American Institute of Mathematical Sciences

Advanced
March  2011, 4(1): 17-40. doi: 10.3934/krm.2011.4.17

Bounded solutions of the Boltzmann equation in the whole space

Radjesvarane Alexandre 1, , Yoshinori Morimoto 2, , Seiji Ukai 3, , Chao-Jiang Xu 4, and  Tong Yang 5,

1. 

Department of Mathematics, Shanghai Jiao Tong University, Shanghai, 200240, China
2. 

Graduate School of Human and Environmental Studies, Kyoto University, Kyoto, 606-8501
3. 

17-26 Iwasaki, Hodogaya, Yokohama 240-0015
4. 

School of Mathematics, Wuhan University, 430072, Wuhan
5. 

Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong

Received  October 2010 Revised  October 2010 Published  January 2011

We construct bounded classical solutions of the Boltzmann equation in the whole space without specifying any limit behaviors at the spatial infinity and without assuming the smallness condition on initial data. More precisely, we show that if the initial data is non-negative and belongs to a uniformly local Sobolev space in the space variable and a standard Sobolev space with Maxwellian type decay property in the velocity variable, then the Cauchy problem of the Boltzmann equation possesses a unique non-negative local solution in the same function space, both for the cutoff and non-cutoff collision cross section with mild singularity. The known solutions such as solutions on the torus (space periodic solutions) and in the vacuum (solutions vanishing at the spatial infinity), and solutions in the whole space having a limit equilibrium state at the spatial infinity are included in our category.
Keywords: local existence, locally uniform Sobolev space, spatial behavior at infinity, pseudo-differential calculus., Boltzmann equation.
Mathematics Subject Classification: Primary: 35A01, 35A02, 35A09, 35S05; Secondary: 76P05, 82C4.
Citation: Radjesvarane Alexandre, Yoshinori Morimoto, Seiji Ukai, Chao-Jiang Xu, Tong Yang. Bounded solutions of the Boltzmann equation in the whole space. Kinetic & Related Models, 2011, 4 (1) : 17-40. doi: 10.3934/krm.2011.4.17
References:
[1]

R. Alexandre, Some solutions of the Boltzmann equation without angular cutoff, J. Stat. Physics, 104 (2001), 327-358. doi: 10.1023/A:1010317913642.  Google Scholar

[2]

R. Alexandre, L. Desvillettes, C. Villani and B. Wennberg, Entropy dissipation and long-range interactions, Arch. Rational Mech. Anal., 152 (2000), 327-355. doi: 10.1007/s002050000083.  Google Scholar

[3]

R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T.Yang, Regularizing effect and local existence for non-cutoff Boltzmann equation, Arch. Rational Mech. Anal., 198 (2010), 39-123. doi: 10.1007/s00205-010-0290-1.  Google Scholar

[4]

R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, Global existence and full regularity of the Boltzmann equation without angular cutoff,, to appear in Comm. Math. Phys., ().   Google Scholar

[5]

R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T.Yang, Global well-posedness theory for the spatially inhomogeneous Boltzmann equation without angular cutoff,, C. R. Math. Acad. Sci. Paris, ().  doi: 10.1016/j.crma.2010.07.008.  Google Scholar

[6]

R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, Boltzmann equation without angular cutoff in the whole space: I. Global existence for soft potential,, Preprint HAL, ().   Google Scholar

[7]

R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T.Yang, The Boltzmann equation without angular cutoff in the whole space: II. Global existence for hard potential,, to appear in Analysis and Applications, ().   Google Scholar

[8]

R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T.Yang, Boltzmann equation without angular cutoff in the whole space: III, Qualitative properties of solutions,, Preprint HAL, ().   Google Scholar

[9]

R. Alexandre and C. Villani, On the Boltzmann equation for long-range interaction, Communications on Pure and Applied Mathematics, 55 (2002), 30-70. doi: 10.1002/cpa.10012.  Google Scholar

[10]

C. Cercignani, "The Boltzmann Equation and its Applications," Applied mathematical sciences, 67, Springer-Verlag, 1988.  Google Scholar

[11]

C. Cercignani, R. Illner and M. Pulvirenti, "The Mathematical Theory of Dilute Gases," Applied mathematical sciences, 106, Springer-Verlag, New York, 1994.  Google Scholar

[12]

R. J. DiPerna and P. L. Lions, On the Cauchy problem for Boltzmann equations: global existence and weak stability, Ann. Math., 130 (1989), 321-366. doi: 10.2307/1971423.  Google Scholar

[13]

H. Grad, Asymptotic Theory of the Boltzmann Equation II, in: Laurmann J. A. (ed.) Rarefied Gas Dynamics, Academic Press, New York, 1 (1963), 26-59.  Google Scholar

[14]

P.-T. Gressman and R.-M. Strain, Global classical solutions of the Boltzmann equation with long-range interactions, Proc. Nat. Acad. Sci., 107 (2010), 5744-5749. doi: 10.1073/pnas.1001185107.  Google Scholar

[15]

Y. Guo, The Boltzmann equation in the whole space, Indiana Univ. Maths. J., 53 (2004), 1081-1094. doi: 10.1512/iumj.2004.53.2574.  Google Scholar

[16]

Y. Guo, Bounded solutions for the Boltzmann equation, Quaterly of Applied Mathematics, LXVIII (2010), 143-148.  Google Scholar

[17]

T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Rational Mech. Anal., 58 (1975), 181-205. doi: 10.1007/BF00280740.  Google Scholar

[18]

P. L. Lions, Regularity and compactness for Boltzmann collision kernels without angular cut-off, C. R. Acad. Sci. Paris Series I, 326 (1998), 37-41.  Google Scholar

[19]

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

[20]

Y. P. Pao, Boltzmann collision operator with inverse power intermolecular potential, I, II Commun. Pure Appl. Math., 27 (1974), 407-–428; ibid., 27 (1974), 559-–581.  Google Scholar

[21]

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

[22]

S. Ukai, Les solutions globales de l'equation de Boltzmann dans l'espace tout entier et dans le demi-espace, C. R. Acad. Sci. Paris Ser. A-B, 282 (1976), Ai, A317-A320.  Google Scholar

[23]

S. Ukai, Local solutions in Gevrey classes to the nonlinear Boltzmann equation without cutoff, Japan J. Appl. Math., 1 (1984), 141-156.  Google Scholar

[24]

S. Ukai, Solutions of the Boltzmann equation, in "Pattern and Waves - Qualitave Analysis of Nonlinear Differential Equations" (eds M. Mimura and T. Nishida), Studies of Mathematics and its Applications 18, pp 37-96. Kinokuniya-North-Holland, Tokyo, 1986.  Google Scholar

[25]

C. Villani, A review of mathematical topics in collisional kinetic theory, in "Handbook of Fluid Mechanics" (eds S. Friedlander and D. Serre), 2002.  Google Scholar

show all references

References:
[1]

R. Alexandre, Some solutions of the Boltzmann equation without angular cutoff, J. Stat. Physics, 104 (2001), 327-358. doi: 10.1023/A:1010317913642.  Google Scholar

[2]

R. Alexandre, L. Desvillettes, C. Villani and B. Wennberg, Entropy dissipation and long-range interactions, Arch. Rational Mech. Anal., 152 (2000), 327-355. doi: 10.1007/s002050000083.  Google Scholar

[3]

R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T.Yang, Regularizing effect and local existence for non-cutoff Boltzmann equation, Arch. Rational Mech. Anal., 198 (2010), 39-123. doi: 10.1007/s00205-010-0290-1.  Google Scholar

[4]

R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, Global existence and full regularity of the Boltzmann equation without angular cutoff,, to appear in Comm. Math. Phys., ().   Google Scholar

[5]

R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T.Yang, Global well-posedness theory for the spatially inhomogeneous Boltzmann equation without angular cutoff,, C. R. Math. Acad. Sci. Paris, ().  doi: 10.1016/j.crma.2010.07.008.  Google Scholar

[6]

R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, Boltzmann equation without angular cutoff in the whole space: I. Global existence for soft potential,, Preprint HAL, ().   Google Scholar

[7]

R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T.Yang, The Boltzmann equation without angular cutoff in the whole space: II. Global existence for hard potential,, to appear in Analysis and Applications, ().   Google Scholar

[8]

R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T.Yang, Boltzmann equation without angular cutoff in the whole space: III, Qualitative properties of solutions,, Preprint HAL, ().   Google Scholar

[9]

R. Alexandre and C. Villani, On the Boltzmann equation for long-range interaction, Communications on Pure and Applied Mathematics, 55 (2002), 30-70. doi: 10.1002/cpa.10012.  Google Scholar

[10]

C. Cercignani, "The Boltzmann Equation and its Applications," Applied mathematical sciences, 67, Springer-Verlag, 1988.  Google Scholar

[11]

C. Cercignani, R. Illner and M. Pulvirenti, "The Mathematical Theory of Dilute Gases," Applied mathematical sciences, 106, Springer-Verlag, New York, 1994.  Google Scholar

[12]

R. J. DiPerna and P. L. Lions, On the Cauchy problem for Boltzmann equations: global existence and weak stability, Ann. Math., 130 (1989), 321-366. doi: 10.2307/1971423.  Google Scholar

[13]

H. Grad, Asymptotic Theory of the Boltzmann Equation II, in: Laurmann J. A. (ed.) Rarefied Gas Dynamics, Academic Press, New York, 1 (1963), 26-59.  Google Scholar

[14]

P.-T. Gressman and R.-M. Strain, Global classical solutions of the Boltzmann equation with long-range interactions, Proc. Nat. Acad. Sci., 107 (2010), 5744-5749. doi: 10.1073/pnas.1001185107.  Google Scholar

[15]

Y. Guo, The Boltzmann equation in the whole space, Indiana Univ. Maths. J., 53 (2004), 1081-1094. doi: 10.1512/iumj.2004.53.2574.  Google Scholar

[16]

Y. Guo, Bounded solutions for the Boltzmann equation, Quaterly of Applied Mathematics, LXVIII (2010), 143-148.  Google Scholar

[17]

T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Rational Mech. Anal., 58 (1975), 181-205. doi: 10.1007/BF00280740.  Google Scholar

[18]

P. L. Lions, Regularity and compactness for Boltzmann collision kernels without angular cut-off, C. R. Acad. Sci. Paris Series I, 326 (1998), 37-41.  Google Scholar

[19]

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

[20]

Y. P. Pao, Boltzmann collision operator with inverse power intermolecular potential, I, II Commun. Pure Appl. Math., 27 (1974), 407-–428; ibid., 27 (1974), 559-–581.  Google Scholar

[21]

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

[22]

S. Ukai, Les solutions globales de l'equation de Boltzmann dans l'espace tout entier et dans le demi-espace, C. R. Acad. Sci. Paris Ser. A-B, 282 (1976), Ai, A317-A320.  Google Scholar

[23]

S. Ukai, Local solutions in Gevrey classes to the nonlinear Boltzmann equation without cutoff, Japan J. Appl. Math., 1 (1984), 141-156.  Google Scholar

[24]

S. Ukai, Solutions of the Boltzmann equation, in "Pattern and Waves - Qualitave Analysis of Nonlinear Differential Equations" (eds M. Mimura and T. Nishida), Studies of Mathematics and its Applications 18, pp 37-96. Kinokuniya-North-Holland, Tokyo, 1986.  Google Scholar

[25]

C. Villani, A review of mathematical topics in collisional kinetic theory, in "Handbook of Fluid Mechanics" (eds S. Friedlander and D. Serre), 2002.  Google Scholar

[1]

Radjesvarane Alexandre, Yoshinori Morimoto, Seiji Ukai, Chao-Jiang Xu, Tong Yang. Local existence with mild regularity for the Boltzmann equation. Kinetic & Related Models, 2013, 6 (4) : 1011-1041. doi: 10.3934/krm.2013.6.1011
[2]

Lanzhe Liu. Mean oscillation and boundedness of Toeplitz Type operators associated to pseudo-differential operators. Communications on Pure & Applied Analysis, 2015, 14 (2) : 627-636. doi: 10.3934/cpaa.2015.14.627
[3]

Liang Huang, Jiao Chen. The boundedness of multi-linear and multi-parameter pseudo-differential operators. Communications on Pure & Applied Analysis, 2021, 20 (2) : 801-815. doi: 10.3934/cpaa.2020291
[4]

JIAO CHEN, WEI DAI, GUOZHEN LU. $L^p$ boundedness for maximal functions associated with multi-linear pseudo-differential operators. Communications on Pure & Applied Analysis, 2017, 16 (3) : 883-898. doi: 10.3934/cpaa.2017042
[5]

Martin Michálek, Dalibor Pražák, Jakub Slavík. Semilinear damped wave equation in locally uniform spaces. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1673-1695. doi: 10.3934/cpaa.2017080
[6]

Seung-Yeal Ha, Ho Lee, Seok Bae Yun. Uniform $L^p$-stability theory for the space-inhomogeneous Boltzmann equation with external forces. Discrete & Continuous Dynamical Systems, 2009, 24 (1) : 115-143. doi: 10.3934/dcds.2009.24.115
[7]

Nicolas Lerner, Yoshinori Morimoto, Karel Pravda-Starov, Chao-Jiang Xu. Phase space analysis and functional calculus for the linearized Landau and Boltzmann operators. Kinetic & Related Models, 2013, 6 (3) : 625-648. doi: 10.3934/krm.2013.6.625
[8]

Md. Rabiul Haque, Takayoshi Ogawa, Ryuichi Sato. Existence of weak solutions to a convection–diffusion equation in a uniformly local lebesgue space. Communications on Pure & Applied Analysis, 2020, 19 (2) : 677-697. doi: 10.3934/cpaa.2020031
[9]

Ildoo Kim. An $L_p$-Lipschitz theory for parabolic equations with time measurable pseudo-differential operators. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2751-2771. doi: 10.3934/cpaa.2018130
[10]

Yan Guo, Juhi Jang, Ning Jiang. Local Hilbert expansion for the Boltzmann equation. Kinetic & Related Models, 2009, 2 (1) : 205-214. doi: 10.3934/krm.2009.2.205
[11]

Rama Ayoub, Aziz Hamdouni, Dina Razafindralandy. A new Hodge operator in discrete exterior calculus. Application to fluid mechanics. Communications on Pure & Applied Analysis, 2021, 20 (6) : 2155-2185. doi: 10.3934/cpaa.2021062
[12]

Nobu Kishimoto, Minjie Shan, Yoshio Tsutsumi. Global well-posedness and existence of the global attractor for the Kadomtsev-Petviashvili Ⅱ equation in the anisotropic Sobolev space. Discrete & Continuous Dynamical Systems, 2020, 40 (3) : 1283-1307. doi: 10.3934/dcds.2020078
[13]

M. Ben Ayed, Abdelbaki Selmi. Asymptotic behavior and existence results for a biharmonic equation involving the critical Sobolev exponent in a five-dimensional domain. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1705-1722. doi: 10.3934/cpaa.2010.9.1705
[14]

Alexander V. Bobylev, Irene M. Gamba. Upper Maxwellian bounds for the Boltzmann equation with pseudo-Maxwell molecules. Kinetic & Related Models, 2017, 10 (3) : 573-585. doi: 10.3934/krm.2017023
[15]

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
[16]

Ralf Kirsch, Sergej Rjasanow. The uniformly heated inelastic Boltzmann equation in Fourier space. Kinetic & Related Models, 2010, 3 (3) : 445-456. doi: 10.3934/krm.2010.3.445
[17]

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
[18]

Seung-Yeal Ha, Eunhee Jeong, Robert M. Strain. Uniform $L^1$-stability of the relativistic Boltzmann equation near vacuum. Communications on Pure & Applied Analysis, 2013, 12 (2) : 1141-1161. doi: 10.3934/cpaa.2013.12.1141
[19]

Luis Barreira, Claudia Valls. Topological conjugacies and behavior at infinity. Communications on Pure & Applied Analysis, 2014, 13 (2) : 687-701. doi: 10.3934/cpaa.2014.13.687
[20]

Jack Schaeffer. Global existence for the Vlasov-Poisson system with steady spatial asymptotic behavior. Kinetic & Related Models, 2012, 5 (1) : 129-153. doi: 10.3934/krm.2012.5.129

2020 Impact Factor: 1.432

Tools

Metrics

  • PDF downloads (73)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences