On the Cauchy problem of the Boltzmann and Landau equations with soft potentials
Authors:
Ling Hsiao and Hongjun Yu
Journal:
Quart. Appl. Math. 65 (2007), 281-315
MSC (2000):
Primary 35Q99; Secondary 35A05
DOI:
https://doi.org/10.1090/S0033-569X-07-01053-8
Published electronically:
April 25, 2007
MathSciNet review:
2330559
Full-text PDF Free Access
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Abstract: Global classical solutions near Maxwellians are constructed for the Boltzmann and Landau equations with soft potentials in the whole space. The construction of global solutions is based on refined energy analysis. Our results generalize the classical results in Ukai and Asano (Publ. Res. Inst. Math. Sci. 18 (1982), 477–519) to the very soft potentials for the Boltzmann equation and also extend the classical results in Caflisch (Comm. Math. Phys. 74 (1980), 97–107), Guo (Comm. Math. Phys. 231 (2002), 391–434), and Guo (Arch. Rat. Mech. Anal. 169 (2003), 305–353) in the periodic box to the whole space for the Boltzmann equation and the Landau equation in the Coulomb interaction.
References
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References
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- Villani, C., On the Landau equation: Weak stability, global existence. Adv. Diff. Eq. 1 (5) (1996), 793-816. MR 1392006 (97e:82048)
- Villani, C., A survey of mathematical topics in kinetic theory. To appear in Handbook of fluid mechanics, S. Friedlander and D. Serre, Eds.
- Yang, T., Yu, H.-J., and Zhao, H.-J., Cauchy problem for the Vlasov-Poisson-Boltzmann system, Arch. Rat. Mech. Anal. 182 (2006), 415–470. MR 2276498
- Yu, H.-J., Global solution of the Vlasov-Poisson-Landau systems near Maxwellians with small amplitude, J. Part. Diff. Eqn. 17 (2) (2004), 173-192. MR 2060788 (2005f:35042)
- Yu, H.-J., Global classical solution of the Vlasov-Maxwell-Landau system near Maxwellians, J. Math. Phys. 11 (45) (2004), 4360-4376. MR 2098143 (2005h:82109)
- Yu, H.-J., Existence and exponential decay of global solution to the Boltzmann equation near Maxwellians, Math. Models Methods Appl. Sci. 15 (3)(2005), 483-505. MR 2126140 (2005m:35038)
- Zhan, M., Local existence of classical solutions to the Landau equations. Transport Theory Statist. Phys. 23 (4) (1994), 479–499. MR 1264848 (95c:35203)
- Zhan, M., Local existence of solutions to the Landau-Maxwell system. Math. Methods Appl. Sci. 17 (8) (1994), 613–641. MR 1280648 (95h:35228)
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Additional Information
Ling Hsiao
Affiliation:
Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, People’s Republic of China
Email:
hsiaol@amss.ac.cn
Hongjun Yu
Affiliation:
School of Mathematical Sciences, South China University of Technology, Guangzhou 510640, People’s Republic of China
Email:
yuhj2002@sina.com
Keywords:
Boltzmann equation,
Landau equation,
soft potentials,
global solutions.
Received by editor(s):
January 15, 2006
Published electronically:
April 25, 2007
Article copyright:
© Copyright 2007
Brown University
The copyright for this article reverts to public domain 28 years after publication.