Uniform L^1-stability of the relativistic Boltzmann equation near vacuum

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March 2013, 12(2): 1141-1161. doi: 10.3934/cpaa.2013.12.1141

Uniform $L^1$-stability of the relativistic Boltzmann equation near vacuum

Seung-Yeal Ha ^1, , Eunhee Jeong ^1, and Robert M. Strain ^2,

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

Received September 2011 Revised February 2012 Published September 2012

Abstract
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We present the uniform $L^1$-stability estimate for the relativistic Boltzmann equation near vacuum. For this, we explicitly construct a relativistic counterpart of the nonlinear functional which is a linear combination of $L^1$-distance and a collision potential. This functional measures the $L^1$-distance between two continuous mild solutions. When the initial data is sufficiently small and decays exponentially fast, we show that the functional satisfies the uniform stability estimate leading to the uniform $L^1$-stability estimate with respect to initial data.

Keywords: Lyapunov functional., asymptotic completeness, The relativistic Boltzmann equation, $L^1$-stability.

Mathematics Subject Classification: 35Q3.

Citation: Seung-Yeal Ha, Eunhee Jeong, Robert M. Strain. Uniform $L^1$-stability of the relativistic Boltzmann equation near vacuum. Communications on Pure and Applied Analysis, 2013, 12 (2) : 1141-1161. doi: 10.3934/cpaa.2013.12.1141

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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