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研究生:許祐承
研究生(外文):Hsu, Yu-Cheng
論文名稱:回顧波茲曼方程的能量估計法
論文名稱(外文):The review of energy method for Boltzmann equation
指導教授:江金城江金城引用關係
指導教授(外文):JIANG, JIN-CHENG
口試委員:蔡東和李明憶
口試委員(外文):TSAI, DONG-HOLI, MING-YI
口試日期:2020-6-29
學位類別:碩士
校院名稱:國立清華大學
系所名稱:數學系
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
論文出版年:2020
畢業學年度:108
語文別:中文
論文頁數:61
中文關鍵詞:波茲曼方程H定理麥克斯威爾狀態巨觀-微觀分解
外文關鍵詞:Boltzmann equationH-TheoremMaxwellian stateMacro-micro decomposition
相關次數:
  • 被引用被引用:0
  • 點閱點閱:112
  • 評分評分:
  • 下載下載:5
  • 收藏至我的研究室書目清單書目收藏:0
在這篇論文中,我們複習波茲曼方程的巨觀-微觀分解,並利用此分解得出能量估計法。我們利用能量估計法來研究全域麥克斯威爾狀態的時間漸進,非線性穩定性問題。
In this paper, we review the macro-micro decomposition of the Boltzmann equation which yields the energy estimate method.We will apply this method to study the problem of time-asymptotic, non-linear stability of the global Maxwellian state.
1.Introduction...1
1.1 Setting...1
1.2 Linearised Collision Operator...4
1.3 The Conservation Laws...6
2.H-Theorem...11
2.1 Microscopic version of H-theorem...11
2.2 Boltzmann inequality...12
2.3 Macroscopic version of H-theorem...14
3.Nonlinear stability of a Maxwellian state...29
3.1 Lower order energy estimates of the fluid variables...29
3.2 Lower order energy estimates of the non-fluid variables...34
3.3 Higher order energy estimates...43
Appendices...54
I Some Calculations...54
II Collision Operator...56
III The proof of estimate on Q({\bf G},{\bf G})...58
Reference...60
1. C.Cercignani. R.Illner and M.Pulvirenti.The Mathematical Theory of Dilute Gases. Springer, New York,(1994).


2. J.C.Maxwell. Illustrations of the dynamical theory of gases. Philosophical Magazine,I.II.III,19,19-32;20,21-32;20,33-37 (1860).


3. J.C.Maxwell. On the dynamical theory of gases.
Phil.Tran.Roy.Soc.London157, 49-88 (1867).



4. J.C.Maxwell. Dynamical theory of gases.The Scientific Papers of James Clerk Maxwell,Cambridge University Press, Cambridge, vol.II, 1980,p.26;
J.C.Maxwell. \emph{Stresses in rarefied gases arising from inequalities of temperature .The Scientific Papers of James Clerk Maxwell, Cambridge University Press,Cambridge,p.681.


5. L.Boltzmann. Lectures on Gas Theory. Dover, New York,1964.


6. D.Hilbert . Grundz{\"u}ge einer allgemeinen Theorie der linearen Integralgleichungen, Chelsea, New York,1953, Chapter 22.


7. T.Carleman . Sur la th{\'e}orie de l'{\'e}quation int{\'e}grodiff{\'e}rentielle de Boltzmann Acta Math.60(1933) 91-142.


8. H.Grad. Asymptotic theory of the Boltzmann equation, J.A.Laurmann(Ed.), Rarefied Gas Dynamics, vol.1,no.26, Academic Press, New York,1963,p.26-59.


9. L.Boltzmann. Weitere studien {\"u}ber das w{\"a}rmegleichgenicht unfer gasmolek{\"u}ler, Stizungsberichte der Akademie der Wissenschaften Wien.66,275-370(1872).


10. 張顯文. 非線性波茲曼方程
,北京:科學出版社,2015.


11. C.Cercignani. Theory and application of the Boltzmann equation
, New York : Elsevier ; Edinburgh : Scottish Academic Press ; London : Distributed by Chatto and Windus, 1975.


12. F.Golse,B.Perthame,C.Sulem}. On a boundary layer problem for the non-linear Boltzmann equation
, Arch.Ration.Mech.Anal.103(1988)81-96


13. Yoshio Sone. Molecular Gas Dynamics Theory,Techniques,and Applications
1st ed., Boston, MA : Birkhuser Boston, 2007.


14. Tai-Ping Liu,Tong Yang,Shih-Hsien Yu. Energy method for Boltzmann equation
,Physica D: Nonlinear Phenomena
Volume 188, Issues 3–4, 1 February 2004, Pages 178-192.
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