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March  2011, 10(2): 435-458. doi: 10.3934/cpaa.2011.10.435

The Boltzmann equation near Maxwellian in the whole space

1. 

College of Mathematics and Information Science, Henan Normal University, 453007, Xinxiang, China

Received  April 2010 Revised  October 2010 Published  December 2010

A recent nonlinear energy method introduced in [19, 20] leads to another construction global solutions near Maxwellian for the Boltzmann equation over the whole space. Moreover, the optimal time decay, uniform stability and the optimal time stability of the solutions to the Boltzmann equation are all obtained via such a energy method.
Citation: Xinkuan Chai. The Boltzmann equation near Maxwellian in the whole space. Communications on Pure and Applied Analysis, 2011, 10 (2) : 435-458. doi: 10.3934/cpaa.2011.10.435
References:
[1]

C. Cercignani, R. Illner and M. Pulvirenti, "The Mathematical Theory of Dilute Gases," Applied Mathematical Sciences, New York, Springer-Verlag, 1994.

[2]

R.-J. Duan, On the Cauchy problem for the Boltzmann equation in the whole space: Global existence and uniform stability in $L^2_\xi(H^N_x)$, J. Differential Equations, 228 (2008), 641-660.

[3]

R.-J. Duan, S. Ukai, T. Yang and H.-J. Zhao, Optimal decay estimates on the linearized Boltzmann equation with time dependent force and their applications, Comm. Math. Phys., 277 (2008), 189-236. doi: doi:10.1007/s00220-007-0366-4.

[4]

R. T. Glassey, "The Cauchy Problem in Kinetic Theory," Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996.

[5]

Y. Guo, The Vlasov-Poisson-Boltzmann system near Maxwellians, Comm. Pure Appl. Math., 55 (2002), 1104-1135. doi: doi:10.1002/cpa.10040.

[6]

Y. Guo, The Vlasov-Maxwell-Boltzmann system near Maxwellians, Invent. Math., 153 (2003), 593-630. doi: doi:10.1007/s00222-003-0301-z.

[7]

Y. Guo, The Boltzmann equation in the whole space, Indiann Univ. Math. J., 53 (2004), 1081-1194. doi: doi:10.1512/iumj.2004.53.2574.

[8]

Y. Guo, Boltzmann diffusive limit beyond the Navier-Stokes approximation, Comm. Pure Appl. Math., 59 (2006), 626-687. doi: doi:10.1002/cpa.20121.

[9]

L. Hsiao and H.-J. Yu, Global classical solutions to the initial value problem for the relativistic Landau equation, J. Differential Equations, 228 (2006), 641-660. doi: doi:10.1016/j.jde.2005.10.022.

[10]

L. Hsiao and H.-J. Yu, On the Cauchy problem of the Boltzmann and Landau equations with soft potentials, Quart. Appl. Math., 65 (2007), 281-315.

[11]

S. Kawashima, The Boltzmann equation and thirteen moments, Japan J. Appl. Math., 7 (1990), 301-320. doi: doi:10.1007/BF03167846.

[12]

T.-P. Liu, T. Yang and S.-H. Yu, Energy method for the Boltzmann equation, Physica D, 188 (2004), 178-192. doi: doi:10.1016/j.physd.2003.07.011.

[13]

T.-P. Liu and S.-H. Yu, Boltzmann equation: micro-macro decompositions and positivity of shock profiles, Comm. Math. Phys., 246 (2004), 133-179. doi: doi:10.1007/s00220-003-1030-2.

[14]

T. Nishida, and K. Imai, Global solutions to the initial value problem for the nonlinear Boltzmann equation,, Publ. Res. Inst. Math. Sci., 12 (): 229. 

[15]

Y. Shizuta, On the classical solutions of the Boltzmann equation, Comm. Pure Appl. Math., 36 (1983), 705-754. doi: doi:10.1002/cpa.3160360602.

[16]

S. Ukai, On the existence of global solutions of mixed problem for non-linear Boltzmann equation, Proc. Japan Acad., 50 (1974), 179-184. doi: doi:10.3792/pja/1195519027.

[17]

S. Ukai, Les solutions globale de l'équation de Boltzmann dans l'espace tout entier et dans le demi-espace, C. R. Acad. Sci. Paris Ser. A, 282 (1976), 317-320.

[18]

S. Ukai, and T. Yang, Mathematical theory of Boltzmann equation, Lecture Notes Series-No. 8, Hongkong: Liu Bie Ju Center of Mathematical Sciences, City University of Hongkong, 2006.

[19]

T. Yang and H.-J. Yu, Hypocoercivity of the relativistic Boltzmann and Landau equations in the whole space, J. Differential Equations, 248 (2010), 1518-1560. doi: doi:10.1016/j.jde.2009.11.027.

[20]

T. Yang and H.-J. Yu, Optimal convergence rates of Landau equation with external forcing in the whole space, Acta Mathematica Scientia, 29 B (2009), 1035-1062.

[21]

T. Yang, H.-J. Yu and H.-J. Zhao, Cauchy problem for the Vlasov-Poisson-Boltzmann system, Arch. Ration. Mech. Anal., 182 (2006), 415-470. doi: doi:10.1007/s00205-006-0009-5.

[22]

T. Yang and H.-J. Zhao, Global existence of classical solutions to the Vlasov-Poisson-Boltzmann system, Comm. Math. Phys., 268 (2006), 569-605. doi: doi:10.1007/s00220-006-0103-4.

[23]

T. Yang, and H.-J. Zhao, A new energy method for the Boltzmann equation, J. Math. Phys., 47 (2006), 053301, 19 pp.

[24]

H.-J. Yu, Existence and exponential decay of global solution to the Boltzmann equation near Maxwellians, Math. Mod. Meth. Appl. Sci., 15 (2005), 483-505. doi: doi:10.1142/S0218202505000443.

[25]

H.-J. Yu, $H^N$ stability of the Vlasov-Poisson-Boltzmann system near Maxwellians, Proc. Royal. Soc. Edinburgh, 137A (2007), 431-446. doi: doi:10.1017/S0308210505001186.

show all references

References:
[1]

C. Cercignani, R. Illner and M. Pulvirenti, "The Mathematical Theory of Dilute Gases," Applied Mathematical Sciences, New York, Springer-Verlag, 1994.

[2]

R.-J. Duan, On the Cauchy problem for the Boltzmann equation in the whole space: Global existence and uniform stability in $L^2_\xi(H^N_x)$, J. Differential Equations, 228 (2008), 641-660.

[3]

R.-J. Duan, S. Ukai, T. Yang and H.-J. Zhao, Optimal decay estimates on the linearized Boltzmann equation with time dependent force and their applications, Comm. Math. Phys., 277 (2008), 189-236. doi: doi:10.1007/s00220-007-0366-4.

[4]

R. T. Glassey, "The Cauchy Problem in Kinetic Theory," Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996.

[5]

Y. Guo, The Vlasov-Poisson-Boltzmann system near Maxwellians, Comm. Pure Appl. Math., 55 (2002), 1104-1135. doi: doi:10.1002/cpa.10040.

[6]

Y. Guo, The Vlasov-Maxwell-Boltzmann system near Maxwellians, Invent. Math., 153 (2003), 593-630. doi: doi:10.1007/s00222-003-0301-z.

[7]

Y. Guo, The Boltzmann equation in the whole space, Indiann Univ. Math. J., 53 (2004), 1081-1194. doi: doi:10.1512/iumj.2004.53.2574.

[8]

Y. Guo, Boltzmann diffusive limit beyond the Navier-Stokes approximation, Comm. Pure Appl. Math., 59 (2006), 626-687. doi: doi:10.1002/cpa.20121.

[9]

L. Hsiao and H.-J. Yu, Global classical solutions to the initial value problem for the relativistic Landau equation, J. Differential Equations, 228 (2006), 641-660. doi: doi:10.1016/j.jde.2005.10.022.

[10]

L. Hsiao and H.-J. Yu, On the Cauchy problem of the Boltzmann and Landau equations with soft potentials, Quart. Appl. Math., 65 (2007), 281-315.

[11]

S. Kawashima, The Boltzmann equation and thirteen moments, Japan J. Appl. Math., 7 (1990), 301-320. doi: doi:10.1007/BF03167846.

[12]

T.-P. Liu, T. Yang and S.-H. Yu, Energy method for the Boltzmann equation, Physica D, 188 (2004), 178-192. doi: doi:10.1016/j.physd.2003.07.011.

[13]

T.-P. Liu and S.-H. Yu, Boltzmann equation: micro-macro decompositions and positivity of shock profiles, Comm. Math. Phys., 246 (2004), 133-179. doi: doi:10.1007/s00220-003-1030-2.

[14]

T. Nishida, and K. Imai, Global solutions to the initial value problem for the nonlinear Boltzmann equation,, Publ. Res. Inst. Math. Sci., 12 (): 229. 

[15]

Y. Shizuta, On the classical solutions of the Boltzmann equation, Comm. Pure Appl. Math., 36 (1983), 705-754. doi: doi:10.1002/cpa.3160360602.

[16]

S. Ukai, On the existence of global solutions of mixed problem for non-linear Boltzmann equation, Proc. Japan Acad., 50 (1974), 179-184. doi: doi:10.3792/pja/1195519027.

[17]

S. Ukai, Les solutions globale de l'équation de Boltzmann dans l'espace tout entier et dans le demi-espace, C. R. Acad. Sci. Paris Ser. A, 282 (1976), 317-320.

[18]

S. Ukai, and T. Yang, Mathematical theory of Boltzmann equation, Lecture Notes Series-No. 8, Hongkong: Liu Bie Ju Center of Mathematical Sciences, City University of Hongkong, 2006.

[19]

T. Yang and H.-J. Yu, Hypocoercivity of the relativistic Boltzmann and Landau equations in the whole space, J. Differential Equations, 248 (2010), 1518-1560. doi: doi:10.1016/j.jde.2009.11.027.

[20]

T. Yang and H.-J. Yu, Optimal convergence rates of Landau equation with external forcing in the whole space, Acta Mathematica Scientia, 29 B (2009), 1035-1062.

[21]

T. Yang, H.-J. Yu and H.-J. Zhao, Cauchy problem for the Vlasov-Poisson-Boltzmann system, Arch. Ration. Mech. Anal., 182 (2006), 415-470. doi: doi:10.1007/s00205-006-0009-5.

[22]

T. Yang and H.-J. Zhao, Global existence of classical solutions to the Vlasov-Poisson-Boltzmann system, Comm. Math. Phys., 268 (2006), 569-605. doi: doi:10.1007/s00220-006-0103-4.

[23]

T. Yang, and H.-J. Zhao, A new energy method for the Boltzmann equation, J. Math. Phys., 47 (2006), 053301, 19 pp.

[24]

H.-J. Yu, Existence and exponential decay of global solution to the Boltzmann equation near Maxwellians, Math. Mod. Meth. Appl. Sci., 15 (2005), 483-505. doi: doi:10.1142/S0218202505000443.

[25]

H.-J. Yu, $H^N$ stability of the Vlasov-Poisson-Boltzmann system near Maxwellians, Proc. Royal. Soc. Edinburgh, 137A (2007), 431-446. doi: doi:10.1017/S0308210505001186.

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