Research of the hydrodynamic limit by probabilistic methods

本文的主要研究成果如下：1.研究结果表明：研究了高维随机晶格气体的平衡波动，在极限下导出了无限维的Ornstein-Uhlenbeck过程。其特征量与在水动力极限下得到的扩散系数一致。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的随机摄动运动运动。对于有效的Ginzburg-Landau△φ界面模型，得到了以表面张力的Hesse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。还讨论了相应的大偏差，并表明在速率函数表示中出现相同的表面张力。研究了两类粒子的粒子系统。该体系是通过对介质进行微观模拟并伴有相的变化而得到的，类似于固液体系。基于水动力极限法，导出了相分离边界宏观演化的自由边界问题，即Stefan问题。本文还从微观角度研究了潜热的影响。利用解析方法，研究了具有分数阶拉普拉斯幂的Burgers方程的整体解和局部解的存在唯一性及其正则性等问题。然后，通过建立混沌的传播，基于概率方法构造了相应的非线性马尔可夫过程，并导出了分数阶拉普拉斯函数的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。

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.