Research of the hydrodynamic limit by probabilistic methods

主要研究结果如下：1.研究了高维随机晶格气体的平衡涨落，得到了无限维Ornstein-Uhlenbeck过程的极限解。其特征量与在水动力极限下得到的扩散系数相吻合。证明了出现在随机反应扩散方程奇异摄动极限的相分离边界是按平均曲率的随机摄动运动的。对于有效的金兹堡-朗道▽φ界面模型，得到了以表面张力海塞矩阵为扩散系数的非线性扩散方程，作为宏观尺度极限下的界面方程。对相应的较大偏差也进行了讨论，结果表明，在速率泛函的表示中出现了相同的表面张力。研究了具有两类粒子的粒子系统。这种体系是通过对伴随相变的介质进行微观建模而得到的，就像固-液体系一样。基于流体力学极限的方法，导出了相分离边界宏观演化的自由边界问题，称为Stefan问题。从微观角度研究了潜热的影响。对于具有拉普拉斯分数次幂的Burgers型方程，利用解析方法研究了其整体解和局部解的存在唯一性及其正则性等问题。然后，基于概率方法，构造了相应的非线性马尔可夫过程，通过建立混沌传播模型，从多粒子系统出发，推导出拉普拉斯分数次幂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.