From homogenization to averaging in cellular flows

We consider an elliptic eigenvalue problem with a fast cellular flow of amplitude $A$, in a two-dimensional domain with $L^2$ cells. For fixed $A$, and $L\to \mathrm{\infty }$, the problem homogenizes, and has been well studied. Also well studied is the limit when $L$ is fixed, and $A\to \mathrm{\infty }$. In this case the solution equilibrates along stream lines. In this paper, we show that if $\mathit{\text{both}}$ $A\to \mathrm{\infty }$ and $L \to \infty$, then a transition between the homogenization and averaging regimes occurs at $A\approx L^4$. When $A\gg L^4$, the principal Dirichlet eigenvalue is approximately constant. On the other hand, when $A\ll L^4$, the principal eigenvalue behaves like $\bar{\sigma }(A)\mathrm{/}{L}^2$, where $\bar \sigma(A) \approx \sqrt{A} I$ is the effective diffusion matrix. A similar transition is observed for the solution of the exit time problem. The proof in the homogenization regime involves bounds on the second correctors. Miraculously, if the slow profile is quadratic, these estimates can be obtained using drift independent $L^p\to L^{\mathrm{\infty }}$ estimates for elliptic equations with an incompressible drift. This provides effective sub and super-solutions for our problem.