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Uniform stability of the Boltzmann equation with an external force near vacuum. (English) Zbl 1421.35239

This study is based on a equation of Boltzmann type, which is one of the most important equations in statistical mechanics, which gives a description of the real behavior of a system of classical particles interacting by short-range forces. As a first result of the paper, the authors obtain the uniform stability of two mild solutions for the Boltzmann equation with an external force for both soft and hard potentials. The idea was “borrowed” from a paper of Y.-K. Cho and B.-J. Yu [J. Differ. Equations 245, No. 12, 3615–3627 (2008; Zbl 1170.35017)] in the references section, where the authors introduced a new method to obtain the uniform stability of solutions for the Boltzmann equation without a force. It is interesting to note that for this result, the authors did not use the Gronwall’s inequality. In the second result of the paper, by using the same method, the authors obtain the stability under a weaker condition than the original condition. Also one can find some details on the force and some examples of the external force satisfying the imposed constructive condition.

MSC:

35Q20 Boltzmann equations
37A60 Dynamical aspects of statistical mechanics
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B05 Classical equilibrium statistical mechanics (general)
82C40 Kinetic theory of gases in time-dependent statistical mechanics

Citations:

Zbl 1170.35017
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Full Text: DOI

References:

[1] R. J. Alonso, Existence of global solutions to the Cauchy problem for the inelastic Boltzmann equation with near-vacuum data,, Indiana Univ. Math. J., 58, 999 (2009) · Zbl 1168.76047 · doi:10.1512/iumj.2009.58.3506
[2] L. Arkeryd, Stability in \(L^1\) for the spatially homogeneous Boltzmann equation,, Arch. Rational Mech. Anal., 103, 151 (1988) · Zbl 0654.76074 · doi:10.1007/BF00251506
[3] N. Bellomo, <em>Mathematical Topics in Nonlinear Kinetic Theory</em>,, World Scientific (1988) · Zbl 0702.76005
[4] N. Bellomo, On the Cauchy problem for the nonlinear Boltzmann equation: Global existence, uniqueness and asymptotic behavior,, J. Math. Phys., 26, 334 (1985) · Zbl 0561.35074 · doi:10.1063/1.526664
[5] N. Bellomo, On the initial value problem for the Boltzmann equation with a force term,, Transport Theory Statist. Phys., 18, 87 (1989) · Zbl 0699.35237 · doi:10.1080/00411458908214500
[6] C. Cercignani, <em>The Boltzmann Equation and Its Applications</em>,, Springer (1988) · Zbl 0646.76001 · doi:10.1007/978-1-4612-1039-9
[7] C. Cercignani, On diffusive equilibria in generalized kinetic theory,, J. Statist. Phys., 105, 337 (2001) · Zbl 1051.82021 · doi:10.1023/A:1012246513712
[8] M. Chae, Stability estimates of the Boltzmann equation with quantum effects,, Contin. Mech. Thermodyn., 17, 511 (2006) · Zbl 1113.76083 · doi:10.1007/s00161-006-0012-y
[9] C. H. Cheng, Uniform stability of solutions of Boltzmann equation for soft potential with external force,, J. Math. Anal. Appl., 352, 724 (2009) · Zbl 1160.35496 · doi:10.1016/j.jmaa.2008.11.027
[10] Y. K. Cho, Uniform stability estimates for solutions and their gradients to the Boltzmann equation: A unified approach,, J. Differ. Eqns., 245, 3615 (2008) · Zbl 1170.35017 · doi:10.1016/j.jde.2008.03.005
[11] R. J. Duan, Global existence to the Boltzmann equation with external force in infinite vacuum,, J. Math. Phys., 46 (2005) · Zbl 1110.82034 · doi:10.1063/1.1899985
[12] R. J. Duan, Boltzmann equation with external force and Vlasov-Poisson-Boltzmann system in infinite vacuum,, Discrete Contin. Dyn. Syst., 16, 253 (2006) · Zbl 1185.35035 · doi:10.3934/dcds.2006.16.253
[13] R. J. Duan, \(L^1\) and BV-type stability of the Boltzmann equation with external forces,, J. Differ. Eqns., 227, 1 (2006) · Zbl 1101.76051 · doi:10.1016/j.jde.2006.01.010
[14] R. J. Duan, \(L^1\) stability for the Vlasov-Poisson-Boltzmann system around vacuum,, Math. Model Meth. Appl. Sci., 16, 1505 (2006) · Zbl 1096.76050 · doi:10.1142/S0218202506001613
[15] R. Glassey, <em>The Cauchy Problem in Kinetic Theory</em>,, SIAM 1996. (1996) · Zbl 0858.76001 · doi:10.1137/1.9781611971477
[16] R. Glassey, Global solutions to the Cauchy problem for the relativistic Boltzmann equation with near-vacuum data,, Comm. Math. Phys., 26, 705 (2006) · Zbl 1107.82047 · doi:10.1007/s00220-006-1522-y
[17] Y. Guo, The Vlasov-Poisson-Boltzmann system near vacuum,, Comm. Math. Phys., 218, 293 (2001) · Zbl 0981.35057 · doi:10.1007/s002200100391
[18] S. Y. Ha, \(L^1\) stability of the Boltzmann equation for the hard sphere model,, Arch. Rational Mech. Anal., 173, 25 (2004) · Zbl 1063.76085 · doi:10.1007/s00205-004-0321-x
[19] S. Y. Ha, Nonlinear functionals of the Boltzmann equation and uniform stability estimates,, J. Differ. Eqns., 215, 178 (2005) · Zbl 1069.76048 · doi:10.1016/j.jde.2004.07.022
[20] K. Hamdache, Thèse de doctorat d’état de Paris VI,, 1986.
[21] K. Hamdache, Existence in the large and asymptotic behaviour for the Boltzmann equation,, Japan. J. Appl. Math., 2, 1 (1984) · Zbl 0653.76053 · doi:10.1007/BF03167035
[22] R. Illner, The Boltzmann equation, global existence for a rare gas in an infinite vacuum,, Comm. Math. Phys., 95, 217 (1984) · Zbl 0599.76088
[23] S. Kaniel, The Boltzmann equation: I. Uniqueness and local existence,, Comm. Math. Phys., 58, 65 (1978) · Zbl 0371.76061
[24] X. G. Lu, Spatial decay solutions of the Boltzmann equation: converse properties of long time limiting behavior,, SIAM J. Math. Anal., 30, 1151 (1999) · Zbl 0948.76072 · doi:10.1137/S0036141098334985
[25] J. Polewczak, Classical solution of the nonlinear Boltzmann equation in all \(R^3\): asymptotic behavior of solutions,, J. Stat. Phys., 50, 611 (1988) · Zbl 1084.82545 · doi:10.1007/BF01026493
[26] M. Tabata, Decay of solutions to the mixed problem for the linearized Boltzmann equation with a potential term in a polyhedral bounded domain,, Rend. Sem. Mat. Univ. Padova, 103, 133 (2000) · Zbl 0982.45006
[27] G. Toscani, H-theorem and asymptotic trend of the solution for a rarefied gas in a vacuum,, Arch. Rational Mech. Anal., 102, 231 (1988) · Zbl 0665.76085 · doi:10.1007/BF00281348
[28] Z. G. Wu, \(L^1\) and BV-type stability of the inelastic Boltzmann equation near vacuum,, Continuum Mech. Thermodyn., 22, 239 (2010) · Zbl 1275.76181 · doi:10.1007/s00161-009-0127-z
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