Hydrodynamic limit and nonlinear PDEs with singularities

Abstract

We shall discuss the derivation, via a scaling limit starting from large-scale interacting systems, of a certain class of nonlinear partial differential equations, especially with singular structures, such as the Stefan problem, the free boundary problem of elliptic type, an evolutionary variational inequality and a stochastic partial differential equation with reflection. A family of independent Brownian particles is first taken up as a simple example to explain the idea behind the scaling limit. Then, we survey several results concerning two kinds of model: the interacting particle systems on $\mathbb{Z}^d$ called lattice gases and the $\nabla \varphi$ interface model, which is a microscopic system for interfaces separating two distinct phases. The entropy method, which plays an essential role in the proof of the hydrodynamic limit, is explained in brief.