We consider an elliptic eigenvalue problem with a fast cellular flow of amplitude A, in a two-dimensional domain with cells. For fixed A, and , the problem homogenizes, and has been well studied. Also well studied is the limit when L is fixed, and . In this case the solution equilibrates along stream lines.In this paper, we show that if both and , then a transition between the homogenization and averaging regimes occurs at . When , the principal Dirichlet eigenvalue is approximately constant. On the other hand, when , the principal eigenvalue behaves like , where 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 estimates for elliptic equations with an incompressible drift. This provides effective sub- and super-solutions for our problem.
@article{AIHPC_2014__31_5_957_0, author = {Iyer, Gautam and Komorowski, Tomasz and Novikov, Alexei and Ryzhik, Lenya}, title = {From homogenization to averaging in cellular flows}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, pages = {957--983}, publisher = {Elsevier}, volume = {31}, number = {5}, year = {2014}, doi = {10.1016/j.anihpc.2013.06.003}, mrnumber = {3258362}, zbl = {1302.35039}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.anihpc.2013.06.003/} }
TY - JOUR AU - Iyer, Gautam AU - Komorowski, Tomasz AU - Novikov, Alexei AU - Ryzhik, Lenya TI - From homogenization to averaging in cellular flows JO - Annales de l'I.H.P. Analyse non linéaire PY - 2014 SP - 957 EP - 983 VL - 31 IS - 5 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.anihpc.2013.06.003/ DO - 10.1016/j.anihpc.2013.06.003 LA - en ID - AIHPC_2014__31_5_957_0 ER -
%0 Journal Article %A Iyer, Gautam %A Komorowski, Tomasz %A Novikov, Alexei %A Ryzhik, Lenya %T From homogenization to averaging in cellular flows %J Annales de l'I.H.P. Analyse non linéaire %D 2014 %P 957-983 %V 31 %N 5 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.anihpc.2013.06.003/ %R 10.1016/j.anihpc.2013.06.003 %G en %F AIHPC_2014__31_5_957_0
Iyer, Gautam; Komorowski, Tomasz; Novikov, Alexei; Ryzhik, Lenya. From homogenization to averaging in cellular flows. Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 5, pp. 957-983. doi : 10.1016/j.anihpc.2013.06.003. http://www.numdam.org/articles/10.1016/j.anihpc.2013.06.003/
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