×

Reaction-diffusion front speed enhancement by flows. (English) Zbl 1328.35105

Summary: 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 B. Audoly et al. [C. R. Acad. Sci., Paris, Sér. II, Fasc. b, Méc. 328, No. 3, 255–262 (2000; Zbl 0992.76097)], H. Berestycki [NATO ASI Ser., Ser. C, Math. Phys. Sci. 569, 11–48 (2002; Zbl 1073.35113)], A. Fannjiang et al. [Geom. Funct. Anal. 16, No. 1, 40–69 (2006; Zbl 1097.35077)] 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.

MSC:

35K57 Reaction-diffusion equations
35B40 Asymptotic behavior of solutions to PDEs
35C07 Traveling wave solutions
80A25 Combustion
35R60 PDEs with randomness, stochastic partial differential equations
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Audoly, B.; Berestycki, H.; Pomeau, Y., Réaction diffusion en écoulement stationnaire rapide, C. R. Acad. Sci. Paris, Série IIb, 328, 255-262 (2000) · Zbl 0992.76097
[2] Berestycki, H.; Hamel, F., Front propagation in periodic excitable media, Comm. Pure Appl. Math., 55, 949-1032 (2002) · Zbl 1024.37054
[3] Berestycki, H., The influence of advection on the propagation of fronts in reaction-diffusion equations, (Berestycki, H.; Pomeau, Y., Nonlinear PDEs in Condensed Matter and Reactive Flows. Nonlinear PDEs in Condensed Matter and Reactive Flows, NATO Science Series C, vol. 569 (2003), Kluwer: Kluwer Doordrecht) · Zbl 1073.35113
[4] Berestycki, H.; Hamel, F.; Nadirashvili, N., The speed of propagation for KPP type problems, I - Periodic framework, J. European Math. Soc., 7, 173-213 (2005) · Zbl 1142.35464
[5] Berestycki, H.; Hamel, F.; Nadirashvili, N., Elliptic eigenvalue problems with large drift and applications to nonlinear propagation phenomena, Comm. Math. Phys., 253, 451-480 (2005) · Zbl 1123.35033
[6] Bhattacharya, R. N.; Gupta, V. K.; Walker, H. F., Asymptotics of solute dispersion in periodic porous media, SIAM J. Appl. Math., 49, 86-98 (1989) · Zbl 0664.60079
[7] Constantin, P.; Kiselev, A.; Oberman, A.; Ryzhik, L., Bulk burning rate in passive-reactive diffusion, Arch. Ration. Mech. Anal., 154, 53-91 (2000) · Zbl 0979.76093
[8] Constantin, P.; Kiselev, A.; Ryzhik, L., Quenching of flames by fluid advection, Comm. Pure Appl. Math., 54, 1320-1342 (2001) · Zbl 1032.35087
[9] Constantin, P.; Kiselev, A.; Ryzhik, L.; Zlatoš, A., Diffusion and mixing in fluid flow, Ann. of Math. (2), 168, 643-674 (2008) · Zbl 1180.35084
[10] El Smaily, M., Min-max formulae for the speeds of pulsating traveling fronts in heterogeneous media, Annali Mat. Pura Appl., 189, 47-66 (2010) · Zbl 1191.35089
[11] Fannjiang, A.; Kiselev, A.; Ryzhik, L., Quenching of reaction by cellular flows, Geom. Funct. Anal., 16, 40-69 (2006) · Zbl 1097.35077
[12] Fannjiang, A.; Papanicolaou, G., Convection enhanced diffusion for periodic flows, SIAM J. Appl. Math., 54, 333-408 (1994) · Zbl 0796.76084
[13] S. Heinze, Large convection limits for KPP fronts, Max Planck Institute for Mathematics, Preprint No. 21/2005, 2005.; S. Heinze, Large convection limits for KPP fronts, Max Planck Institute for Mathematics, Preprint No. 21/2005, 2005.
[14] Jikov, V. V.; Kozlov, S. M.; Oleinik, O. A., Homogenization of Differential Operators and Integral Functionals (1994), Springer-Verlag: Springer-Verlag Berlin, Chapter 2 · Zbl 0801.35001
[15] Kagan, L.; Ronney, P. D.; Sivashinsky, G., Activation energy effect on flame propagation in large-scale vortical flows, Combust. Theory Model., 6, 479-485 (2002) · Zbl 1068.80516
[16] Kagan, L.; Sivashinsky, G., Flame propagation and extinction in large-scale vortical flows, Combust. Flame, 120, 222-232 (2000)
[17] Kiselev, A.; Ryzhik, L., Enhancement of the traveling front speeds in reaction-diffusion equations with advection, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18, 309-358 (2001) · Zbl 1002.35069
[18] Kiselev, A.; Zlatoš, A., Quenching of combustion by shear flows, Duke Math. J., 132, 49-72 (2006) · Zbl 1103.35048
[19] Kolmogorov, A. N.; Petrovskii, I. G.; Piskunov, N. S., Étude de lʼéquation de la chaleur de matière et son application à un problème biologique, Bull. Moskov. Gos. Univ. Mat. Mekh., 1, 1-25 (1937)
[20] Koralov, L., Random perturbations of 2-dimensional Hamiltonian flows, Probab. Theory Related Fields, 129, 37-62 (2004) · Zbl 1103.60068
[21] Kramer, P.; Majda, A., Simplified models for turbulent diffusion: theory, numerical modelling, and physical phenomena, Phys. Rep., 314, 237-574 (1999), (Chapter 2)
[22] Norris, J. R., Long-time behaviour of heat flow: global estimates and exact asymptotics, Arch. Ration. Mech. Anal., 140, 161-195 (1997) · Zbl 0899.35015
[23] Novikov, A.; Ryzhik, L., Bounds on the speed of propagation of the KPP fronts in a cellular flow, Arch. Ration. Mech. Anal., 184, 23-48 (2007) · Zbl 1109.76064
[24] Øksendal, B., Stochastic Differential Equations (1995), Springer-Verlag: Springer-Verlag Berlin
[25] Ryzhik, L.; Zlatoš, A., KPP pulsating front speed-up by flows, Commun. Math. Sci., 5, 575-593 (2007) · Zbl 1152.35055
[26] Smoller, J., Shock Waves and Reaction-Diffusion Equations (1994), Springer-Verlag: Springer-Verlag New York · Zbl 0807.35002
[27] Vladimirova, N.; Constantin, P.; Kiselev, A.; Ruchayskiy, O.; Ryzhik, L., Flame enhancement and quenching in fluid flows, Combust. Theory Model., 7, 487-508 (2003) · Zbl 1068.76570
[28] Xin, J., Existence of planar flame fronts in convective-diffusive media, Arch. Ration. Mech. Anal., 121, 205-233 (1992) · Zbl 0764.76074
[29] Zlatoš, A., Quenching and propagation of combustion without ignition temperature cutoff, Nonlinearity, 18, 1463-1475 (2005) · Zbl 1116.35069
[30] Zlatoš, A., Pulsating front speed-up and quenching of reaction by fast advection, Nonlinearity, 20, 2907-2921 (2007) · Zbl 1149.35308
[31] Zlatoš, A., Diffusion in fluid flow: Dissipation enhancement by flows in 2D, Comm. Partial Differential Equations, 35, 496-534 (2010) · Zbl 1201.35106
[32] Zlatoš, A., Sharp asymptotics for KPP pulsating front speed-up and diffusion enhancement by flows, Arch. Ration. Mech. Anal., 195, 441-453 (2010) · Zbl 1185.35205
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.