Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-12-01T03:56:49.694Z Has data issue: false hasContentIssue false
  • English
  • Français

HN stability of the Vlasov–Poisson–Boltzmann system near Maxwellians

Published online by Cambridge University Press:  26 March 2007

Hongjun Yu
Hongjun Yu*
Affiliation:
School of Mathematical Sciences, South China University of Technology, Guangzhou 510640, People's Republic of China ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the HN stability of the Vlasov–Poisson Boltzmann system near Maxwellians. Under a suitable smallness assumption on initial data, we show that the global classical solutions constructed by Guo are HN stable. For a stability estimate, we employ the energy methods of Guo.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 137 , Issue 2 , 2007 , pp. 431 - 446
Copyright
Copyright © Royal Society of Edinburgh 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1 Arkeryd, L.. Stability in L 1 for the spatially homogeneous Boltzmann equation. Arch. Ration. Mech. Analysis 103 (1988), 151–167.Google Scholar
2 Duan, R.-J., Yang, T. and Zhu, C.-J.. Boltzmann equation with external force and Vlasov–Poisson–Boltzmann system in infinite vacuum. Discrete Contin. Dynam. Syst. 16 (2006), 253–277.Google Scholar
3 Glassey, R.. The Cauchy problem in kinetic theory (Philadelphia, PA: SIAM, 1996).CrossRefGoogle Scholar
4 Golse, F. and Poupaud, F.. Stationary solutions of the linearized Boltzmann equation in a half-space. Math. Meth. Appl. Sci. 11 (1989), 483–502.Google Scholar
5 Guo, Y.. The Vlasov–Poisson–Boltzmann system near vacuum. Commun. Math. Phys. 218 (2001), 293–313.Google Scholar
6 Guo, Y.. The Vlasov–Poisson–Boltzmann system near Maxwellians. Commun. Pure Appl. Math. 55 (2002), 1104–1135.Google Scholar
7 Guo, Y.. Classical solutions to the Boltzmann equation for molecules with angular cutoff. Arch. Ration. Mech. Analysis 169 (2003), 305–353.CrossRefGoogle Scholar
8 Guo, Y.. The Vlasov Maxwell–Boltzmann system near Maxwellians. Invent. Math. 153 (2003), 593–630.Google Scholar
9 Guo, Y.. The Boltzmann equation in the whole space. Indiana Univ. Math. J. 53 (2004), 1081–1094.Google Scholar
10 Ha, S.-Y.. L 1 stability of the Boltzmann equation for the hard-sphere model. Arch. Ration. Mech. Analysis 173 (2004), 279–296.Google Scholar
11 Ha, S.-Y.. Nonlinear functionals of the Boltzmann equation and uniform stability estimates. J. Diff. Eqns 215 (2005), 178–205.Google Scholar
12 Ha, S.-Y.. L 1 stability of the Boltzmann equation for Maxwellian molecules. NonlinearUy 18 (2005), 981–1001.Google Scholar
13 Ha, S.-Y.. L 1 stability of the Vlasov–Poisson–Boltzmann system. Preprint (2005).Google Scholar
14 Lions, P.-L.. Compactness in Boltzmann's equation via Fourier integral operators and applications. I, II. J. Math. Kyoto Univ. 34 (1994), 391–461.Google Scholar
15 Liu, T.-P., Yang, T. and Yu, S.-H.. Energy method for the Boltzmann equation. Physica D188 (2004), 178–192.Google Scholar
16 Lu, X.. Spatial decay solutions of the Boltzmann equation: converse properties of long time limiting behavior. SIAM J. Math. Analysis 30 (1999), 1151–1174.Google Scholar
17 Mischler, S.. On the initial boundary value problem for the Vlasov–Poisson–Boltzmann system. Commun. Math. Phys. 210 (2000), 447–466.Google Scholar
18 Nishida, T. and Imai, K.. Global solutions to the initial value problem for the nonlinear Boltzmann equation. Publ. RIMS Kyoto 12 (1976), 229–239.Google Scholar
19 Shizuta, Y.. On the classical solutions of the Boltzmann equation. Commun. Pure Appl. Math. 36 (1983), 705–754.Google Scholar
20 Ukai, S.. On the existence of global solutions of mixed problem for non-linear Boltzmann equation. Proc. Jpn Acad. 50 (1974), 179–184.Google Scholar
21 Ukai, S. and Asano, K.. On the Cauchy problem of Boltzmann equation with a soft potential. Publ. RIMS Kyoto 18 (1982), 477–519.CrossRefGoogle Scholar
22 Ukai, S., Yang, T. and Zhao, H.-J.. Global solutions to the Boltzmann equation with external forces. Analysis Apphc. 3 (2005), 157–193.Google Scholar
23 Yang, T., Yu, H.-J. and Zhao, H.-J.. Cauchy problem for the Vlasov–Poisson–Boltzmann system. Arch. Ration. Mech. Analysis 182 (2006), 415–470.Google Scholar
24 Yu, H.-J.. Existence and exponential decay of global solution to the Boltzmann equation near Maxwellians. Math. Models Meth. Appl Sci. 15 (2005), 483–505.Google Scholar