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Quenching of combustion by shear flows. (English) Zbl 1103.35048

The authors consider a model describing premixed combustion in the presence of fluid flow: a reaction-diffusion equation with passive advection and ignition-type nonlinearity. What kinds of velocity profiles are capable for quenching (suppressing) any given flame, provided the velocity’s amplitude is adequately large? Even for shear flows, the solution turns out to be surprisingly subtle. In this paper, they provide a sharp characterization of quenching for shear flows; the flow can quench any initial data if and only if the velocity profile does not have an interval larger than a certain critical size where it is identically constant. The efficiency of quenching depends strongly on the geometry and scaling of the flow. They discuss the cases of slowly and quickly varying flows, proving rigorously scaling laws that have been observed earlier in numerical experiments. The results require new estimates on the behavior of the solutions to the advection-enhanced diffusion equation, a classical model describing a wealth of phenomena in nature. The technique involves probabilistic and PDE estimates, in particular, applications of Malliavin calculus and the central limit theorem.

MSC:

35K57 Reaction-diffusion equations
80A25 Combustion
60H07 Stochastic calculus of variations and the Malliavin calculus
76V05 Reaction effects in flows
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[1] H. Berestycki, “The influence of advection on the propagation of fronts in reaction-diffusion equations” in Nonlinear PDEs in Condensed Matter and Reactive Flows (Cargese, France, 1999) , NATO Adv. Sci. Ser. C Math. Phys. Sci. 569 , Kluwer, Dordrecht, 2002, 11–48. · Zbl 1073.35113 · doi:10.1007/978-94-010-0307-0_2
[2] H. Berestycki, B. Larrouturou, and P.-L. Lions, Multi-dimensional travelling-wave solutions of a flame propagation model , Arch. Rational Mech. Anal. 111 (1990), 33–49. · Zbl 0711.35066 · doi:10.1007/BF00375699
[3] P. Clavin and F. A. Williams, Theory of premixed-flame propagation in large-scale turbulence , J. Fluid Mech. 90 (1979), 589–604. · Zbl 0434.76052 · doi:10.1017/S002211207900241X
[4] P. Constantin, A. Kiselev, and L. Ryzhik, Quenching of flames by fluid advection , Comm. Pure Appl. Math. 54 (2001), 1320–1342. · Zbl 1032.35087 · doi:10.1002/cpa.3000
[5] R. Durrett, Probability: Theory and Examples , 2nd ed., Duxbury Press, Belmont, Calif., 1996. · Zbl 1202.60001
[6] L. C. Evans, Partial Differential Equations , Grad. Stud. Math. 19 , Amer. Math. Soc., Providence, 1998. · Zbl 0902.35002
[7] A. Fannjiang, A. Kiselev, and L. Ryzhik, Quenching of reaction by cellular flows , to appear in Geom. Funct. Anal., · Zbl 1097.35077 · doi:10.1007/s00039-006-0554-y
[8] Ja. I. Kanel, Stabilization of the solutions of the equations of combustion theory with finite initial functions (in Russian), Mat. Sb. (N.S.) 65 (1964), 398–413.
[9] J. Nash, Continuity of solutions of parabolic and elliptic equations , Amer. J. Math. 80 (1958), 931–954. JSTOR: · Zbl 0096.06902 · doi:10.2307/2372841
[10] J. R. Norris, Long-time behaviour of heat flow: Global estimates and exact asymptotics , Arch. Rational Mech. Anal. 140 (1997), 161–195. · Zbl 0899.35015 · doi:10.1007/s002050050063
[11] D. Nualart, The Malliavin Calculus and Related Topics, Probab. Appl. (N.Y.), Springer, New York, 1995. · Zbl 0837.60050
[12] B. ØKsendal, Stochastic Differential Equations: An Introduction with Applications , 4th ed., Universitext, Springer, Berlin, 1995.
[13] J.-M. Roquejoffre, Eventual monotonicity and convergence to travelling fronts for the solutions of parabolic equations in cylinders , Ann. Inst. H. Poincaré Anal. Non Linéaire 14 (1997), 499–552. · Zbl 0884.35013 · doi:10.1016/S0294-1449(97)80137-0
[14] A. N. Shiryayev, Probability , Grad. Texts in Math. 95 , Springer, New York, 1984.
[15] J. Smoller, Shock Waves and Reaction-Diffusion Equations , 2nd ed., Grundlehren Math. Wiss. 258 , Springer, New York, 1994. · Zbl 0807.35002
[16] N. Vladimirova, P. Constantin, A. Kiselev, O. Ruchayskiy, and L. Ryzhik, Flame enhancement and quenching in fluid flows , Combust. Theory Model. 7 (2003), 487–508. · Zbl 1068.76570 · doi:10.1088/1364-7830/7/3/303
[17] J. X. Xin, Existence and nonexistence of traveling waves and reaction-diffusion front propagation in periodic media , J. Statist. Phys. 73 (1993), 893–926. · Zbl 1102.35340 · doi:10.1007/BF01052815
[18] -, Front propagation in heterogeneous media , SIAM Rev. 42 (2000), 161–230. JSTOR: · Zbl 0951.35060 · doi:10.1137/S0036144599364296
[19] A. Zlatoš, Quenching and propagation of combustion without ignition temperature cutoff , Nonlinearity 18 (2005), 1463–1475. · Zbl 1116.35069 · doi:10.1088/0951-7715/18/4/003
[20] -, Sharp transition between extinction and propagation of reaction , J. Amer. Math. Soc. 19 (2006), 251–263. · Zbl 1081.35011 · doi:10.1090/S0894-0347-05-00504-7
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