TY - JOUR

T1 - From homogenization to averaging in cellular flows

AU - Iyer, Gautam

AU - Komorowski, Tomasz

AU - Novikov, Alexei

AU - Ryzhik, Lenya

N1 - Funding Information:
G.I. was partially supported by NSF grant DMS-1007914 , Center for Nonlinear Analysis ( NSF DMS-0405343 and DMS-0635983 ) and PIRE grant OISE-0967140 . T.K. was partially supported by Polish Ministry of Science and Higher Education grant NN 201419139 . A.N. was partially supported by NSF grant DMS-0908011 . L.R. was partially supported by NSF grant DMS-0908507 , and NSSEFF fellowship .
Publisher Copyright:
© 2013 Elsevier Masson SAS.

PY - 2014/9/1

Y1 - 2014/9/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84908477489&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84908477489&partnerID=8YFLogxK

U2 - 10.1016/j.anihpc.2013.06.003

DO - 10.1016/j.anihpc.2013.06.003

M3 - Article

AN - SCOPUS:84908477489

SN - 0294-1449

VL - 31

SP - 957

EP - 983

JO - Annales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis

JF - Annales de l'Institut Henri Poincare. Annales: Analyse Non Lineaire/Nonlinear Analysis

IS - 5

ER -