Effective boundary conditions of diffusion equations on domains containing thin layers
Scientia Sinica Mathematica · 2016
This is a survey paper on a class of diffusion equations. Of concern is the following scenario: A domain $\Omega$ consists of two parts $\Omega_1$ and $\Omega_2$, with the latter being a thin layer; on $\Omega$ we have a diffusion equation, one of whose physical parameters such as diffusion coefficient has different size scales on different subdomains. This kind of problem arises in real applications such as thermal barrier coatings of turbine engine blades, and effects of roads on the spreading of populations and epidemics. Numerical computation of such a diffusion equation on such a domain is time-consuming because we would need very fine grids on the thin layer; moreover, we cannot see easily the effects of the thin layer on the dynamics of the diffusion equation. A good resolution to this problem is to think of the thin layer as a thickness surface/curve, on which we impose an ``effective boundary condition'' (EBC); then with ease we can solve, numerically, the diffusion equation on $\Omega_1$ with the EBC; furthermore, we can see the effects of the thin layer via EBC (for example, if the EBC is Neumann, then the mass does not penetrate through the thin layer). We will review classic theorems, and introduce results obtained in recent years, as well as results that have not been published. These results not only are of practical significance, but also bring some new challenges to pure PDE-research.