Research of the hydrodynamic limit by probabilistic methods

概率方法研究水动力极限

基本信息

批准号：
08454036
负责人：
FUNAKI Tadahisa
金额：
$ 4.8万
依托单位：
The University of Tokyo
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
1996
资助国家：
日本
起止时间：
1996 至 1998
项目状态：
已结题

项目摘要

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.

本研究取得的主要成果如下： 1．研究了高维随机晶格气体的平衡涨落，推导了极限条件下的无限维Ornstein-Uhlenbeck过程，其特征量与流体力学极限条件下得到的扩散系数相吻合。 2．证明了随机反应扩散方程奇异摄动极限处的相分离边界按平均曲率随机摄动运动。3．对于有效的Ginzburg-Landau ▽φ界面模型，得到了以表面张力Hesse矩阵为扩散系数的扩散方程作为宏观尺度极限下的界面方程，并讨论了相应的大偏差，结果表明，在宏观尺度极限下，存在相同的表面张力。 4. 研究了具有两种类型粒子的粒子系统，该系统是通过对伴随相变的介质进行微观建模而获得的，如固液系统的宏观演化的自由边界问题。基于流体力学极限的方法推导了相分离边界，即Stefan问题，并从微观角度研究了潜热的影响。 5．利用解析方法研究了局部解及其正则性质等，然后基于概率方法构造了相应的非线性马尔可夫过程，并通过建立多粒子系统推导了具有拉普拉斯分次幂的Burgers方程。基于这些结果，我们将从几个角度对相分离问题进行研究。