Carnegie Mellon

We consider an elliptic eigenvalue problem in the presence a fast cellular flow in a two-dimensional domain. It is well known that when the amplitude, A, is fixed, and the number of cells, $L^2$, increases to infinity, the problem `homogenizes' -- that is, can be approximated by the solution of an effective (homogeneous) problem. On the other hand, if the number of cells, $L^2$, is fixed and the amplitude $A$ increases to infinity, the solution ``averages''. In this case, the solution equilibrates along stream lines, and it's behaviour across stream lines is given by an averaged equation.
In this talk we study what happens if we simultaneously send both the amplitude $A$, and the number of cells $L^2$ to infinity. It turns out that if $A \ll L^4$, the problem homogenizes, and if $A \gg L^4$, the problem averages. The transition at $A \approx L^4$ can quickly predicted by matching the effective diffusivity of the homogenized problem, to that of the averaged problem. However a rigorous proof is much harder, in part because the effective diffusion matrix is unbounded. I will provide the essential ingredients for the proofs in both the averaging and homogenization regimes. This is joint work with T. Komorowski, A. Novikov and L. Ryzhik.