From homogenization to averaging in cellular flows
Annales de l'I.H.P. Analyse non linéaire, Volume 31 (2014) no. 5, p. 957-983

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, 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 AL 4 . When AL 4 , the principal Dirichlet eigenvalue is approximately constant. On the other hand, when AL 4 , the principal eigenvalue behaves like σ ¯(A)/L 2 , 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 L p L 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},
     publisher = {Elsevier},
     volume = {31},
     number = {5},
     year = {2014},
     pages = {957-983},
     doi = {10.1016/j.anihpc.2013.06.003},
     zbl = {1302.35039},
     mrnumber = {3258362},
     language = {en},
     url = {http://www.numdam.org/item/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/item/AIHPC_2014__31_5_957_0/

[1] G. Allaire, Y. Capdeboscq, Homogenization of a spectral problem in neutronic multigroup diffusion, Comput. Methods Appl. Mech. Eng. 187 no. 1–2 (2000), 91 -117 | MR 1765549 | Zbl 1126.82346

[2] G. Allaire, R. Orive, Homogenization of periodic non self-adjoint problems with large drift and potential, ESAIM Control Optim. Calc. Var. 13 no. 4 (2007), 735 -749 | Numdam | MR 2351401 | Zbl 1130.35307

[3] A. Bensoussan, J.-L. Lions, G. Papanicolaou, Asymptotic Analysis for Periodic Structures, Stud. Math. Appl. vol. 5 , North-Holland Publishing Co., Amsterdam (1978) | MR 503330

[4] H. Berestycki, F. Hamel, N. Nadirashvili, Elliptic eigenvalue problems with large drift and applications to nonlinear propagation phenomena, Commun. Math. Phys. 253 no. 2 (2005), 451 -480 | MR 2140256 | Zbl 1123.35033

[5] H. Berestycki, A. Kiselev, A. Novikov, L. Ryzhik, Explosion problem in a flow, J. Anal. Math. 110 (2010), 31 -65 | MR 2753290 | Zbl 1204.35131

[6] H. Berestycki, L. Nirenberg, S.R.S. Varadhan, The principal eigenvalue and maximum principle for second-order elliptic operators in general domains, Commun. Pure Appl. Math. 47 no. 1 (1994), 47 -92 | Zbl 0806.35129

[7] Y. Capdeboscq, Homogenization of a diffusion equation with drift, C. R. Acad. Sci. Paris Sér. I Math. 327 no. 9 (1998), 807 -812 | MR 1663726 | Zbl 0918.35135

[8] Y. Capdeboscq, Homogenization of a neutronic critical diffusion problem with drift, Proc. R. Soc. Edinb., Sect. A 132 no. 3 (2002), 567 -594 | MR 1912416 | Zbl 1066.82530

[9] S. Childress, Alpha-effect influx ropes and sheets, Phys. Earth Planet. Inter. 20 (1979), 172 -180

[10] A. Fannjiang, G. Papanicolaou, Convection enhanced diffusion for periodic flows, SIAM J. Appl. Math. 54 no. 2 (1994), 333 -408 | MR 1265233 | Zbl 0796.76084

[11] A. Fannjiang, A. Kiselev, L. Ryzhik, Quenching of reaction by cellular flows, Geom. Funct. Anal. 16 no. 1 (2006), 40 -69 | MR 2221252 | Zbl 1097.35077

[12] M.I. Freidlin, A.D. Wentzell, Random Perturbations of Dynamical Systems, Grundlehren Math. Wiss. vol. 260 , Springer-Verlag, New York (1998) | MR 1652127 | Zbl 0922.60006

[13] T. Godoy, J.-P. Gossez, S. Paczka, On the asymptotic behavior of the principal eigenvalues of some elliptic problems, Ann. Mat. Pura Appl. (4) 189 no. 3 (2010), 497 -521 | MR 2657422 | Zbl 1205.35172

[14] Y. Gorb, D. Nam, A. Novikov, Numerical simulations of diffusion in cellular flows at high Peclet numbers, Discrete Contin. Dyn. Syst., Ser. B 15 no. 1 (2011), 75 -92 | MR 2746477 | Zbl 1308.76256

[15] S. Heinze, Diffusion–advection in cellular flows with large Peclet numbers, Arch. Ration. Mech. Anal. 168 no. 4 (2003), 329 -342 | MR 1994746 | Zbl 1044.76061

[16] C.J. Holland, A minimum principle for the principal eigenvalue for second-order linear elliptic equations with natural boundary conditions, Commun. Pure Appl. Math. 31 no. 4 (1978), 509 -519 | Zbl 0388.35053

[17] G. Iyer, A. Novikov, L. Ryzhik, A. Zlatoš, Exit times for diffusions with incompressible drift, SIAM J. Math. Anal. 42 no. 6 (2010), 2484 -2498 | MR 2733257 | Zbl 05936757

[18] V.V. Jikov, S.M. Kozlov, O.A. Oleĭnik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin (1994) | MR 1329546

[19] S. Kesavan, Homogenization of elliptic eigenvalue problems. I, Appl. Math. Optim. 5 no. 2 (1979), 153 -167 | MR 533617 | Zbl 0415.35061

[20] S. Kesavan, Homogenization of elliptic eigenvalue problems. II, Appl. Math. Optim. 5 no. 3 (1979), 197 -216 | MR 546068 | Zbl 0428.35062

[21] Y. Kifer, Random Perturbations of Dynamical Systems, Progr. Probab. Stat. vol. 16 , Birkhäuser Boston Inc., Boston, MA (1988) | MR 1015933 | Zbl 0659.58003

[22] A. Kiselev, L. Ryzhik, Enhancement of the traveling front speeds in reaction–diffusion equations with advection, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 18 no. 3 (2001), 309 -358 | Numdam | MR 1831659 | Zbl 1002.35069

[23] L. Koralov, Random perturbations of 2-dimensional Hamiltonian flows, Probab. Theory Relat. Fields 129 no. 1 (2004), 37 -62 | MR 2052862 | Zbl 1103.60068

[24] A.J. Majda, P.R. Kramer, Simplified models for turbulent diffusion: theory, numerical modelling, and physical phenomena, Phys. Rep. 314 no. 4–5 (1999), 237 -574 | MR 1699757

[25] A. Novikov, G. Papanicolaou, L. Ryzhik, Boundary layers for cellular flows at high Peclet numbers, Commun. Pure Appl. Math. 58 no. 7 (2005), 867 -922 | MR 2142878 | Zbl 1112.76023

[26] G.A. Pavliotis, A.M. Stuart, Multiscale Methods: Averaging and Homogenization, Texts Appl. Math. vol. 53 , Springer, New York (2008) | MR 2382139 | Zbl 1160.35006

[27] M.N. Rosenbluth, H.L. Berk, I. Doxas, W. Horton, Effective diffusion in laminar convective flows, Phys. Fluids 30 (1987), 2636 -2647 | Zbl 0636.76089

[28] F. Santosa, M. Vogelius, First-order corrections to the homogenized eigenvalues of a periodic composite medium, SIAM J. Appl. Math. 53 no. 6 (1993), 1636 -1668 | Zbl 0808.35085

[29] F. Santosa, M. Vogelius, Erratum to the paper: “First-order corrections to the homogenized eigenvalues of a periodic composite medium” [SIAM J. Appl. Math. 53 (6) (1993) 1636–1668, MR1247172 (94h:35188)], SIAM J. Appl. Math. 55 no. 3 (1995), 864 | MR 1247172

[30] B. Shraiman, Diffusive transport in a Raleigh–Bernard convection cell, Phys. Rev. A 36 (1987), 261 -267

[31] P.B. Rhines, W.R. Young, How rapidly is passive scalar mixed within closed streamlines?, J. Fluid Mech. 133 (1983), 135 -145 | Zbl 0576.76088

[32] A. Zlatoš, Reaction–diffusion front speed enhancement by flows, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 28 no. 5 (2011), 711 -726 | Numdam | MR 2838397 | Zbl 1328.35105