Quarterly of Applied Mathematics Quarterly of Applied Mathematics
Online ISSN: 1552-4485 Print ISSN: 0033-569X

     

Stability estimates of the Boltzmann equation in a half space

Author(s): Myeongju Chae; Seung-Yeal Ha
Journal: Quart. Appl. Math. 65 (2007), 653-682.
MSC (2000): Primary 35Q35
Posted: August 24, 2007
Retrieve article in: PDF DVI PostScript

Abstract | References | Similar articles | Additional information

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:

1.
Arkeryd, L., Cercignani, C.: A global existence theorem for the initial-boundary value problem for the Boltzmann equation when the boundaries are not isothermal. Arch. Rational Mech. Anal. 125 (1993) 271-288. MR 1245073 (94j:82059)

2.
Arkeryd, L., Cercignani, C., Illner, R.: Measure solutions of the steady Boltzmann equation in a slab. Commun. Math. Phys. 142 (1991) 285-296. MR 1137065 (92m:82109)

3.
Arkeryd, L., Maslova, N.: On diffusive reflection at the boundary for the Boltzmann equation and related equations. J. Stat. Phys. 77 (1994), 1051-1077. MR 1301931 (95i:82080)

4.
Asano, K.: Local solutions to the initial and initial boundary value problem for the Boltzmann equation with an external force I. J. Math. Kyoto Univ. 24 (1984) 225-228. MR 751698 (86a:45011)

5.
Bellomo, N., Palczewski. A., Toscani, G.: Mathematical topics in nonlinear kinetic theory. World Scientific Publishing Co., Singapore, 1988. MR 996631 (90m:82030)

6.
Blouza, A., Le Dret, H.: An up-to-boundary version of Friedrichs's lemma and applications to the linear Koiter shell model. SIAM J. Math. Anal. 33 (2001) 877-895. MR 1884727 (2003a:35185)

7.
Bellomo, N., Toscani, G.: On the Cauchy problem for the nonlinear Boltzmann equation: global existence, uniqueness and asymptotic behavior. J. Math. Phys. 26 (1985) 334-338. MR 776503 (86h:82038)

8.
Cercignani, C., Illner, R.: Global weak solutions of the Boltzmann equation in a slab with diffusive bounary conditions. Arch. Rationa Mech. Anal. 134 (1996), 1-16. MR 1392307 (97k:82037)

9.
Cercignani, C., Illner, R., Pulvirenti, M.: The mathematical theory of dilute gases. Applied Mathematical Sciences 106, Springer-Verlag, New York 1994. MR 1307620 (96g:82046)

10.
DiPerna, R., Lions, P.L.: On the Cauchy problem for Boltzmann equations: global existence and weak stability. Ann Math. 130 (1989) 321-366. MR 1014927 (90k:82045)

11.
Glikson, A.: On the existence of general solutions of the initial-value problem for the nonlinear Boltzmann equation with a cut-off. Arch. Rational Mech. Anal. 45 (1972) 35-46. MR 0337235 (49:2006a)

12.
Guiraud, J.P.: An $ H$-theorem for a gas of rigid spheres in a bounded domain. Colloq. Int. CNRS, 236 (1975), 29-58. MR 0449410 (56:7713)

13.
Ha, S.-Y.: $ L^1$ stability of the Boltzmann equation for Maxwellian molecules. Nonlinearity. 18 (2005) 981-1001. MR 2134080 (2006b:82106)

14.
Ha, S.-Y.: Nonlinear functionals of the Boltzmann equation and uniform stability estimates. J. Differential Equations 215 (2005) 178-205. MR 2146347 (2006b:35033)

15.
Hamdache, K.: Initial-boundary value problems for the Boltzmann equation: Global existence of weak solutions. Arch. Rational Mech. Anal. 119 (1992) 309-353. MR 1179690 (93g:35132)

16.
Hamdache, K.: Problemes aux limites pour l'equation de Boltzmann: Existence globale de solutions. Comm. PDE 13 (1988) 813-845. MR 940959 (89f:45016)

17.
Hamdache, K.: Existence in the large and asymptotic behavior for the Boltzmann equation. Japan. J. Appl. Math. 2 (1984) 1-15. MR 839316 (87j:35322)

18.
Illner, R., Shinbrot, M.: Global existence for a rare gas in an infinite vacuum. Commun. Math. Phys. 95 (1984) 217-226. MR 760333 (86a:82019)

19.
Kaniel, S., Shinbrot, M.: The Boltzmann equation 1: Uniqueness and local existence. Commun. Math. Phys. 58 (1978) 65-84. MR 0475532 (57:15133)

20.
Lu, X. : Spatial decay solutions of the Boltzmann equation: converse properties of long time limiting behavior. SIAM J. Math. Anal. 30 (1999) 1151-1174. MR 1709791 (2000k:35274)

21.
Mischler, S., Perthame, B.: Boltzmann equation with infinite energy: renormalized solutions and distributional solutions for small initial data and initial data close to a Maxwellian. SIAM J. Math. Anal. 28 (1997) 1015-1027. MR 1466666 (98g:35197)

22.
Polewczak, J.: Classical solution of the nonlinear Boltzmann equation in all $ R^3$: asymptotic behavior of solutions. J. Stat. Phys. 50 (1988) 611-632. MR 939503 (89h:82021)

23.
Shinbrot, M.: The Boltzmann equation: flow of a cloud of gas past a body. Transport Theory Statist. Phys. 15 (1986) 317-332. MR 839634 (87f:76089)

24.
Shizuta, Y., Asano, K.: Global solutions of the Boltzmann equation in a bounded convex domain. Proc. Japan Acad. 53 (1977), 3-5. MR 0466988 (57:6861)

25.
Sone, Y., Takata, S.: Discontinuity of the velocity distribution function in a rarefied gas around a convex body and the $ s$ layer at the bottom of the Knudsen layer. Transport Theory Statist. Phys. 21 (1992) 501-530.

26.
Toscani, G.: Global solution of the initial value problem for the Boltzmann equation near a local Maxwellian. Arch. Rational Mech. Anal. 102 (1988) 231-241. MR 944547 (90b:35225)

27.
Toscani, G.: $ H$-theorem and asymptotic trend of the solution for a rarefied gas in a vacuum. Arch. Rational Mech. Anal. 100 (1987) 1-12. MR 906131 (89b:82053)

28.
Toscani, G.: On the nonlinear Boltzmann equation in unbounded domains, Arch. Rational Mech. Anal. 95 (1986) 37-49. MR 849403 (87j:82060)

29.
Toscani, G., Protopopescu, V.: The nonlinear Boltzmann equation with partially absorbing boundary conditions: Global existence and uniqueness results.

30.
Ukai, S.: Solutions of the Boltzmann equation. Patterns and waves. Stud. Math. Appl., 18, North-Holland, Amsterdam, 1986. MR 882376 (88g:35187)

31.
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)

32.
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)

Similar Articles:

Retrieve articles in Quarterly of Applied Mathematics with MSC (2000): 35Q35

Retrieve articles in all Journals with MSC (2000): 35Q35

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
Email: syha@math.snu.ac.kr

PII: S0033-569X-07-01060-4
Received by editor(s): April 13, 2006
Posted: August 24, 2007
Dedicated: This paper is dedicated to Tai-Ping Liu on the occasion of his sixtieth birthday.
Copyright of article: Copyright 2007, Brown University
The copyright for this article reverts to public domain after 28 years from publication.


Brown University The Quarterly of Applied Mathematics
is distributed by the American Mathematical Society
for Brown University
Online ISSN 1552-4485; Print ISSN 0033-569X
© 2008 Brown University
Comments: qam-query@ams.org		 AMS Website