American Institute of Mathematical Sciences

Advanced
June  2010, 3(2): 281-297. doi: 10.3934/krm.2010.3.281

Rigorous validity of the Boltzmann equation for a thin layer of a rarefied gas

Raffaele Esposito 1, and  Mario Pulvirenti 2,

1. 

Dipartimento di Matematica pura ed Applicata, Università dell’Aquila, Via Vetoio - Coppito, L’Aquila, 67100, Italy
2. 

Dipartimento di Matematica Guido Castelnuovo, Università La Sapienza, P.e Aldo Moro, 2, 67100 Roma, Italy

Received  January 2010 Revised  February 2010 Published  May 2010

We consider a thin layer of a rarified gas modeled by a large hard-sphere system and show that, as long as the thickness of the layer is much larger than the interaction length, the limiting behavior is described, at least for short times, by a Boltzmann equation with two-dimensional position variable and three-dimensional velocity. By the analysis of the Lorentz gas we argue that, if the thickness of the layer is of the same order of the interaction length, this is not the case.
Keywords: Boltzmann Equation, Rarified Gases, Thin Films..
Mathematics Subject Classification: Primary: 76P05, 82C40; Secondary: 76A2.
Citation: Raffaele Esposito, Mario Pulvirenti. Rigorous validity of the Boltzmann equation for a thin layer of a rarefied gas. Kinetic & Related Models, 2010, 3 (2) : 281-297. doi: 10.3934/krm.2010.3.281
[1]

Rejeb Hadiji, Ken Shirakawa. Asymptotic analysis for micromagnetics of thin films governed by indefinite material coefficients. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1345-1361. doi: 10.3934/cpaa.2010.9.1345
[2]

Xiao-Ping Wang, Ke Wang, Weinan E. Simulations of 3-D domain wall structures in thin films. Discrete & Continuous Dynamical Systems - B, 2006, 6 (2) : 373-389. doi: 10.3934/dcdsb.2006.6.373
[3]

Yang Liu, Wenke Li. A class of fourth-order nonlinear parabolic equations modeling the epitaxial growth of thin films. Discrete & Continuous Dynamical Systems - S, 2021, 14 (12) : 4367-4381. doi: 10.3934/dcdss.2021112
[4]

Tai-Ping Liu, Shih-Hsien Yu. Boltzmann equation, boundary effects. Discrete & Continuous Dynamical Systems, 2009, 24 (1) : 145-157. doi: 10.3934/dcds.2009.24.145
[5]

Leif Arkeryd, Anne Nouri. On a Boltzmann equation for Haldane statistics. Kinetic & Related Models, 2019, 12 (2) : 323-346. doi: 10.3934/krm.2019014
[6]

Marina Chugunova, Roman M. Taranets. New dissipated energy for the unstable thin film equation. Communications on Pure & Applied Analysis, 2011, 10 (2) : 613-624. doi: 10.3934/cpaa.2011.10.613
[7]

Eric A. Carlen, Süleyman Ulusoy. Localization, smoothness, and convergence to equilibrium for a thin film equation. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4537-4553. doi: 10.3934/dcds.2014.34.4537
[8]

Richard S. Laugesen. New dissipated energies for the thin fluid film equation. Communications on Pure & Applied Analysis, 2005, 4 (3) : 613-634. doi: 10.3934/cpaa.2005.4.613
[9]

Changchun Liu, Jingxue Yin, Juan Zhou. Existence of weak solutions for a generalized thin film equation. Communications on Pure & Applied Analysis, 2007, 6 (2) : 465-480. doi: 10.3934/cpaa.2007.6.465
[10]

Jian-Guo Liu, Jinhuan Wang. Global existence for a thin film equation with subcritical mass. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1461-1492. doi: 10.3934/dcdsb.2017070
[11]

Claude Bardos, François Golse, Ivan Moyano. Linear Boltzmann equation and fractional diffusion. Kinetic & Related Models, 2018, 11 (4) : 1011-1036. doi: 10.3934/krm.2018039
[12]

Yan Guo, Juhi Jang, Ning Jiang. Local Hilbert expansion for the Boltzmann equation. Kinetic & Related Models, 2009, 2 (1) : 205-214. doi: 10.3934/krm.2009.2.205
[13]

Raffaele Esposito, Yan Guo, Rossana Marra. Validity of the Boltzmann equation with an external force. Kinetic & Related Models, 2011, 4 (2) : 499-515. doi: 10.3934/krm.2011.4.499
[14]

El Miloud Zaoui, Marc Laforest. Stability and modeling error for the Boltzmann equation. Kinetic & Related Models, 2014, 7 (2) : 401-414. doi: 10.3934/krm.2014.7.401
[15]

Alexander Bobylev, Åsa Windfäll. Boltzmann equation and hydrodynamics at the Burnett level. Kinetic & Related Models, 2012, 5 (2) : 237-260. doi: 10.3934/krm.2012.5.237
[16]

Radjesvarane Alexandre. A review of Boltzmann equation with singular kernels. Kinetic & Related Models, 2009, 2 (4) : 551-646. doi: 10.3934/krm.2009.2.551
[17]

Igor Kukavica, Mohammed Ziane. Regularity of the Navier-Stokes equation in a thin periodic domain with large data. Discrete & Continuous Dynamical Systems, 2006, 16 (1) : 67-86. doi: 10.3934/dcds.2006.16.67
[18]

Huiqiang Jiang. Energy minimizers of a thin film equation with born repulsion force. Communications on Pure & Applied Analysis, 2011, 10 (2) : 803-815. doi: 10.3934/cpaa.2011.10.803
[19]

Daniel Ginsberg, Gideon Simpson. Analytical and numerical results on the positivity of steady state solutions of a thin film equation. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1305-1321. doi: 10.3934/dcdsb.2013.18.1305
[20]

M. Ben Ayed, K. El Mehdi, M. Hammami. Nonexistence of bounded energy solutions for a fourth order equation on thin annuli. Communications on Pure & Applied Analysis, 2004, 3 (4) : 557-580. doi: 10.3934/cpaa.2004.3.557

2020 Impact Factor: 1.432

Tools

Metrics

  • PDF downloads (47)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy