| 非線形解析セミナーの記録（２００９年度） |
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| 日時 | ２月２７日（土） １４時００分～１７時００分 |
| 場所 | ３６棟２０５／２０７（２階セミナー室） |
| 講演者 | 大堀 祐司 氏 （慶應・理工） |
| 講演題目 | Mathematical Analysis of a Model Equation for Aeolian Sand Ripples |
| 講演者 | 尾花 高宏 氏 （慶應・理工） |
| 講演題目 | Mathematical Analysis of Bubble Rising |
| 講演者 | 相木 雅次 氏 （慶應・理工） |
| 講演題目 | Motion of a Vortex Filament in the Half Space |
| 日時 | １月１９日（火） １６時３０分～１８時００分 |
| 場所 | １４棟２１６（創想館２階ディスカッションルーム６） |
| 講演者 | 矢崎 成俊 氏（宮崎大・工） |
| 講演題目 | 時間に依存した隙間を持つセル中のHele-Shaw流れについて |
| 日時 | １２月１１日（金） １６時３０分～１８時００分 |
| 場所 | １４棟２１６（創想館２階ディスカッションルーム６） |
| 講演者 | Professor Chongchun Zeng （Georgia Inst. of Tech.） |
| 講演題目 | Euler equation with fixed or free boundaries: from a Lagrangian point of view |
| 講演要旨 | In this talk, we discuss 1.) the nonlinear instability and unstable manifolds of steady solutions of the Euler equation with fixed domains and 2.) the evolution of free (inviscid) fluid surfaces, which may involve vorticity, gravity, surface tension, or magnetic fields. These problems can be formulated in a Lagrangian formulation on infinite dimensional manifolds of volume preserving diffeomorphisms with an invariant Lie group action. In this setting, the physical pressure turns out to come from the combination of the gravity, surface tension, and the Lagrangian multiplier. The vorticity is naturally related to an invariant group action. In the absence of surface tension, the well-known Rayleigh-Taylor and Kelvin-Helmholtz instabilities appear naturally related to the signs of the curvatures of those infinite dimensional manifolds. Based on these considerations, we obtain 1.) the existence of unstable manifolds and $L^2$ nonlinear instability in the cases of the fixed domains and 2.) in the free boundary cases, the local well-posedness with surface tension in a rather uniform energy method. In particular, for the cases without surface tension which do not involve hydrodynamical instabilities, we obtain the local existence of solutions by taking the vanishing surface tension limit. |
| 日時 | １２月１日（火） １６時３０分～１８時００分 |
| 場所 | １４棟２１２（創想館２階ディスカッションルーム２） |
| 講演者 | Professor Kumbakonam R. Rajagopal （Texas A&M University） |
| 講演題目 | A hierarchy of models for the flow of fluids through porous solids |
| 講演要旨 | The celebrated partial differential equations due to Fick and Darcy are approximations that can be obtained systematically on the basis of numerous assumptions within the context of Mixture Theory; these equations however not having been developed in such a manner by Fick or Darcy. Relaxing the assumptions made in deriving these equations via mixture theory, selectively, leads to a hierarchy of mathematical models and it can be shown that popular models due to Forchheimer, Brinkman, Biot and many others can be obtained via appropriate approximations to the equations governing the flow of interacting continua. It is shown that a variety of other generalizations are possible in addition to those that are currently in favor, and these might be appropriate for describing numerous interesting technological applications, e.g., enhanced oil recovery. Moreover, these generalizations lead to very interesting and novel partial differential equations that have not yet been analysed and present interesting challenges. The numerical solutions of some of the simplest of these equations show that the structure of the solution present very interesting features such as pronounced boundary layers. |
| 日時 | １０月２８日（水） １６時３０分～１８時００分 |
| 場所 | １４棟２１６（創想館２階ディスカッションルーム６） |
| 講演者 | 高田 滋 氏（京大・工・機械理工） |
| 講演題目 | 定常系における線形化ボルツマン方程式の対称性とその応用 |
| 講演要旨 | 微小系における気体流は非平衡流であるが，外部が系に与える温度や圧力の平衡状態からの摂動は多くの場合に小さく，線形化ボルツマン方程式による取り扱いが許される．この講演では線形化ボルツマン方程式に内在する対称性を紹介し，それをもとに定常な微小系内の種々の気体流の間で一般的に成り立つ相反関係を報告する．これはオンサーガの相反関係の一般化ととらえてもよい．考え方の骨子を明確に伝えるために，なるべく簡単な状況に限定して説明するが，多くの具体例を紹介して，報告する相反関係の一般性と物理的な有用性を伝えたい． |
| 日時 | １０月２３日（金） １６時３０分～１８時００分 |
| 場所 | １４棟２１２（創想館２階ディスカッションルーム２） |
| 講演者 | 井口 達雄 氏（慶應・理工） |
| 講演題目 | A mathematical analysis of tsunami generation in shallow water due to seabed deformation |