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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 |
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. |
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. |
[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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