2003 Fiscal Year Final Research Report Summary
Multi-Scale Analysis of Differential Equations for Many Particle System
| Project/Area Number |
13640207
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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 |
Global analysis
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| Research Institution | YOKOHAMA NATIONAL UNIVERSITY |
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Principal Investigator |
UKAI Seiji Yokohama National Univ., Faculty of Engineering, Prof., 大学院・工学研究院, 教授 (30047170)
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| Co-Investigator(Kenkyū-buntansha) |
SHIOJI Naoki Yokohama National Univ., Faculty of Environment Info.Sci., associate Prof., 大学院・環境情報研究院, 助教授 (50215943)
HIRANO Norimiti Yokohama National Univ., Faculty of Environment Info.Sci., Prof., 大学院・環境情報研究院, 教授 (80134815)
KONNO Norio Yokohama National Univ., Faculty of Engineering, Associate Prof., 大学院・工学研究院, 助教授 (80205575)
MORIMOTO Hirtoko Keio Univ., Faculty of Engineering, Prof., 理工学部, 教授 (50061974)
TANI Atusi Keio Univ., Faculty of Engineering, Prof., 理工学部, 教授 (90118969)
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| Project Period (FY) |
2001 – 2003
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| Keywords | Boltzmann-Grad Limit / Cauchy-Kovalevskaya Theorem / Boltzmann equation / fluid equation / multi-scale analysis / asymptotic analysis / boundary layer solution / solvability condition |
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| Research Abstract |
1.Derivation of Uniform Estimates for the Boltzmann-Grad Limits : The Newton equation of motion for the many-particle system gives rise to the Boltzmann equation in the limit of two scale parameters, the number of particles N and the radius of the particle r, as N→∞ and r→0, under the condition Nr^2=constant. The mathematical proof of this convergence was proven by O.Lanford (1975), the most crucial part of which is the uniform estimates in N and r for the solutions of the Newton equation. We showed that the technique of the abstract version of Cauchy-Kovalevskaya theorem can give improved estimates.
2.Establishment of the solvability condition of the nonlinear boundary layer problem of the Boltzmann Equation : The most basic boundary layer is the solution of the boundary value problem in the half-space. However, the problem is not unconditionally solvable because the boundary condition at infinity is over-determined. We established the solvability condition on the boundary data. More precisely, we showed that the number of restrictions on the boundary data depends on the Mach number M at infinity, as 5 for M>1,4 for <M<1,1 for -1<M<0 and 0 for M<-1. The proof relies on sharp a priori estimates of solutions, which is obtained by use of a proper weight function and by introduction of a new artificial damping term.
3.Proof of the stability of the nonlinear boundary layer. We proved that the stationary solutions obtained above are exponentially stable for the case M<-1,. First, the exponential decay is established for the linearized equation, using the energy method, and then the nonlinear stability is established by the contraction mapping principle.
4.Asymptotic analysis of Fluid equations : The uniform estimates of solutions needed in establishing asymptotic relations between various fluid equations are derived by a unified method based on the abstract Cauchy-Kovalevskaya technique introduced in 1.
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