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題名 波茲曼方程式柔和解的存在性
Existence result of mild solution of Boltzmann equation作者 魏照誠
Chao-Cheng Wei貢獻者 李明融
魏照誠
Chao-Cheng Wei關鍵詞 波茲曼方程
柔和解的存在性
均勻穩定性日期 2009 上傳時間 4-Sep-2013 15:15:35 (UTC+8) 摘要 我們利用Banach固定點定理證明了波茲曼方程式的柔和解在一個加權魯貝格空間上的存在性，並證明其解在如此的範數下之均勻穩定性。關鍵詞: 波茲曼方程，馬氏分佈函數，柔和解的存在性，均勻穩定。
In this thesis, we consider the initial-value problem for the Boltzmann equation.We prove that the existence of mild solution in the weighted Lebesgue space by using Banach`s fixed point theorem, and that the uniform stability of solution with respect to the weighted norm.Key words: Boltzmann equation, Maxwellian, existence of mild solutions, uniform stability.參考文獻 @BOOK{Marsden, title = {A Mathematical Introduction to Fluid Mechanics}, publisher = {Springer}, year = {1992}, editor = {3rd}, author = {A.J.Chorin, J.E.Marsden}, owner = {FUJITSU}, timestamp = {2010.03.21}}@BOOK{Serge, title = {Hyperbolic Patial Differential Equations}, publisher = {Springer}, year = {2009}, author = {Serge Alinhac}, owner = {FUJITSU}, timestamp = {2010.03.21}}@BOOK{Bird, title = {Molecular Gas Dynamics and the Direct simulation of Gas Flows}, publisher = {Oxfold}, year = {1994}, author = {G.A. Bird}, owner = {FUJITSU}, timestamp = {2010.03.21}}@BOOK{Evans, title = {Partial Differential Equations}, publisher = {American Mathematical Society}, year = {1998}, author = {Lawrence C.Evans}, owner = {FUJITSU}, timestamp = {2010.03.20}}@BOOK{Mario, title = {The Mathematical Theory of Dilute Gases}, publisher = {Springer}, year = {1994}, author = {Carlo Cercignani, Reinhard llner, Mario Pulvirenti}, owner = {FUJITSU}, timestamp = {2010.03.20}}@BOOK{Carlo, title = {The Boltzmann Equation and Its Applications}, publisher = {Springer}, year = {1988}, author = {Carlo Cercignani}, owner = {FUJITSU}, timestamp = {2010.03.20}}@ARTICLE{Hilbert, author = {D.Hilbert}, title = {Begr\\"{u}ndung der Kinetischen Gastheorie}, journal = {Mathematische Annalen 72}, year = {1916/1917}, pages = {562-577}, owner = {FUJITSU}, timestamp = {2010.04.29}}@MISC{Diperna, author = {R.J. Diperna and P.L. Lions}, title = {On the {C}auchy problem for {B}oltzmann equation: {G}lobal existence and weak stability}, howpublished = {Ann. of Math. 130}, year = {1989}, note = {pp. 321–366}, owner = {FUJITSU}, timestamp = {2010.04.09}}@MISC{Duan, author = {Renjun Duan}, title = {On the {C}auchy problem for the {B}oltzmann equation in the whole space: {G}lobal existence and uniform stability in ${L}^2_{\\xi}({H}^{N}_{x})$}, howpublished = {Journal of Differential Equations Volume 244, Issue 12}, month = {15 June}, year = {2008}, owner = {FUJITSU}, timestamp = {2010.04.09}}@BOOK{Evans, title = {Partial Differential Equations}, publisher = {Providence, R.I. : American Mathematical Society}, year = {1998}, author = {Lawrence C. Evans}, owner = {FUJITSU}, timestamp = {2010.04.06}}@MISC{Guo, author = {Y. Guo}, title = {The {B}oltzmann equation in the whole space}, howpublished = {Indiana Univ. Math. J. 53}, year = {2004}, note = {pp. 1081–1094}, owner = {FUJITSU}, timestamp = {2010.04.09}}@MISC{Guo3, author = {Y. Guo}, title = {The {V}lasov - {M}axwell - {B}oltzmann system near {M}axwellians}, howpublished = {, Invent. Math. 153}, year = {2003}, note = {pp. 593–630}, owner = {FUJITSU}, timestamp = {2010.04.09}}@MISC{Guo5, author = {Y. Guo}, title = {Classical solutions to the {B}oltzmann equation for molecules with an angular cutoff}, howpublished = {Arch. Ration. Mech. Anal. 169}, year = {2003}, note = {pp. 305–353}, owner = {FUJITSU}, timestamp = {2010.04.09}}@MISC{Guo2, author = {Y. Guo}, title = {The {L}andau equation in a periodic box}, howpublished = {Comm. Math. Phys. 231}, year = {2002}, note = {pp. 391–434}, owner = {FUJITSU}, timestamp = {2010.04.09}}@MISC{Guo4, author = {Y. Guo}, title = {The {V}lasov -{P}oisson-{B}oltzmann system near {M}axwellians}, howpublished = {Comm. Pure Appl. Math. 55 (9)}, year = {2002}, note = {pp. 1104–1135}, owner = {FUJITSU}, timestamp = {2010.04.09}}@ARTICLE{Grad, author = {H.Grad}, title = {Asymptotic equivalence of the {N}avier-{S}tokes and nonlinear {B}oltzmann equations}, journal = {Proc.Symp.Appl.Math. 17}, year = {1965}, pages = {p.154-183}, owner = {FUJITSU}, timestamp = {2010.04.11}}@MISC{Yu2, author = {T.-P. Liu and S.-H. Yu}, title = {Green`s function of {B}oltzmann equation, 3-{D} waves}, howpublished = {Bull. Inst. Math. Acad. Sin. (N.S.) 1}, year = {2006}, note = {pp. 1–78}, owner = {FUJITSU}, timestamp = {2010.04.09}}@MISC{Liu2, author = {T.-P. Liu and S.-H. Yu}, title = {Boltzmann equation: {M}icro -{M}acro decompositions and positivity of shock profiles}, howpublished = {Comm. Math. Phys. 246 (1)}, year = {2004}, note = {pp. 133–179.}, owner = {FUJITSU}, timestamp = {2010.04.09}}@MISC{Nishida, author = {T. Nishida and K. Imai}, title = {Global solutions to the initial value problem for the nonlinear {B}oltzmann equation}, howpublished = {Publ. Res. Inst. Math. Sci. 12}, year = {1976/77}, note = {pp. 229–239}, owner = {FUJITSU}, timestamp = {2010.04.09}}@MISC{Shizuta, author = {Y. Shizuta}, title = {On the classical solutions of the {B}oltzmann equation}, howpublished = {Comm. Pure Appl. Math. 36}, year = {1983}, note = {pp. 705–754}, owner = {FUJITSU}, timestamp = {2010.04.09}}@BOOK{Boltzmann, title = {Lectures On Gas Theory}, publisher = {Dover Publications}, year = {1995}, author = {Ludwig Boltzmann ; translated by Stephen G. Brush}, owner = {FUJITSU}, timestamp = {2010.03.21}}@BOOK{Chapman, title = {The Mathematical Theory of Non-Uniform Gases}, publisher = {Cambridge University Press}, year = {1970}, editor = {3rd}, author = {Sydney Chapman, T.G.Cowling}, owner = {FUJITSU}, timestamp = {2010.03.21}}@MISC{Liu, author = {T.-P. Liu, T. Yang and S.-H. Yu}, title = {Energy method for the {B}oltzmann equation}, howpublished = {Phys. D 188 (3–4)}, year = {2004}, note = {pp. 178–192}, owner = {FUJITSU}, timestamp = {2010.04.09}}@MISC{Yu, author = {T.-P. Liu, T. Yang, S.-H. Yu and H. Zhao}, title = {Nonlinear stability of rarefaction waves for {B}oltzmann equation}, howpublished = {Arch. Ration. Mech. Anal. 181 (2)}, year = {2006}, note = {pp. 333–371}, owner = {FUJITSU}, timestamp = {2010.04.09}}@ARTICLE{Carleman, author = {T.S.Carleman}, title = {Sur la th\\`{e}orie de l`\\`{e}quation int\\`{e}gro-differentielle de {B}oltzmann}, journal = {Acta.Math,60}, year = {1932}, pages = {p.91-146}, owner = {FUJITSU}, timestamp = {2010.04.10}}@MISC{Ukai, author = {S. Ukai}, title = {On the existence of global solutions of mixed problem for non-linear {B}oltzmann equation}, howpublished = {Proc. Japan Acad. 50}, year = {1974}, note = {pp. 179–184}, owner = {FUJITSU}, timestamp = {2010.04.09}}@MISC{Yang2, author = {S. Ukai and T. Yang}, title = {The {B}oltzmann equation in the space ${L}^2\\cap {L}^{\\infty}_{\\beta}$ : Global and time-periodic solutions}, howpublished = {Anal. Appl. 4}, year = {2006}, note = {pp. 263–310}, owner = {FUJITSU}, timestamp = {2010.04.09}}@BOOK{Yang3, title = {Mathematical Theory of Boltzmann Equation}, publisher = {Liu Bie Ju Centre for Mathematical Sciences, Lecture Notes Series, No. 8}, year = {2006}, author = {S. Ukai and T. Yang}, owner = {FUJITSU}, timestamp = {2010.05.15}}@MISC{Yang, author = {T. Yang and H.-J. Zhao}, title = {A new energy method for the {B}oltzmann equation}, howpublished = {J. Math. Phys. 47}, year = {2006}, note = {p. 053301}, owner = {FUJITSU}, timestamp = {2010.04.09}}@ARTICLE{Yu3, author = {Shih-Hsien Yu}, title = {Hydrodynamic limits with Shock Waves of the {B}oltzmann equation}, journal = {Commun. Pur. Appl. Math, 58}, year = {2005}, pages = {409-443}, owner = {FUJITSU}, timestamp = {2010.05.02}} 描述 碩士
國立政治大學
應用數學研究所
97751004
98資料來源 http://thesis.lib.nccu.edu.tw/record/#G0097751004 資料類型 thesis dc.contributor.advisor 李明融 zh_TW dc.contributor.author (Authors) 魏照誠 zh_TW dc.contributor.author (Authors) Chao-Cheng Wei en_US dc.creator (作者) 魏照誠 zh_TW dc.creator (作者) Chao-Cheng Wei en_US dc.date (日期) 2009 en_US dc.date.accessioned 4-Sep-2013 15:15:35 (UTC+8) - dc.date.available 4-Sep-2013 15:15:35 (UTC+8) - dc.date.issued (上傳時間) 4-Sep-2013 15:15:35 (UTC+8) - dc.identifier (Other Identifiers) G0097751004 en_US dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/60082 - dc.description (描述) 碩士 zh_TW dc.description (描述) 國立政治大學 zh_TW dc.description (描述) 應用數學研究所 zh_TW dc.description (描述) 97751004 zh_TW dc.description (描述) 98 zh_TW dc.description.abstract (摘要) 我們利用Banach固定點定理證明了波茲曼方程式的柔和解在一個加權魯貝格空間上的存在性，並證明其解在如此的範數下之均勻穩定性。關鍵詞: 波茲曼方程，馬氏分佈函數，柔和解的存在性，均勻穩定。 zh_TW dc.description.abstract (摘要) In this thesis, we consider the initial-value problem for the Boltzmann equation.We prove that the existence of mild solution in the weighted Lebesgue space by using Banach`s fixed point theorem, and that the uniform stability of solution with respect to the weighted norm.Key words: Boltzmann equation, Maxwellian, existence of mild solutions, uniform stability. en_US dc.description.tableofcontents 1.Introduction ............................................12.Basic Kinetic Theory of Gases and the Boltzmann Equation.43.Element Properties of the Collision Operator............154.Hydrodynamical Limits:Expansions and Decomposition......225.Solutions in a Weighted Lebesgue Space..................35Reference.................................................44 zh_TW dc.format.extent 561545 bytes - dc.format.mimetype application/pdf - dc.language.iso en_US - dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0097751004 en_US dc.subject (關鍵詞) 波茲曼方程 zh_TW dc.subject (關鍵詞) 柔和解的存在性 zh_TW dc.subject (關鍵詞) 均勻穩定性 zh_TW dc.title (題名) 波茲曼方程式柔和解的存在性 zh_TW dc.title (題名) Existence result of mild solution of Boltzmann equation en_US dc.type (資料類型) thesis en dc.relation.reference (參考文獻) @BOOK{Marsden, title = {A Mathematical Introduction to Fluid Mechanics}, publisher = {Springer}, year = {1992}, editor = {3rd}, author = {A.J.Chorin, J.E.Marsden}, owner = {FUJITSU}, timestamp = {2010.03.21}}@BOOK{Serge, title = {Hyperbolic Patial Differential Equations}, publisher = {Springer}, year = {2009}, author = {Serge Alinhac}, owner = {FUJITSU}, timestamp = {2010.03.21}}@BOOK{Bird, title = {Molecular Gas Dynamics and the Direct simulation of Gas Flows}, publisher = {Oxfold}, year = {1994}, author = {G.A. Bird}, owner = {FUJITSU}, timestamp = {2010.03.21}}@BOOK{Evans, title = {Partial Differential Equations}, publisher = {American Mathematical Society}, year = {1998}, author = {Lawrence C.Evans}, owner = {FUJITSU}, timestamp = {2010.03.20}}@BOOK{Mario, title = {The Mathematical Theory of Dilute Gases}, publisher = {Springer}, year = {1994}, author = {Carlo Cercignani, Reinhard llner, Mario Pulvirenti}, owner = {FUJITSU}, timestamp = {2010.03.20}}@BOOK{Carlo, title = {The Boltzmann Equation and Its Applications}, publisher = {Springer}, year = {1988}, author = {Carlo Cercignani}, owner = {FUJITSU}, timestamp = {2010.03.20}}@ARTICLE{Hilbert, author = {D.Hilbert}, title = {Begr\\"{u}ndung der Kinetischen Gastheorie}, journal = {Mathematische Annalen 72}, year = {1916/1917}, pages = {562-577}, owner = {FUJITSU}, timestamp = {2010.04.29}}@MISC{Diperna, author = {R.J. Diperna and P.L. Lions}, title = {On the {C}auchy problem for {B}oltzmann equation: {G}lobal existence and weak stability}, howpublished = {Ann. of Math. 130}, year = {1989}, note = {pp. 321–366}, owner = {FUJITSU}, timestamp = {2010.04.09}}@MISC{Duan, author = {Renjun Duan}, title = {On the {C}auchy problem for the {B}oltzmann equation in the whole space: {G}lobal existence and uniform stability in ${L}^2_{\\xi}({H}^{N}_{x})$}, howpublished = {Journal of Differential Equations Volume 244, Issue 12}, month = {15 June}, year = {2008}, owner = {FUJITSU}, timestamp = {2010.04.09}}@BOOK{Evans, title = {Partial Differential Equations}, publisher = {Providence, R.I. : American Mathematical Society}, year = {1998}, author = {Lawrence C. Evans}, owner = {FUJITSU}, timestamp = {2010.04.06}}@MISC{Guo, author = {Y. Guo}, title = {The {B}oltzmann equation in the whole space}, howpublished = {Indiana Univ. Math. J. 53}, year = {2004}, note = {pp. 1081–1094}, owner = {FUJITSU}, timestamp = {2010.04.09}}@MISC{Guo3, author = {Y. Guo}, title = {The {V}lasov - {M}axwell - {B}oltzmann system near {M}axwellians}, howpublished = {, Invent. Math. 153}, year = {2003}, note = {pp. 593–630}, owner = {FUJITSU}, timestamp = {2010.04.09}}@MISC{Guo5, author = {Y. Guo}, title = {Classical solutions to the {B}oltzmann equation for molecules with an angular cutoff}, howpublished = {Arch. Ration. Mech. Anal. 169}, year = {2003}, note = {pp. 305–353}, owner = {FUJITSU}, timestamp = {2010.04.09}}@MISC{Guo2, author = {Y. Guo}, title = {The {L}andau equation in a periodic box}, howpublished = {Comm. Math. Phys. 231}, year = {2002}, note = {pp. 391–434}, owner = {FUJITSU}, timestamp = {2010.04.09}}@MISC{Guo4, author = {Y. Guo}, title = {The {V}lasov -{P}oisson-{B}oltzmann system near {M}axwellians}, howpublished = {Comm. Pure Appl. Math. 55 (9)}, year = {2002}, note = {pp. 1104–1135}, owner = {FUJITSU}, timestamp = {2010.04.09}}@ARTICLE{Grad, author = {H.Grad}, title = {Asymptotic equivalence of the {N}avier-{S}tokes and nonlinear {B}oltzmann equations}, journal = {Proc.Symp.Appl.Math. 17}, year = {1965}, pages = {p.154-183}, owner = {FUJITSU}, timestamp = {2010.04.11}}@MISC{Yu2, author = {T.-P. Liu and S.-H. Yu}, title = {Green`s function of {B}oltzmann equation, 3-{D} waves}, howpublished = {Bull. Inst. Math. Acad. Sin. (N.S.) 1}, year = {2006}, note = {pp. 1–78}, owner = {FUJITSU}, timestamp = {2010.04.09}}@MISC{Liu2, author = {T.-P. Liu and S.-H. Yu}, title = {Boltzmann equation: {M}icro -{M}acro decompositions and positivity of shock profiles}, howpublished = {Comm. Math. Phys. 246 (1)}, year = {2004}, note = {pp. 133–179.}, owner = {FUJITSU}, timestamp = {2010.04.09}}@MISC{Nishida, author = {T. Nishida and K. Imai}, title = {Global solutions to the initial value problem for the nonlinear {B}oltzmann equation}, howpublished = {Publ. Res. Inst. Math. Sci. 12}, year = {1976/77}, note = {pp. 229–239}, owner = {FUJITSU}, timestamp = {2010.04.09}}@MISC{Shizuta, author = {Y. Shizuta}, title = {On the classical solutions of the {B}oltzmann equation}, howpublished = {Comm. Pure Appl. Math. 36}, year = {1983}, note = {pp. 705–754}, owner = {FUJITSU}, timestamp = {2010.04.09}}@BOOK{Boltzmann, title = {Lectures On Gas Theory}, publisher = {Dover Publications}, year = {1995}, author = {Ludwig Boltzmann ; translated by Stephen G. Brush}, owner = {FUJITSU}, timestamp = {2010.03.21}}@BOOK{Chapman, title = {The Mathematical Theory of Non-Uniform Gases}, publisher = {Cambridge University Press}, year = {1970}, editor = {3rd}, author = {Sydney Chapman, T.G.Cowling}, owner = {FUJITSU}, timestamp = {2010.03.21}}@MISC{Liu, author = {T.-P. Liu, T. Yang and S.-H. Yu}, title = {Energy method for the {B}oltzmann equation}, howpublished = {Phys. D 188 (3–4)}, year = {2004}, note = {pp. 178–192}, owner = {FUJITSU}, timestamp = {2010.04.09}}@MISC{Yu, author = {T.-P. Liu, T. Yang, S.-H. Yu and H. Zhao}, title = {Nonlinear stability of rarefaction waves for {B}oltzmann equation}, howpublished = {Arch. Ration. Mech. Anal. 181 (2)}, year = {2006}, note = {pp. 333–371}, owner = {FUJITSU}, timestamp = {2010.04.09}}@ARTICLE{Carleman, author = {T.S.Carleman}, title = {Sur la th\\`{e}orie de l`\\`{e}quation int\\`{e}gro-differentielle de {B}oltzmann}, journal = {Acta.Math,60}, year = {1932}, pages = {p.91-146}, owner = {FUJITSU}, timestamp = {2010.04.10}}@MISC{Ukai, author = {S. Ukai}, title = {On the existence of global solutions of mixed problem for non-linear {B}oltzmann equation}, howpublished = {Proc. Japan Acad. 50}, year = {1974}, note = {pp. 179–184}, owner = {FUJITSU}, timestamp = {2010.04.09}}@MISC{Yang2, author = {S. Ukai and T. Yang}, title = {The {B}oltzmann equation in the space ${L}^2\\cap {L}^{\\infty}_{\\beta}$ : Global and time-periodic solutions}, howpublished = {Anal. Appl. 4}, year = {2006}, note = {pp. 263–310}, owner = {FUJITSU}, timestamp = {2010.04.09}}@BOOK{Yang3, title = {Mathematical Theory of Boltzmann Equation}, publisher = {Liu Bie Ju Centre for Mathematical Sciences, Lecture Notes Series, No. 8}, year = {2006}, author = {S. Ukai and T. Yang}, owner = {FUJITSU}, timestamp = {2010.05.15}}@MISC{Yang, author = {T. Yang and H.-J. Zhao}, title = {A new energy method for the {B}oltzmann equation}, howpublished = {J. Math. Phys. 47}, year = {2006}, note = {p. 053301}, owner = {FUJITSU}, timestamp = {2010.04.09}}@ARTICLE{Yu3, author = {Shih-Hsien Yu}, title = {Hydrodynamic limits with Shock Waves of the {B}oltzmann equation}, journal = {Commun. Pur. Appl. Math, 58}, year = {2005}, pages = {409-443}, owner = {FUJITSU}, timestamp = {2010.05.02}} zh_TW
