Research of the hydrodynamic limit by probabilistic methods

本研究取得的主要成果如下：1.研究了高维随机格子气的平衡涨落，在极限下导出了无穷维Ornstein-Uhlenbeck过程。其特征量与水动力极限下的扩散系数一致.证明了随机反应扩散方程在奇摄动极限下的相分离边界是按平均曲率随机摄动运动移动的.在Ginzburg-Landau界面模型的宏观标度极限下，得到了以表面张力的黑森矩阵为扩散系数的非线性扩散方程.讨论了相应的较大偏差，表明在速率泛函表示中出现了相同的表面张力.研究了含有两类粒子的粒子系统。这种系统是通过显微镜模拟伴随着相变的介质而获得的，如固-液系统。基于流体力学极限方法，导出了描述相分离边界宏观演化的自由边界问题，即Stefan问题。并从微观角度研究了潜热的影响.利用解析方法研究了具有Laplacian分数次幂的Burgers方程整体解和局部解的存在唯一性及其正则性等问题。然后，基于概率论方法，从多粒子系统出发，通过建立混沌的传播过程，构造了相应的非线性Markov过程，导出了具有Laplacian分数次幂的Burgers方程，并在此基础上，从多个角度对相分离问题进行了研究。

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.