Stability estimates of the Boltzmann equation in a half space
Authors:
Myeongju Chae and Seung-Yeal Ha
Journal:
Quart. Appl. Math. 65 (2007), 653-682
MSC (2000):
Primary 35Q35
DOI:
https://doi.org/10.1090/S0033-569X-07-01060-4
Published electronically:
August 24, 2007
MathSciNet review:
2370355
Full-text PDF Free Access
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Abstract: In this paper, we study the large-time behavior and the stability of continuous mild solutions to the Boltzmann equation in a half space. For this, we introduce two nonlinear functionals measuring future binary collisions and $L^1$-distance. Through the time-decay estimates of these functionals and the pointwise estimate of the gain part of the collision operator, we show that continuous mild solutions approach to collision free flows time-asymptotically in $L^1$, and $L^1$-distance at time $t$ is uniformly bounded by that of corresponding initial data, when initial datum is a small perturbation of the vacuum.
References
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References
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- Kaniel, S., Shinbrot, M.: The Boltzmann equation 1: Uniqueness and local existence. Commun. Math. Phys. 58 (1978) 65-84. MR 0475532 (57:15133)
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- Toscani, G.: On the nonlinear Boltzmann equation in unbounded domains, Arch. Rational Mech. Anal. 95 (1986) 37-49. MR 849403 (87j:82060)
- Toscani, G., Protopopescu, V.: The nonlinear Boltzmann equation with partially absorbing boundary conditions: Global existence and uniqueness results.
- Ukai, S.: Solutions of the Boltzmann equation. Patterns and waves. Stud. Math. Appl., 18, North-Holland, Amsterdam, 1986. MR 882376 (88g:35187)
- Ukai, S., Asano, K.: Steady solutions of the Boltzmann equation for a gas flow past an obstacle I. Existence. Arch. Rational Mech. Anal. 84 (1983) 249-291. MR 714977 (85a:76079)
- Ukai, S., Asano, K.: Steady solutions of the Boltzmann equation for a gas flow past an obstacle II. Stability. Publ. Res. Inst. Math. Sci. 22 (1986) 1035-1062. MR 879996 (88f:76033)
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Additional Information
Myeongju Chae
Affiliation:
Department of Applied Mathematics, Hankyong National University, Ansung 456-749, Korea
Email:
mchae@kias.re.kr
Seung-Yeal Ha
Affiliation:
Department of Mathematics, Seoul National University, Seoul 151-747, Korea
MR Author ID:
684438
Email:
syha@math.snu.ac.kr
Received by editor(s):
April 13, 2006
Published electronically:
August 24, 2007
Dedicated:
This paper is dedicated to Tai-Ping Liu on the occasion of his sixtieth birthday.
Article copyright:
© Copyright 2007
Brown University
The copyright for this article reverts to public domain 28 years after publication.