| Co-Investigator(Kenkyū-buntansha) |
TANI Atsusi Keio University, Department of Mathematics, Faculty of Science and Technology, Professor, 理工学部, 教授 (90118969)
MATSUMURA Akitaka Osaka University, Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Professor, 大学院・情報科学研究科, 教授 (60115938)
今野 紀雄 国立大学法人横浜国立大学, 大学院・工学研究院, 助教授 (80205575)
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| Budget Amount *help |
¥1,600,000 (Direct Cost: ¥1,600,000)
Fiscal Year 2005: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2004: ¥800,000 (Direct Cost: ¥800,000)
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| Research Abstract |
The purpose of the present study is to clarify some properties of transit layers of the Boltzmann equation. In general, nonlinear partial differential equations, which describe complex phenomena in various fields of mathematical sciences, are investigated on fundamental mathematical structures of solutions, including the existence, uniqueness and asymptotic behavior, with the help of functional analysis, harmonic analysis, operator theory, theory of bifurcation and so on. Applications are made for the Navier-Stokes, Boltzmann and related equations which govern the motion of fluids, on the time-global existence of solutions, multi-scale analysis which establishes the asymptotic relations between these equations, bifurcating solutions, shock wave profiles, and mathematical mechanism of the development of transit layers. The theory of chaos is also investigated, which aims at qualitative and quantitative descriptions of complexity of behaviors of solutions to nonlinear equations. The chaos implies the difficulty of prediction of phenomena governed by the deterministic (non-probabilistic) law of motion, as shown by the famous Lorenz equation. The theory of chaos is now well-established for systems of finite degree. In particular, it is known that the existence of scrambled sets of Li-Yorke type implies the chaos. However, no concrete examples having scrambled sets are known of systems of infinite degree such as nonlinear partial differential equations. The condition for systems of finite degree with infinite dimensional compact perturbations to have the scrambled set is studied. Along this line, we have investigated transit layers of the Boltzmann equation. In addition, relations of between nonlinear differential equations and quantum walks were also discussed from various aspects.
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