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

show all references

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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