2001 Fiscal Year Final Research Report Summary
Problems of Hydrodynamic Limits and Related Subjects
| Project/Area Number |
11440026
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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 | Tokyo Institute of Technology |
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Principal Investigator |
UCHIYAMA Kohei Graduate School of Science and Engineering, Tokyo Institute of Technology, Professor, 大学院・理工学研究科, 教授 (00117566)
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| Co-Investigator(Kenkyū-buntansha) |
SHIGA Hirosige Graduate School of Science and Engineering, Tokyo Institute of Technology, Professor, 大学院・理工学研究科, 教授 (10154189)
MURATA Minoru Graduate School of Science and Engineering, Tokyo Institute of Technology, Professor, 大学院・理工学研究科, 教授 (50087079)
FUNAKI Tadahisa The University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (60112174)
SHIRAI Tomoyuki Graduate School of Science and Engineering, Tokyo Institute of Technology, Assistant, 大学院・理工学研究科, 助手 (70302932)
MORITA Takehiko Hiroshima University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (00192782)
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| Project Period (FY) |
1999 – 2001
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| Keywords | hydrodynamic limit / local equilibrium / scaling limit / relative entropy / large deviation |
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| Research Abstract |
1. The hydrodynamic limit for a system of multi-dimensional Brownian particles which interact one another through a two-body potential having a finite range and satisfying the super-stability condition is derived under the hypothesis that the third moments of the empirical densities are uniformly bounded. For the derivation we also show a kind of virial theorem in the classical theory of Gibbs statistical mechanics: the pressure is represented as a limit of normalized sums of the virials in the law of large numbers in every space dimension [K. Uchiyama, Pressure in classical statistical mechanics and interacting Brownian particles in multi-dimensions, Annales Henri Poincare (J. Theor. Math. Phys.) 1,1159-1202 (2000)]. Although such a theorem has already been proved if the uniqueness for Gibbs states holds (as in the cases of one-dimension or of high temperature ), this is for the first time of the proof without assuming the uniqueness.
2. A class of one-dimensional evolution equations which are nonlocal and derived in the scaling limits from systems of interacting particles obeying the classical law of motion is studied. As fundamental theorems there are established the existence and the uniqueness of solutions, the maximum principle, a comparison theorem for integrated solutions, and the convergences to the 'Barenblatt' solutions, etc. In the particular case when the interacting force in the particle system is given by a logarithmic potential, quite fine properties of the solutions are obtained. [K.Uchiyama, Behavior of solutions to the initial value problem for a class of integro-differential equations, Nonlinear Analysis (2001).] These results if combined with former results on the scaling limits give a clear macroscopic view about the behavior of the underlying microscopic model of particles when the interaction potential has a long range.
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