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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 & 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, (1994). Google Scholar

[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. Google Scholar

[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. doi: doi:10.1007/s00220-007-0366-4. Google Scholar

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[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. doi: doi:10.1016/j.jde.2005.10.022. Google Scholar

[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. Google Scholar

[11]

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

[12]

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

[13]

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

[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. Google Scholar

[15]

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

[16]

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

[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. Google Scholar

[18]

S. Ukai, and T. Yang, Mathematical theory of Boltzmann equation,, Lecture Notes Series-No. \textbf{8}, 8 (2006). Google Scholar

[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. doi: doi:10.1016/j.jde.2009.11.027. Google Scholar

[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. Google Scholar

[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. doi: doi:10.1007/s00205-006-0009-5. Google Scholar

[22]

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

[23]

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

[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. doi: doi:10.1142/S0218202505000443. Google Scholar

[25]

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

show all references

References:
[1]

C. Cercignani, R. Illner and M. Pulvirenti, "The Mathematical Theory of Dilute Gases,", Applied Mathematical Sciences, (1994). Google Scholar

[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. Google Scholar

[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. doi: doi:10.1007/s00220-007-0366-4. Google Scholar

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[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. doi: doi:10.1016/j.jde.2005.10.022. Google Scholar

[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. Google Scholar

[11]

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

[12]

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

[13]

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

[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. Google Scholar

[15]

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

[16]

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

[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. Google Scholar

[18]

S. Ukai, and T. Yang, Mathematical theory of Boltzmann equation,, Lecture Notes Series-No. \textbf{8}, 8 (2006). Google Scholar

[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. doi: doi:10.1016/j.jde.2009.11.027. Google Scholar

[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. Google Scholar

[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. doi: doi:10.1007/s00205-006-0009-5. Google Scholar

[22]

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

[23]

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

[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. doi: doi:10.1142/S0218202505000443. Google Scholar

[25]

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

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