|TITLE||Initial-Boundary Value Problems in Kinetic Theory and Fluid Dynamics|
In this talk, I will introduce the boundary phenomenon in the kinetic theory caused by a moving boundary and a boundary with temperature changes. To investigate these boundary effects, we use the Boltzmann equation of the hard sphere model to describe the gas dynamics, and use the diffuse reflection boundary conditions to describe the interaction between the particles and the boundary. Then we show that the short time solution consists of the free molecular flow and its perturbation, which exhibits logarithmic singularities along the characteristic line and on the boundary. On the other hand, there are wave patterns combining the fast decaying boundary layer and slow varying fluid-like waves in the gas dynamic problems. In order to study these problems, it is important and fundamental to develop quantitative as well as qualitative theories on initial-boundary value problems for those fluid equations and kinetic equations. I will introduce a new method to solve the initial-boundary value problem for hyperbolic- dissipative PDEs based on the spirit of LY algorithm. Then we will establish the Green’s functions for some basic PDEs such as the heat equation, the wave equation and the damped wave equation, in a half space and a quarter plane with various boundary conditions. Moreover, we will also construct the complete representations of the Green’s functions for the convection-diffusion equation and the drifted wave equation in a half space with various boundary conditions.