2012 Fiscal Year Final Research Report
The Boltzmann equation and nonlinear microlocal analysis
| Project/Area Number |
22540187
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | Kyoto University |
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Principal Investigator |
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| Co-Investigator(Kenkyū-buntansha) |
UEKI Naomasa 京都大学, 大学院・人間・環境学研究科, 准教授 (80211069)
ASAKURA Fumioki 大阪電気通信大学, 金融経済学部, 教授 (20140238)
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| Research Collaborator |
UKAI Seiji 東京工業大学, 名誉教授
ALEXANDRE Radjesvarane French Naval Academy, 教授 仏
LERNER Nicolas パリ 6 大学, 数学科, 教授 仏
PRAVDA-STAROV Karel Cergy-Pointoise 大学, 数学科, 講師 仏
XU Chao-jiang Rouen 大学, 数学科, 教授 仏
YANG Tong City University of Hong Kong, 数学科, 教授
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| Project Period (FY) |
2010 – 2012
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| Keywords | ボルツマン方程式 / 衝突積分項 / 非切断近似 / 平滑効果 / 大域解 / 安定性 / 解の一意性 / 測度解 |
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| Research Abstract |
W e considered the Cauchy problem for the Boltzmann equation without angular cutoff approximation. To this Cauchy problem, we showed the structure of solutions, such as, the existence of solutions, the smoothing effect of solutions, the convergence of the equilibrium state, the uniqueness and the non-negativityof solutions, under the physically reasonable assumptions of the collision cross sections. The key ingredient of this study was to give precise lower and upper estimates forthe Boltzmann collision integral operator which contains the singularity with respect to the angular variable between collision particles, by means of the microlocal analysis developed in the theory of linear partial differential equations.
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