# American Institute of Mathematical Sciences

May  2015, 14(3): 811-823. doi: 10.3934/cpaa.2015.14.811

## Uniform stability of the Boltzmann equation with an external force near vacuum

 1 Department of Applied Mathematics, Donghua University, Shanghai 201620, China 2 College of Science, University of Shanghai for Science and Technology, Shanghai 200093

Received  August 2010 Revised  January 2015 Published  March 2015

The temporal uniform $L^1(x,v)$ stability of mild solutions for the Boltzmann equation with an external force is considered. We give a unified proof of the stability for two kinds of the forces. Firstly, we extend the soft potential case in [9] to both soft and hard cases. Secondly, we weaken the condition on the force in [13]. Furthermore, we give some new examples satisfying the constructive conditions on the force in [11].
Citation: Zhigang Wu, Wenjun Wang. Uniform stability of the Boltzmann equation with an external force near vacuum. Communications on Pure & Applied Analysis, 2015, 14 (3) : 811-823. doi: 10.3934/cpaa.2015.14.811
##### References:
 [1] R. J. Alonso, Existence of global solutions to the Cauchy problem for the inelastic Boltzmann equation with near-vacuum data,, \emph{Indiana Univ. Math. J.}, 58 (2009), 999.  doi: 10.1512/iumj.2009.58.3506.  Google Scholar [2] L. Arkeryd, Stability in $L^1$ for the spatially homogeneous Boltzmann equation,, \emph{Arch. Rational Mech. Anal.}, 103 (1988), 151.  doi: 10.1007/BF00251506.  Google Scholar [3] N. Bellomo, A. Palczewski and G. Toscani, Mathematical Topics in Nonlinear Kinetic Theory,, World Scientific, (1988).   Google Scholar [4] N. Bellomo and G. Toscani, On the Cauchy problem for the nonlinear Boltzmann equation: Global existence, uniqueness and asymptotic behavior,, \emph{J. Math. Phys.}, 26 (1985), 334.  doi: 10.1063/1.526664.  Google Scholar [5] N. Bellomo, M. Lachowicz, A. Palczewski and G. Toscani, On the initial value problem for the Boltzmann equation with a force term,, \emph{Transport Theory Statist. Phys.}, 18 (1989), 87.  doi: 10.1080/00411458908214500.  Google Scholar [6] C. Cercignani, The Boltzmann Equation and Its Applications,, Springer, (1988).  doi: 10.1007/978-1-4612-1039-9.  Google Scholar [7] C. Cercignani, R. Illner and C. Stoica, On diffusive equilibria in generalized kinetic theory,, \emph{J. Statist. Phys.}, 105 (2001), 337.  doi: 10.1023/A:1012246513712.  Google Scholar [8] M. Chae and S. Y. Ha, Stability estimates of the Boltzmann equation with quantum effects,, \emph{Contin. Mech. Thermodyn., 17 (2006), 511.  doi: 10.1007/s00161-006-0012-y.  Google Scholar [9] C. H. Cheng, Uniform stability of solutions of Boltzmann equation for soft potential with external force,, \emph{J. Math. Anal. Appl.}, 352 (2009), 724.  doi: 10.1016/j.jmaa.2008.11.027.  Google Scholar [10] Y. K. Cho and B. J. Yu, Uniform stability estimates for solutions and their gradients to the Boltzmann equation: A unified approach,, \emph{J. Differ. Eqns.}, 245 (2008), 3615.  doi: 10.1016/j.jde.2008.03.005.  Google Scholar [11] R. J. Duan, T. Yang and C. J. Zhu, Global existence to the Boltzmann equation with external force in infinite vacuum,, \emph{J. Math. Phys.}, 46 (2005).  doi: 10.1063/1.1899985.  Google Scholar [12] R. J. Duan, T. Yang and C. J. Zhu, Boltzmann equation with external force and Vlasov-Poisson-Boltzmann system in infinite vacuum,, \emph{Discrete Contin. Dyn. Syst.}, 16 (2006), 253.  doi: 10.3934/dcds.2006.16.253.  Google Scholar [13] R. J. Duan, T. Yang and C. J. Zhu, $L^1$ and BV-type stability of the Boltzmann equation with external forces,, \emph{J. Differ. Eqns.}, 227 (2006), 1.  doi: 10.1016/j.jde.2006.01.010.  Google Scholar [14] R. J. Duan, M. Zhang and C. J. Zhu, $L^1$ stability for the Vlasov-Poisson-Boltzmann system around vacuum,, \emph{Math. Model Meth. Appl. Sci.}, 16 (2006), 1505.  doi: 10.1142/S0218202506001613.  Google Scholar [15] R. Glassey, The Cauchy Problem in Kinetic Theory,, SIAM 1996., (1996).  doi: 10.1137/1.9781611971477.  Google Scholar [16] R. Glassey, Global solutions to the Cauchy problem for the relativistic Boltzmann equation with near-vacuum data,, \emph{Comm. Math. Phys., 26 (2006), 705.  doi: 10.1007/s00220-006-1522-y.  Google Scholar [17] Y. Guo, The Vlasov-Poisson-Boltzmann system near vacuum,, \emph{Comm. Math. Phys.}, 218 (2001), 293.  doi: 10.1007/s002200100391.  Google Scholar [18] S. Y. Ha, $L^1$ stability of the Boltzmann equation for the hard sphere model,, \emph{Arch. Rational Mech. Anal., 173 (2004), 25.  doi: 10.1007/s00205-004-0321-x.  Google Scholar [19] S. Y. Ha, Nonlinear functionals of the Boltzmann equation and uniform stability estimates,, \emph{J. Differ. Eqns., 215 (2005), 178.  doi: 10.1016/j.jde.2004.07.022.  Google Scholar [20] K. Hamdache, Thèse de doctorat d'état de Paris VI,, 1986., ().   Google Scholar [21] K. Hamdache, Existence in the large and asymptotic behaviour for the Boltzmann equation,, \emph{Japan. J. Appl. Math.}, 2 (1984), 1.  doi: 10.1007/BF03167035.  Google Scholar [22] R. Illner and M. Shinbrot, The Boltzmann equation, global existence for a rare gas in an infinite vacuum,, \emph{Comm. Math. Phys.}, 95 (1984), 217.   Google Scholar [23] S. Kaniel and M. Shinbrot, The Boltzmann equation: I. Uniqueness and local existence,, \emph{Comm. Math. Phys.}, 58 (1978), 65.   Google Scholar [24] X. G. Lu, Spatial decay solutions of the Boltzmann equation: converse properties of long time limiting behavior,, \emph{SIAM J. Math. Anal.}, 30 (1999), 1151.  doi: 10.1137/S0036141098334985.  Google Scholar [25] J. Polewczak, Classical solution of the nonlinear Boltzmann equation in all $R^3$: asymptotic behavior of solutions,, \emph{J. Stat. Phys.}, 50 (1988), 611.  doi: 10.1007/BF01026493.  Google Scholar [26] M. Tabata and N. Eshima, Decay of solutions to the mixed problem for the linearized Boltzmann equation with a potential term in a polyhedral bounded domain,, \emph{Rend. Sem. Mat. Univ. Padova}, 103 (2000), 133.   Google Scholar [27] G. Toscani, H-theorem and asymptotic trend of the solution for a rarefied gas in a vacuum,, \emph{Arch. Rational Mech. Anal., 102 (1988), 231.  doi: 10.1007/BF00281348.  Google Scholar [28] Z. G. Wu, $L^1$ and BV-type stability of the inelastic Boltzmann equation near vacuum,, \emph{Continuum Mech. Thermodyn.}, 22 (2010), 239.  doi: 10.1007/s00161-009-0127-z.  Google Scholar

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##### References:
 [1] R. J. Alonso, Existence of global solutions to the Cauchy problem for the inelastic Boltzmann equation with near-vacuum data,, \emph{Indiana Univ. Math. J.}, 58 (2009), 999.  doi: 10.1512/iumj.2009.58.3506.  Google Scholar [2] L. Arkeryd, Stability in $L^1$ for the spatially homogeneous Boltzmann equation,, \emph{Arch. Rational Mech. Anal.}, 103 (1988), 151.  doi: 10.1007/BF00251506.  Google Scholar [3] N. Bellomo, A. Palczewski and G. Toscani, Mathematical Topics in Nonlinear Kinetic Theory,, World Scientific, (1988).   Google Scholar [4] N. Bellomo and G. Toscani, On the Cauchy problem for the nonlinear Boltzmann equation: Global existence, uniqueness and asymptotic behavior,, \emph{J. Math. Phys.}, 26 (1985), 334.  doi: 10.1063/1.526664.  Google Scholar [5] N. Bellomo, M. Lachowicz, A. Palczewski and G. Toscani, On the initial value problem for the Boltzmann equation with a force term,, \emph{Transport Theory Statist. Phys.}, 18 (1989), 87.  doi: 10.1080/00411458908214500.  Google Scholar [6] C. Cercignani, The Boltzmann Equation and Its Applications,, Springer, (1988).  doi: 10.1007/978-1-4612-1039-9.  Google Scholar [7] C. Cercignani, R. Illner and C. Stoica, On diffusive equilibria in generalized kinetic theory,, \emph{J. Statist. Phys.}, 105 (2001), 337.  doi: 10.1023/A:1012246513712.  Google Scholar [8] M. Chae and S. Y. Ha, Stability estimates of the Boltzmann equation with quantum effects,, \emph{Contin. Mech. Thermodyn., 17 (2006), 511.  doi: 10.1007/s00161-006-0012-y.  Google Scholar [9] C. H. Cheng, Uniform stability of solutions of Boltzmann equation for soft potential with external force,, \emph{J. Math. Anal. Appl.}, 352 (2009), 724.  doi: 10.1016/j.jmaa.2008.11.027.  Google Scholar [10] Y. K. Cho and B. J. Yu, Uniform stability estimates for solutions and their gradients to the Boltzmann equation: A unified approach,, \emph{J. Differ. Eqns.}, 245 (2008), 3615.  doi: 10.1016/j.jde.2008.03.005.  Google Scholar [11] R. J. Duan, T. Yang and C. J. Zhu, Global existence to the Boltzmann equation with external force in infinite vacuum,, \emph{J. Math. Phys.}, 46 (2005).  doi: 10.1063/1.1899985.  Google Scholar [12] R. J. Duan, T. Yang and C. J. Zhu, Boltzmann equation with external force and Vlasov-Poisson-Boltzmann system in infinite vacuum,, \emph{Discrete Contin. Dyn. Syst.}, 16 (2006), 253.  doi: 10.3934/dcds.2006.16.253.  Google Scholar [13] R. J. Duan, T. Yang and C. J. Zhu, $L^1$ and BV-type stability of the Boltzmann equation with external forces,, \emph{J. Differ. Eqns.}, 227 (2006), 1.  doi: 10.1016/j.jde.2006.01.010.  Google Scholar [14] R. J. Duan, M. Zhang and C. J. Zhu, $L^1$ stability for the Vlasov-Poisson-Boltzmann system around vacuum,, \emph{Math. Model Meth. Appl. Sci.}, 16 (2006), 1505.  doi: 10.1142/S0218202506001613.  Google Scholar [15] R. Glassey, The Cauchy Problem in Kinetic Theory,, SIAM 1996., (1996).  doi: 10.1137/1.9781611971477.  Google Scholar [16] R. Glassey, Global solutions to the Cauchy problem for the relativistic Boltzmann equation with near-vacuum data,, \emph{Comm. Math. Phys., 26 (2006), 705.  doi: 10.1007/s00220-006-1522-y.  Google Scholar [17] Y. Guo, The Vlasov-Poisson-Boltzmann system near vacuum,, \emph{Comm. Math. Phys.}, 218 (2001), 293.  doi: 10.1007/s002200100391.  Google Scholar [18] S. Y. Ha, $L^1$ stability of the Boltzmann equation for the hard sphere model,, \emph{Arch. Rational Mech. Anal., 173 (2004), 25.  doi: 10.1007/s00205-004-0321-x.  Google Scholar [19] S. Y. Ha, Nonlinear functionals of the Boltzmann equation and uniform stability estimates,, \emph{J. Differ. Eqns., 215 (2005), 178.  doi: 10.1016/j.jde.2004.07.022.  Google Scholar [20] K. Hamdache, Thèse de doctorat d'état de Paris VI,, 1986., ().   Google Scholar [21] K. Hamdache, Existence in the large and asymptotic behaviour for the Boltzmann equation,, \emph{Japan. J. Appl. Math.}, 2 (1984), 1.  doi: 10.1007/BF03167035.  Google Scholar [22] R. Illner and M. Shinbrot, The Boltzmann equation, global existence for a rare gas in an infinite vacuum,, \emph{Comm. Math. Phys.}, 95 (1984), 217.   Google Scholar [23] S. Kaniel and M. Shinbrot, The Boltzmann equation: I. Uniqueness and local existence,, \emph{Comm. Math. Phys.}, 58 (1978), 65.   Google Scholar [24] X. G. Lu, Spatial decay solutions of the Boltzmann equation: converse properties of long time limiting behavior,, \emph{SIAM J. Math. Anal.}, 30 (1999), 1151.  doi: 10.1137/S0036141098334985.  Google Scholar [25] J. Polewczak, Classical solution of the nonlinear Boltzmann equation in all $R^3$: asymptotic behavior of solutions,, \emph{J. Stat. Phys.}, 50 (1988), 611.  doi: 10.1007/BF01026493.  Google Scholar [26] M. Tabata and N. Eshima, Decay of solutions to the mixed problem for the linearized Boltzmann equation with a potential term in a polyhedral bounded domain,, \emph{Rend. Sem. Mat. Univ. Padova}, 103 (2000), 133.   Google Scholar [27] G. Toscani, H-theorem and asymptotic trend of the solution for a rarefied gas in a vacuum,, \emph{Arch. Rational Mech. Anal., 102 (1988), 231.  doi: 10.1007/BF00281348.  Google Scholar [28] Z. G. Wu, $L^1$ and BV-type stability of the inelastic Boltzmann equation near vacuum,, \emph{Continuum Mech. Thermodyn.}, 22 (2010), 239.  doi: 10.1007/s00161-009-0127-z.  Google Scholar
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