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Multi-Scale Analysis of Differential Equations for Many Particle System

多粒子系统微分方程的多尺度分析

基本信息

批准号：
13640207
负责人：
UKAI Seiji
金额：
$ 2.24万
依托单位：
YOKOHAMA NATIONAL UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2003
项目状态：
已结题

项目摘要

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.

1.玻尔兹曼-格拉德极限的一致估计的推导：在Nr^2=常数的条件下，多粒子系统的牛顿运动方程在粒子数N和粒子半径r这两个尺度参数的极限下，当N→∞和r→0时，得到玻尔兹曼方程。这种收敛性的数学证明由O.Lanford（1975）证明，其中最关键的部分是牛顿方程解的N和r的一致估计。我们证明了Cauchy-Kovalevskaya定理的抽象形式的技巧可以给出改进的估计。2. Boltzmann方程非线性边界层问题可解性条件的建立：最基本的边界层是半空间中边值问题的解。然而，该问题并不是无条件可解的，因为无穷远处的边界条件是超定的。我们建立了边界数据的可解性条件。更精确地说，我们证明了边界数据上的限制数取决于无穷远处的马赫数M，当M> 1时为5，<M<1时为4，-1 <M<0时为1，M<-1时为0。证明依赖于解的精确先验估计，该估计通过使用适当的权函数和引入新的人工阻尼项而获得。3.非线性边界层稳定性的证明。我们证明了当M<-1时，所得到的平稳解是指数稳定的.首先，利用能量方法建立线性化方程的指数衰减性，然后利用压缩映射原理建立非线性稳定性。4.流体方程的渐近分析：基于1中介绍的抽象Cauchy-Kovalevskaya技巧，用统一的方法导出了建立各种流体方程之间渐近关系所需解的一致估计。