From homogenization to averaging in cellular flows
Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) no. 5, pp. 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.

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     title = {From homogenization to averaging in cellular flows},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {957--983},
     publisher = {Elsevier},
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Iyer, Gautam; Komorowski, Tomasz; Novikov, Alexei; Ryzhik, Lenya. From homogenization to averaging in cellular flows. Annales de l'I.H.P. Analyse non linéaire, Tome 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/

[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 | Zbl

[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 | EuDML | Numdam | MR | Zbl

[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

[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 | Zbl

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

[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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

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

[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 | Zbl

[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 | Zbl

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

[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

[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 | Zbl

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

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

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

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

[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 | EuDML | Numdam | MR | Zbl

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

[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

[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 | Zbl

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

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

[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

[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

[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

[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 | Zbl

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