| Project/Area Number |
08454036
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | The University of Tokyo |
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Principal Investigator |
FUNAKI Tadahisa Univ. of Tokyo. Grad. Sch. Math. Sci., Prof., 大学院・数理科学研究科, 教授 (60112174)
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| Co-Investigator(Kenkyū-buntansha) |
TSUTUSUMI Yoshio Tohoku Univ., Grad. Sch. Sci., Assosi. Prof., 大学院・数理科学研究科, 助教授 (10180027)
YAJIMA Kenji Univ. of Tokyo. Grad. Sch. Math. Sci., Prof., 大学院・数理科学研究科, 教授 (80011758)
KUSUOKA Shigeo Univ. of Tokyo. Grad. Sch. Math. Sci., Prof., 大学院・数理科学研究科, 教授 (00114463)
OSADA Hirofumi Nagoya Univ.., Grad. Sch. Polymath., Prof., 大学院・多元数理科学研究科, 教授 (20177207)
MIMURA Masayasu Hiroshima Univ., Grad. Sch. Sci., Prof., 大学院・理学研究科, 教授 (50068128)
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| Project Period (FY) |
1996 – 1998
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| Keywords | Hydrodynamic limit / Infinite particle system / Stochastic analysis / Nonlinear phenomena / Nonlinear partial differential equation / Phase separation / Interface / Ginzburg-Landau model |
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| Research Abstract |
Main results obtained by this research are the following :
1. Equilibrium fluctuations for higher dimensional stochastic lattice gas are studied and infinite-dimensional Ornstein-Uhlenbeck process is derived in the limit. Its characteristic quantity coincides with the diffusion coefficient obtained in the hydrodynamic limit.
2. It is proved that the phase separating boundary appearing in the limit of singular perturbation for stochastic reaction-diffusion equation moves according to the randomly perturbed motion by mean curvature.
3. Nonlinear diffusion equation with Hesse matrix of surface tension as its diffusion coefficient is obtained as an interface equation in the macroscopic scaling limit for effective Ginzburg-Landau ▽φ interface model. Corresponding large deviation is also discussed and it is shown that the same surface tension appears in the representation of rate functional.
4. Particle system with two types of particles is investigated. Such system is obtained by microscopically modeling medium accompanied by a change of phases, like a solid-liquid system. The free boundary problem for the macroscopic evolution of the phase separating boundary, called Stefan problem, is derived based on the method of the hydrodynamic limit. The effect of latent heat is also studied from the microscopic point of view.
5. For Burgers equation with fractional power of Laplacian, existence and uniqueness of global and local solutions, their regularity property, and others are investigated using analytic method. Then, based on probabilistic method, the corresponding nonlinear Markov process is constructed and Burgers equation with fractional power of Laplacian is derived from many particles' system by establishing the propagation of chaos.
Based on these results, we shall proceed the research on the problem of phase separation from several points of view.
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