Date of Original Version
This is the author’s version of a work that was accepted for publication. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version is available at http://dx.doi.org/10.1016/j.anihpc.2013.06.003
Abstract or Description
We consider an elliptic eigenvalue problem with a fast cellular flow of amplitude A , in a two-dimensional domain with L2 cells. For fixed A , and L→∞, the problem homogenizes, and has been well studied. Also well studied is the limit when L is fixed, and A→∞. In this case the solution equilibrates along stream lines.
In this paper, we show that if both A→∞ and L→∞, then a transition between the homogenization and averaging regimes occurs at A≈L4. When A≫L4, the principal Dirichlet eigenvalue is approximately constant. On the other hand, when A≪L4, the principal eigenvalue behaves like σ(A)/L2 , where σ(A) ≈ √ AI 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 Lp→L∞ estimates for elliptic equations with an incompressible drift. This provides effective sub- and super-solutions for our problem.
Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 31, 5, 957-983.