# American Institute of Mathematical Sciences

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

## 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

Received  September 2011 Revised  February 2012 Published  September 2012

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.
Citation: Seung-Yeal Ha, Eunhee Jeong, Robert M. Strain. Uniform $L^1$-stability of the relativistic Boltzmann equation near vacuum. Communications on Pure & 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. doi: 10.1007/s10955-009-9873-3. Google Scholar [2] D. Bancel, Problème de Cauchy pour l'équation de Boltzmann en relativité générale,, Ann. Inst. Henri Poincare\'e, XVIII 3 (1973), 263. Google Scholar [3] D. Bancel and Y. Choquet-Bruhat, Uniqureness and local stability for the Einstein-Maxwell-Boltzmann system,, Comm. Math. Phys., 33 (1973), 83. Google Scholar [4] K. Bichteler, On the Cauchy problem of the relativistic Boltzmann equation,, Comm. Math. Phys., 4 (1967), 352. Google Scholar [5] S. Calogero, The Newtonian limit of the relativistic Botlzmann equation,, J. Math. Phys., 45 (2004), 4042. doi: 10.1063/1.1793328. Google Scholar [6] M. Dudyński and M. Ekiel Jezewska, On the linearized Relativistic Boltzmann equation,, Comm. Math. Phys., 115 (1988), 607. Google Scholar [7] M. Dudyński and M. Ekiel Jezewska, Global existence proof for relativistic Boltzmann equation,, J. Stat. Phys., 66 (1992), 991. Google Scholar [8] M. Dudyński and M. Ekiel Jezewska, The relativistic Boltzmann equation-mathematical and physical aspects,, J. Tech. Phys., 48 (2007), 39. Google Scholar [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. Google Scholar [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. doi: 10.1007/s00220-006-1522-y. Google Scholar [11] R. T. Glassey and W. Strauss, Asymptotic stability of the relativistic Maxwellian via fourteen moments,, Trans. Th. Stat. Phys., 24 (1995), 657. Google Scholar [12] R. T. Glassey and W. Strauss, Asymptotic stability of the relativistic Maxwellian,, Publ. R.I.M.S. Kyoto Univ., 29 (1993), 301. Google Scholar [13] J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations,, Comm. Pure Appl. Math., 18 (1965), 697. Google Scholar [14] S.-Y. Ha, Nonlinear functionals of the Boltzmann equation and uniform stability estimates,, J. Differential Equations, 215 (2005), 178. doi: 10.1016/j.jde.2004.07.022. Google Scholar [15] S.-Y Ha, $L_1$-stability of the Boltzmann equation for the hard-sphere model,, Arch. Ration. Mech. Anal., 173 (2004), 279. doi: 10.1007/s00205-004-0321-x. Google Scholar [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. Google Scholar [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. doi: 10.1016/j.physd.2006.06.011. Google Scholar [18] L. Hsiao and H. Yu, Asymptotic stability of the relativistic Maxwellian,, Math. Methods Appl. Sci., 29 (2006), 1481. doi: 10.1002/mma.736. Google Scholar [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. Google Scholar [20] S. Kaniel and M. Shinbrot, The Boltzmann equation 1. Uniqueness and local existence,, Comm. Math. Phys., 58 (1978), 65. Google Scholar [21] A. Lichnerowich and R. Marrot, Propriés statistiques des ensembles de particules en relativité restreinte,, F. R. Acad. Sci. Paris, 210 (1940), 759. Google Scholar [22] J. Polewczak, Classical Solution of the nonlinear Boltzmann equation in all $\bbr^3$ Asymptotic behavior of solutions,, J. Stat. Phys., 50 (1988), 611. Google Scholar [23] R. M. Strain, Global newtonian limit for the relativistic Boltzmann equation near vacuum,, SIAM J. Math. Anal., 42 (2010), 1568. doi: 10.1137/090762695. Google Scholar [24] R. M. Strain, Coordinates in the relativistic Boltzmann theory,, Kinetic and Related Models, 4 (2011), 345. doi: 10.3934/krm.2011.4.345. Google Scholar [25] R. M. Strain and K. Zhu, Large-time decay of the soft potential relativistic Boltzmann equation in $\bbr^3_x$,, Kinetic and Related Models, 5 (2012), 383. doi: 3934/krm.2012.5.383. Google Scholar [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. Google Scholar [27] T. Yang and H. Yu, Hypocoercivity of the relativistic Boltzmann and Landau equations in the whole space,, J. Differential Equations, 248 (2010), 1518. doi: 10.1016/j.jde.2009.11.027. Google Scholar

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##### 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. doi: 10.1007/s10955-009-9873-3. Google Scholar [2] D. Bancel, Problème de Cauchy pour l'équation de Boltzmann en relativité générale,, Ann. Inst. Henri Poincare\'e, XVIII 3 (1973), 263. Google Scholar [3] D. Bancel and Y. Choquet-Bruhat, Uniqureness and local stability for the Einstein-Maxwell-Boltzmann system,, Comm. Math. Phys., 33 (1973), 83. Google Scholar [4] K. Bichteler, On the Cauchy problem of the relativistic Boltzmann equation,, Comm. Math. Phys., 4 (1967), 352. Google Scholar [5] S. Calogero, The Newtonian limit of the relativistic Botlzmann equation,, J. Math. Phys., 45 (2004), 4042. doi: 10.1063/1.1793328. Google Scholar [6] M. Dudyński and M. Ekiel Jezewska, On the linearized Relativistic Boltzmann equation,, Comm. Math. Phys., 115 (1988), 607. Google Scholar [7] M. Dudyński and M. Ekiel Jezewska, Global existence proof for relativistic Boltzmann equation,, J. Stat. Phys., 66 (1992), 991. Google Scholar [8] M. Dudyński and M. Ekiel Jezewska, The relativistic Boltzmann equation-mathematical and physical aspects,, J. Tech. Phys., 48 (2007), 39. Google Scholar [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. Google Scholar [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. doi: 10.1007/s00220-006-1522-y. Google Scholar [11] R. T. Glassey and W. Strauss, Asymptotic stability of the relativistic Maxwellian via fourteen moments,, Trans. Th. Stat. Phys., 24 (1995), 657. Google Scholar [12] R. T. Glassey and W. Strauss, Asymptotic stability of the relativistic Maxwellian,, Publ. R.I.M.S. Kyoto Univ., 29 (1993), 301. Google Scholar [13] J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations,, Comm. Pure Appl. Math., 18 (1965), 697. Google Scholar [14] S.-Y. Ha, Nonlinear functionals of the Boltzmann equation and uniform stability estimates,, J. Differential Equations, 215 (2005), 178. doi: 10.1016/j.jde.2004.07.022. Google Scholar [15] S.-Y Ha, $L_1$-stability of the Boltzmann equation for the hard-sphere model,, Arch. Ration. Mech. Anal., 173 (2004), 279. doi: 10.1007/s00205-004-0321-x. Google Scholar [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. Google Scholar [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. doi: 10.1016/j.physd.2006.06.011. Google Scholar [18] L. Hsiao and H. Yu, Asymptotic stability of the relativistic Maxwellian,, Math. Methods Appl. Sci., 29 (2006), 1481. doi: 10.1002/mma.736. Google Scholar [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. Google Scholar [20] S. Kaniel and M. Shinbrot, The Boltzmann equation 1. Uniqueness and local existence,, Comm. Math. Phys., 58 (1978), 65. Google Scholar [21] A. Lichnerowich and R. Marrot, Propriés statistiques des ensembles de particules en relativité restreinte,, F. R. Acad. Sci. Paris, 210 (1940), 759. Google Scholar [22] J. Polewczak, Classical Solution of the nonlinear Boltzmann equation in all $\bbr^3$ Asymptotic behavior of solutions,, J. Stat. Phys., 50 (1988), 611. Google Scholar [23] R. M. Strain, Global newtonian limit for the relativistic Boltzmann equation near vacuum,, SIAM J. Math. Anal., 42 (2010), 1568. doi: 10.1137/090762695. Google Scholar [24] R. M. Strain, Coordinates in the relativistic Boltzmann theory,, Kinetic and Related Models, 4 (2011), 345. doi: 10.3934/krm.2011.4.345. Google Scholar [25] R. M. Strain and K. Zhu, Large-time decay of the soft potential relativistic Boltzmann equation in $\bbr^3_x$,, Kinetic and Related Models, 5 (2012), 383. doi: 3934/krm.2012.5.383. Google Scholar [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. Google Scholar [27] T. Yang and H. Yu, Hypocoercivity of the relativistic Boltzmann and Landau equations in the whole space,, J. Differential Equations, 248 (2010), 1518. doi: 10.1016/j.jde.2009.11.027. Google Scholar
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