Homogenization of a diffusion convection equation, with random source terms periodically distributed

Abstract

We are interested to study $u(x,t)$ , the evolution in time of the concentration, which is transported by diffusion and convection from a "sources site" made of a large number of similar "local sources". For this we consider a "local model" based on a general diffusion convection equation: \begin{eqnarray} \label{intro_eq} \partial_t u^\eps-\mathrm {div}(a(x)\nabla u^\eps)+\mathrm {div}(b(x) u^\eps)=f^\eps;\qquad{ }\\ u^\eps\big|_{t=0}=0,\qquad \frac{\partial}{\partial n_a}u^\eps\cdot n(x)-b(x)\cdot n(x)u^\eps+ \lambda u^\eps=0 .\qquad{ } \end{eqnarray} where the sources density f^\eps comes from a set of "local sources" periodically repeated and lying on a same plan $\Sigma$; f^\eps(x,t)= \bigcup\limits_{\textbf{j}\in\mathbb Z^2}f_\textbf{j}(x,t). Assuming the release curve ( source emission vs. space and time),$f_\textbf{j}(,.)$, of each local source, being random, our aim is to give a mathematical model describing the global evolution of such a system