Reaction–diffusion front speed enhancement by flows
Annales de l'I.H.P. Analyse non linéaire, Volume 28 (2011) no. 5, p. 711-726

We study flow-induced enhancement of the speed of pulsating traveling fronts for reaction–diffusion equations, and quenching of reaction by fluid flows. We prove, for periodic flows in two dimensions and any combustion-type reaction, that the front speed is proportional to the square root of the (homogenized) effective diffusivity of the flow. We show that this result does not hold in three and more dimensions. We also prove conjectures from Audoly, Berestycki and Pomeau (2000) [1], Berestycki (2003) [3], Fannjiang, Kiselev and Ryzhik (2006) [11] for cellular flows, concerning the rate of speed-up of fronts and the minimal flow amplitude necessary to quench solutions with initial data of a fixed (large) size.

@article{AIHPC_2011__28_5_711_0,
author = {Zlato\v s, Andrej},
title = {Reaction--diffusion front speed enhancement by flows},
journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
publisher = {Elsevier},
volume = {28},
number = {5},
year = {2011},
pages = {711-726},
doi = {10.1016/j.anihpc.2011.05.004},
zbl = {1328.35105},
mrnumber = {2838397},
language = {en},
url = {http://www.numdam.org/item/AIHPC_2011__28_5_711_0}
}

Zlatoš, Andrej. Reaction–diffusion front speed enhancement by flows. Annales de l'I.H.P. Analyse non linéaire, Volume 28 (2011) no. 5, pp. 711-726. doi : 10.1016/j.anihpc.2011.05.004. http://www.numdam.org/item/AIHPC_2011__28_5_711_0/

[1] B. Audoly, H. Berestycki, Y. Pomeau, Réaction diffusion en écoulement stationnaire rapide, C. R. Acad. Sci. Paris, Série IIb 328 (2000), 255-262 | Zbl 0992.76097

[2] H. Berestycki, F. Hamel, Front propagation in periodic excitable media, Comm. Pure Appl. Math. 55 (2002), 949-1032 | MR 1900178 | Zbl 1024.37054

[3] H. Berestycki, The influence of advection on the propagation of fronts in reaction–diffusion equations, H. Berestycki, Y. Pomeau (ed.), Nonlinear PDEs in Condensed Matter and Reactive Flows, NATO Science Series C vol. 569, Kluwer, Doordrecht (2003) | Zbl 1073.35113

[4] H. Berestycki, F. Hamel, N. Nadirashvili, The speed of propagation for KPP type problems, I – Periodic framework, J. European Math. Soc. 7 (2005), 173-213 | MR 2127993 | Zbl 1142.35464

[5] H. Berestycki, F. Hamel, N. Nadirashvili, Elliptic eigenvalue problems with large drift and applications to nonlinear propagation phenomena, Comm. Math. Phys. 253 (2005), 451-480 | MR 2140256 | Zbl 1123.35033

[6] R.N. Bhattacharya, V.K. Gupta, H.F. Walker, Asymptotics of solute dispersion in periodic porous media, SIAM J. Appl. Math. 49 (1989), 86-98 | MR 978827 | Zbl 0664.60079

[7] P. Constantin, A. Kiselev, A. Oberman, L. Ryzhik, Bulk burning rate in passive-reactive diffusion, Arch. Ration. Mech. Anal. 154 (2000), 53-91 | MR 1778121 | Zbl 0979.76093

[8] P. Constantin, A. Kiselev, L. Ryzhik, Quenching of flames by fluid advection, Comm. Pure Appl. Math. 54 (2001), 1320-1342 | MR 1846800 | Zbl 1032.35087

[9] P. Constantin, A. Kiselev, L. Ryzhik, A. Zlatoš, Diffusion and mixing in fluid flow, Ann. of Math. (2) 168 (2008), 643-674 | MR 2434887 | Zbl 1180.35084

[10] M. El Smaily, Min–max formulae for the speeds of pulsating traveling fronts in heterogeneous media, Annali Mat. Pura Appl. 189 (2010), 47-66 | MR 2556759 | Zbl 1191.35089

[11] A. Fannjiang, A. Kiselev, L. Ryzhik, Quenching of reaction by cellular flows, Geom. Funct. Anal. 16 (2006), 40-69 | MR 2221252 | Zbl 1097.35077

[12] A. Fannjiang, G. Papanicolaou, Convection enhanced diffusion for periodic flows, SIAM J. Appl. Math. 54 (1994), 333-408 | MR 1265233 | Zbl 0796.76084

[13] S. Heinze, Large convection limits for KPP fronts, Max Planck Institute for Mathematics, Preprint No. 21/2005, 2005.

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

[15] L. Kagan, P.D. Ronney, G. Sivashinsky, Activation energy effect on flame propagation in large-scale vortical flows, Combust. Theory Model. 6 (2002), 479-485 | MR 1934205 | Zbl 1068.80516

[16] L. Kagan, G. Sivashinsky, Flame propagation and extinction in large-scale vortical flows, Combust. Flame 120 (2000), 222-232

[17] A. Kiselev, L. Ryzhik, Enhancement of the traveling front speeds in reaction–diffusion equations with advection, Ann. Inst. H. Poincaré Anal. Non Linéaire 18 (2001), 309-358 | Numdam | MR 1831659 | Zbl 1002.35069

[18] A. Kiselev, A. Zlatoš, Quenching of combustion by shear flows, Duke Math. J. 132 (2006), 49-72 | MR 2219254 | Zbl 1103.35048

[19] A.N. Kolmogorov, I.G. Petrovskii, N.S. Piskunov, Étude de lʼéquation de la chaleur de matière et son application à un problème biologique, Bull. Moskov. Gos. Univ. Mat. Mekh. 1 (1937), 1-25

[20] L. Koralov, Random perturbations of 2-dimensional Hamiltonian flows, Probab. Theory Related Fields 129 (2004), 37-62 | MR 2052862 | Zbl 1103.60068

[21] P. Kramer, A. Majda, Simplified models for turbulent diffusion: theory, numerical modelling, and physical phenomena, Phys. Rep. 314 (1999), 237-574 | MR 1699757

[22] J.R. Norris, Long-time behaviour of heat flow: global estimates and exact asymptotics, Arch. Ration. Mech. Anal. 140 (1997), 161-195 | MR 1482931 | Zbl 0899.35015

[23] A. Novikov, L. Ryzhik, Bounds on the speed of propagation of the KPP fronts in a cellular flow, Arch. Ration. Mech. Anal. 184 (2007), 23-48 | MR 2289862 | Zbl 1109.76064

[24] B. Øksendal, Stochastic Differential Equations, Springer-Verlag, Berlin (1995) | MR 1411679

[25] L. Ryzhik, A. Zlatoš, KPP pulsating front speed-up by flows, Commun. Math. Sci. 5 (2007), 575-593 | MR 2352332 | Zbl 1152.35055

[26] J. Smoller, Shock Waves and Reaction–Diffusion Equations, Springer-Verlag, New York (1994) | MR 1301779 | Zbl 0807.35002

[27] N. Vladimirova, P. Constantin, A. Kiselev, O. Ruchayskiy, L. Ryzhik, Flame enhancement and quenching in fluid flows, Combust. Theory Model. 7 (2003), 487-508 | MR 2007570 | Zbl 1068.76570

[28] J. Xin, Existence of planar flame fronts in convective-diffusive media, Arch. Ration. Mech. Anal. 121 (1992), 205-233 | MR 1188981 | Zbl 0764.76074

[29] A. Zlatoš, Quenching and propagation of combustion without ignition temperature cutoff, Nonlinearity 18 (2005), 1463-1475 | MR 2150338 | Zbl 1116.35069

[30] A. Zlatoš, Pulsating front speed-up and quenching of reaction by fast advection, Nonlinearity 20 (2007), 2907-2921 | MR 2368331 | Zbl 1149.35308

[31] A. Zlatoš, Diffusion in fluid flow: Dissipation enhancement by flows in 2D, Comm. Partial Differential Equations 35 (2010), 496-534 | MR 2748635 | Zbl 1201.35106

[32] A. Zlatoš, Sharp asymptotics for KPP pulsating front speed-up and diffusion enhancement by flows, Arch. Ration. Mech. Anal. 195 (2010), 441-453 | MR 2592283 | Zbl 1185.35205