekiel Jezewska, Global existence proof for relativistic Boltzmann equation, J. Stat. Phys., 66 (1992), 991-1001. |
[8] |
M. Dudyński and M. Ekiel Jezewska, The relativistic Boltzmann equation-mathematical and physical aspects, J. Tech. Phys., 48 (2007), 39-47. |
[9] |
R. J. DiPerna and P.-L. Lions, On the Cauchy problem for Boltzmann equations: global existence and weak stability, Ann. of Math., 130 (1989), 321-366. |
[10] |
R. T. Glassey, Global solutioins 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. |
[11] |
R. T. Glassey and W. Strauss, Asymptotic stability of the relativistic Maxwellian via fourteen moments, Trans. Th. Stat. Phys., 24 (1995), 657-678. |
[12] |
R. T. Glassey and W. Strauss, Asymptotic stability of the relativistic Maxwellian, Publ. R.I.M.S. Kyoto Univ., 29 (1993), 301-347. |
[13] |
J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math., 18 (1965), 697-715. |
[14] |
S.-Y. Ha, Nonlinear functionals of the Boltzmann equation and uniform stability estimates, J. Differential Equations, 215 (2005), 178-205.
doi: 10.1016/j.jde.2004.07.022. |
[15] |
S.-Y Ha, $L_1$-stability of the Boltzmann equation for the hard-sphere model, Arch. Ration. Mech. Anal., 173 (2004), 279-296.
doi: 10.1007/s00205-004-0321-x. |
[16] |
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. |
[17] |
S.-Y. Ha and S.-B. Yun, Uniform $L^1$-stability estmate of the Bolzmann equation near a local Maxwellian, Physica D, 220 (2006), 79-97.
doi: 10.1016/j.physd.2006.06.011. |
[18] |
L. Hsiao and H. Yu, Asymptotic stability of the relativistic Maxwellian, Math. Methods Appl. Sci., 29 (2006), 1481-1499.
doi: 10.1002/mma.736. |
[19] |
R. Illner and M. Shinbrot, The Boltzmann equation, global existence for a rare gas in an infinite vacuum, Comm. Math. Phys., 95 (1984), 217-226. |
[20] |
S. Kaniel and M. Shinbrot, The Boltzmann equation 1. Uniqueness and local existence, Comm. Math. Phys., 58 (1978), 65-84. |
[21] |
A. Lichnerowich and R. Marrot, Propriés statistiques des ensembles de particules en relativité restreinte, F. R. Acad. Sci. Paris, 210 (1940), 759-761. |
[22] |
J. Polewczak, Classical Solution of the nonlinear Boltzmann equation in all $\mathbb{R}^3$ Asymptotic behavior of solutions, J. Stat. Phys., 50 (1988), 611-632. |
[23] |
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. |
[24] |
R. M. Strain, Coordinates in the relativistic Boltzmann theory, Kinetic and Related Models, 4 (2011), 345-359.
doi: 10.3934/krm.2011.4.345. |
[25] |
R. M. Strain and K. Zhu, Large-time decay of the soft potential relativistic Boltzmann equation in $\mathbb{R}_{x}^{3}$, Kinetic and Related Models, 5 (2012), 383-415.
doi: 3934/krm.2012.5.383. |
[26] |
G. Toscani, H-thoerem and asymptotic trend of the solution for a rarefied gas in a vacuum, Arch. Rational Mech. Anal., 100 (1987), 1-12. |
[27] |
T. Yang and H. Yu, Hypocoercivity of the relativistic Boltzmann and Landau equations in the whole space, J. Differential Equations, 248 (2010), 1518-1560.
doi: 10.1016/j.jde.2009.11.027. |
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