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Remarks on the full dispersion Kadomtsev-Petviashvli equation
Local existence with mild regularity for the Boltzmann equation
1. | Department of Mathematics, Shanghai Jiao Tong University, Shanghai, 200240 |
2. | Graduate School of Human and Environmental Studies, Kyoto University, Kyoto, 606-8501 |
3. | 17-26 Iwasaki, Hodogaya, Yokohama 240-0015 |
4. | Université de Rouen, UMR 6085-CNRS, Mathématiques, Avenue de l’Université, BP.12, 76801 Saint Etienne du Rouvray |
5. | Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong |
References:
[1] |
R. Alexandre, Some solutions of the Boltzmann equation without angular cutof, 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, Comm. Math. Phys., 304 (2011), 513-581.
doi: 10.1007/s00220-011-1242-9. |
[5] |
R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, The Boltzmann equation without angular cutoff in the whole space: I. Global existence for soft potential, J. Funct. Anal., 262 (2012), 915-1010.
doi: 10.1016/j.jfa.2011.10.007. |
[6] |
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, Anal. Appl.(Singap.), 9 (2011), 113-134.
doi: 10.1142/S0219530511001777. |
[7] |
R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, The Boltzmann equation without angular cutoff in the whole space: Qualitative properties of solutions, Arch. Ration. Mech. Anal., 202 (2011), 599-661.
doi: 10.1007/s00205-011-0432-0. |
[8] |
R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, Bounded solutions of the Boltzmann equation in the whole space, Kinet. Relat. Models, 4 (2011), 17-40.
doi: 10.3934/krm.2011.4.17. |
[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, New York, 1988.
doi: 10.1007/978-1-4612-1039-9. |
[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 Rarefied Gas Dynamics, (ed. J. A. Laurmann ), 1, Academic Press, New York, (1963), 26-59. |
[14] |
P.-T. Gressman and R.-M. Strain, Global classical solutions of the Boltzmann equation without angular cut-off, J. Amer. Math. Soc., 24 (2011), 771-847.
doi: 10.1090/S0894-0347-2011-00697-8. |
[15] |
Y. Guo, The Landau equation in a periodic box, Comm. Math. Phys., 231 (2002), 391-434.
doi: 10.1007/s00220-002-0729-9. |
[16] |
Y. Guo, Bounded solutions for the Boltzmann equationn, Quaterly of Applied Mathematics, 68 (2010), 143-148. |
[17] |
P. L. Lions, Régularité et compacité pour des noyaux de collision de Boltzmann sans troncature angulaire,(French) [Regularity and compactness for Boltzmann collision kernels without angular cutoff], C. R. Acad. Sci. Paris Series I Math, 326 (1998), 37-41.
doi: 10.1016/S0764-4442(97)82709-7. |
[18] |
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. |
[19] |
Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, Regularity of solutions to the spatially homogeneous Boltzmann equation without angular cutoff, Discrete and Continuous Dynamical Systems - Series A, 24 (2009), 187-212.
doi: 10.3934/dcds.2009.24.187. |
[20] |
Y. P. Pao, Boltzmann collision operator with inverse power intermolecular potential, I, II, Commun. Pure Appl. Math., 27 (1974), 559-581.
doi: 10.1002/cpa.3160270402. |
[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), 317-320. |
[23] |
S. Ukai, Local solutions in Gevrey classes to the nonlinear Boltzmann equation without cutoff, Japan J. Appl. Math., 1 (1984), 141-156.
doi: 10.1007/BF03167864. |
[24] |
S. Ukai, Solutions of the Boltzmann equation, Patterns and waves, Stud. Math. Appl., North-Holland, Amsterdam, 18 (1986), 37-96.
doi: 10.1016/S0168-2024(08)70128-0. |
[25] |
S. Ukai and T. Yang, The Boltzmann equation in the space $L^2\cap L^\infty_\beta$: Global and time-periodic solutions, Analysis and Applications, 4 (2006), 263-310.
doi: 10.1142/S0219530506000784. |
[26] |
C. Villani, A review of mathematical topics in collisional kinetic theory, Handbook of mathematical fluid dynamics, North-Holland, Amsterdam, I (2002), 71-305.
doi: 10.1016/S1874-5792(02)80004-0. |
show all references
References:
[1] |
R. Alexandre, Some solutions of the Boltzmann equation without angular cutof, 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, Comm. Math. Phys., 304 (2011), 513-581.
doi: 10.1007/s00220-011-1242-9. |
[5] |
R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, The Boltzmann equation without angular cutoff in the whole space: I. Global existence for soft potential, J. Funct. Anal., 262 (2012), 915-1010.
doi: 10.1016/j.jfa.2011.10.007. |
[6] |
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, Anal. Appl.(Singap.), 9 (2011), 113-134.
doi: 10.1142/S0219530511001777. |
[7] |
R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, The Boltzmann equation without angular cutoff in the whole space: Qualitative properties of solutions, Arch. Ration. Mech. Anal., 202 (2011), 599-661.
doi: 10.1007/s00205-011-0432-0. |
[8] |
R. Alexandre, Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, Bounded solutions of the Boltzmann equation in the whole space, Kinet. Relat. Models, 4 (2011), 17-40.
doi: 10.3934/krm.2011.4.17. |
[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, New York, 1988.
doi: 10.1007/978-1-4612-1039-9. |
[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 Rarefied Gas Dynamics, (ed. J. A. Laurmann ), 1, Academic Press, New York, (1963), 26-59. |
[14] |
P.-T. Gressman and R.-M. Strain, Global classical solutions of the Boltzmann equation without angular cut-off, J. Amer. Math. Soc., 24 (2011), 771-847.
doi: 10.1090/S0894-0347-2011-00697-8. |
[15] |
Y. Guo, The Landau equation in a periodic box, Comm. Math. Phys., 231 (2002), 391-434.
doi: 10.1007/s00220-002-0729-9. |
[16] |
Y. Guo, Bounded solutions for the Boltzmann equationn, Quaterly of Applied Mathematics, 68 (2010), 143-148. |
[17] |
P. L. Lions, Régularité et compacité pour des noyaux de collision de Boltzmann sans troncature angulaire,(French) [Regularity and compactness for Boltzmann collision kernels without angular cutoff], C. R. Acad. Sci. Paris Series I Math, 326 (1998), 37-41.
doi: 10.1016/S0764-4442(97)82709-7. |
[18] |
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. |
[19] |
Y. Morimoto, S. Ukai, C.-J. Xu and T. Yang, Regularity of solutions to the spatially homogeneous Boltzmann equation without angular cutoff, Discrete and Continuous Dynamical Systems - Series A, 24 (2009), 187-212.
doi: 10.3934/dcds.2009.24.187. |
[20] |
Y. P. Pao, Boltzmann collision operator with inverse power intermolecular potential, I, II, Commun. Pure Appl. Math., 27 (1974), 559-581.
doi: 10.1002/cpa.3160270402. |
[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), 317-320. |
[23] |
S. Ukai, Local solutions in Gevrey classes to the nonlinear Boltzmann equation without cutoff, Japan J. Appl. Math., 1 (1984), 141-156.
doi: 10.1007/BF03167864. |
[24] |
S. Ukai, Solutions of the Boltzmann equation, Patterns and waves, Stud. Math. Appl., North-Holland, Amsterdam, 18 (1986), 37-96.
doi: 10.1016/S0168-2024(08)70128-0. |
[25] |
S. Ukai and T. Yang, The Boltzmann equation in the space $L^2\cap L^\infty_\beta$: Global and time-periodic solutions, Analysis and Applications, 4 (2006), 263-310.
doi: 10.1142/S0219530506000784. |
[26] |
C. Villani, A review of mathematical topics in collisional kinetic theory, Handbook of mathematical fluid dynamics, North-Holland, Amsterdam, I (2002), 71-305.
doi: 10.1016/S1874-5792(02)80004-0. |
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