Local existence with mild regularity for the Boltzmann equation

Radjesvarane Alexandre; Yoshinori Morimoto; Seiji Ukai; Chao-Jiang Xu; Tong Yang

doi:10.3934/krm.2013.6.1011

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December 2013, 6(4): 1011-1041. doi: 10.3934/krm.2013.6.1011

Local existence with mild regularity for the Boltzmann equation

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
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

Received August 2013 Revised September 2013 Published November 2013

Abstract
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Without Grad's angular cutoff assumption, the local existence of classical solutions to the Boltzmann equation is studied. There are two new improvements: the index of Sobolev spaces for the solution is related to the parameter of the angular singularity; moreover, we do not assume that the initial data is close to a global equilibrium. Using the energy method, one important step in the analysis is the study of fractional derivatives of the collision operator and related commutators.

Keywords: existence of solution, energy estimates, fractional derivatives., Boltzmann equation.

Mathematics Subject Classification: 35A05, 35B65, 35D10, 35H20, 76P05, 84C4.

Citation: 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

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. 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, Comm. Math. Phys., 304 (2011), 513-581. doi: 10.1007/s00220-011-1242-9. Google Scholar
[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. Google Scholar
[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. 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: Qualitative properties of solutions, Arch. Ration. Mech. Anal., 202 (2011), 599-661. doi: 10.1007/s00205-011-0432-0. Google Scholar
[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. 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, New York, 1988. doi: 10.1007/978-1-4612-1039-9. 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 Rarefied Gas Dynamics, (ed. J. A. Laurmann ), 1, Academic Press, New York, (1963), 26-59. Google Scholar
[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. Google Scholar
[15]	Y. Guo, The Landau equation in a periodic box, Comm. Math. Phys., 231 (2002), 391-434. doi: 10.1007/s00220-002-0729-9. Google Scholar
[16]	Y. Guo, Bounded solutions for the Boltzmann equationn, Quaterly of Applied Mathematics, 68 (2010), 143-148. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. 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), 317-320. 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. doi: 10.1007/BF03167864. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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. 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, Comm. Math. Phys., 304 (2011), 513-581. doi: 10.1007/s00220-011-1242-9. Google Scholar
[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. Google Scholar
[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. 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: Qualitative properties of solutions, Arch. Ration. Mech. Anal., 202 (2011), 599-661. doi: 10.1007/s00205-011-0432-0. Google Scholar
[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. 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, New York, 1988. doi: 10.1007/978-1-4612-1039-9. 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 Rarefied Gas Dynamics, (ed. J. A. Laurmann ), 1, Academic Press, New York, (1963), 26-59. Google Scholar
[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. Google Scholar
[15]	Y. Guo, The Landau equation in a periodic box, Comm. Math. Phys., 231 (2002), 391-434. doi: 10.1007/s00220-002-0729-9. Google Scholar
[16]	Y. Guo, Bounded solutions for the Boltzmann equationn, Quaterly of Applied Mathematics, 68 (2010), 143-148. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. 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), 317-320. 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. doi: 10.1007/BF03167864. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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