From homogenization to averaging in cellular flows

Gautam Iyer, Tomasz Komorowski, Alexei Novikov, Lenya Ryzhik

Research output: Contribution to journalArticle

4 Scopus citations

Abstract

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 蠑 L4, the principal Dirichlet eigenvalue is approximately constant. On the other hand, when A 蠐 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.

Original languageEnglish (US)
Pages (from-to)957-983
Number of pages27
JournalAnnales de l'Institut Henri Poincare (C) Analyse Non Lineaire
Volume31
Issue number5
DOIs
StatePublished - Sep 1 2014

    Fingerprint

All Science Journal Classification (ASJC) codes

  • Analysis
  • Mathematical Physics
  • Applied Mathematics

Cite this