# American Institute of Mathematical Sciences

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

## Bounded solutions of the Boltzmann equation in the whole space

 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.
Citation: Radjesvarane Alexandre, Yoshinori Morimoto, Seiji Ukai, Chao-Jiang Xu, Tong Yang. Bounded solutions of the Boltzmann equation in the whole space. Kinetic and 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. [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. [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. [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., (). [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. [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, (). [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, (). [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, (). [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. [10] C. Cercignani, "The Boltzmann Equation and its Applications," Applied mathematical sciences, 67, Springer-Verlag, 1988. [11] C. Cercignani, R. Illner and M. Pulvirenti, "The Mathematical Theory of Dilute Gases," Applied mathematical sciences, 106, Springer-Verlag, New York, 1994. [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. [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. [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. [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. [16] Y. Guo, Bounded solutions for the Boltzmann equation, Quaterly of Applied Mathematics, LXVIII (2010), 143-148. [17] T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Rational Mech. Anal., 58 (1975), 181-205. doi: 10.1007/BF00280740. [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. [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. [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. [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. [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. [23] S. Ukai, Local solutions in Gevrey classes to the nonlinear Boltzmann equation without cutoff, Japan J. Appl. Math., 1 (1984), 141-156. [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. [25] C. Villani, A review of mathematical topics in collisional kinetic theory, in "Handbook of Fluid Mechanics" (eds S. Friedlander and D. Serre), 2002.

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##### 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. [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. [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. [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., (). [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. [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, (). [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, (). [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, (). [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. [10] C. Cercignani, "The Boltzmann Equation and its Applications," Applied mathematical sciences, 67, Springer-Verlag, 1988. [11] C. Cercignani, R. Illner and M. Pulvirenti, "The Mathematical Theory of Dilute Gases," Applied mathematical sciences, 106, Springer-Verlag, New York, 1994. [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. [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. [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. [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. [16] Y. Guo, Bounded solutions for the Boltzmann equation, Quaterly of Applied Mathematics, LXVIII (2010), 143-148. [17] T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Rational Mech. Anal., 58 (1975), 181-205. doi: 10.1007/BF00280740. [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. [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. [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. [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. [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. [23] S. Ukai, Local solutions in Gevrey classes to the nonlinear Boltzmann equation without cutoff, Japan J. Appl. Math., 1 (1984), 141-156. [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. [25] C. Villani, A review of mathematical topics in collisional kinetic theory, in "Handbook of Fluid Mechanics" (eds S. Friedlander and D. Serre), 2002.
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