From homogenization to averaging in cellular flows

TL;DR

This paper investigates the transition between homogenization and averaging regimes in a 2D elliptic eigenvalue problem with cellular flows, revealing a critical scaling at A ≈ L^4 and analyzing eigenvalue behavior.

Contribution

It identifies the transition point between homogenization and averaging regimes in cellular flows when both amplitude and cell size grow, and develops new estimates for elliptic equations with drift.

Findings

Eigenvalue remains constant when A ≫ L^4.

Eigenvalue scales as σ̄(A)/L^2 when A ≪ L^4.

Transition occurs at A ≈ L^4.

Abstract

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 $L \to \infty$, the problem homogenizes, and has been well studied. Also well studied is the limit when $L$ is fixed, and $A \to \infty$. In this case the solution equilibrates along stream lines. In this paper, we show that if \textit{both} $A \to \infty$ 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 ${\bar \sigma(A)}/L^2$, 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 $L^p \to L^\infty$ estimates for elliptic equations with an incompressible drift. This provides effective sub and super-solutions for our problem.