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Enhancement of the traveling front speeds in reaction-diffusion equations with advection. (English) Zbl 1002.35069
The authors investigate the following question: which characteristic of the ambient fluid flow is responsible for burning rate enhancement? Note that the bulk burning rate $V(t)=\int_\Omega T_t(x,y,t) {dxdy \over H}$ and its time average $\langle V\rangle_t={1\over t}\int^t_0 V(s)ds,$ where $$T$$ denotes the temperature, satisfy the corresponding PDE. Here they consider a general class of reaction rates $$f(T)$$ that are either of the ignition or general KPP type, and establish lower bounds for $$V(t)$$ for percolating flows that are periodic in space. Another interesting result of the paper concerns cellular flows with closed streamlines.

##### MSC:
 35K57 Reaction-diffusion equations 80A32 Chemically reacting flows 35K40 Second-order parabolic systems 35B45 A priori estimates in context of PDEs 35B10 Periodic solutions to PDEs
##### Keywords:
ambient fluid flow; percolating flows; cellular flows
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##### References:
 [1] Audoly B., Berestycki H., Pomeau Y., Reaction-diffusion in fast steady flow, C.R. Acad. Sci., Ser. IIB, 328, 255-262 · Zbl 0992.76097 [2] Bensoussan, A; Lions, J.L; Papanicolaou, G, Asymptotic analysis for periodic structures, (1978), North-Holland Amsterdam · Zbl 0411.60078 [3] Berestycki H., Hamel F., Propagation of flames through vortical cells, preprint [4] Berestycki, H; Larrouturou, B; Lions, P.L, Multi-dimensional traveling wave solutions of a flame propagation model, Arch. rational mech. anal, 111, 33-49, (1990) · Zbl 0711.35066 [5] Berestycki, H; Larrouturou, B; Roquejoffre, J.-M, Stability of traveling fronts in a model for flame propagation I: linear stability, Arch. rational mech. anal., 117, 97-117, (1992) · Zbl 0763.76033 [6] Berestycki, H; Nirenberg, L, Some qualitative properties of solutions of semilinear equations in cylindrical domains, (), 115-164 · Zbl 0705.35004 [7] Berestycki, H; Nirenberg, L, Traveling fronts in cylinders, Annales de l’IHP, analyse non linéare, 9, 497-572, (1992) · Zbl 0799.35073 [8] Clavin, P; Williams, F.A, Theory of pre-mixed flame propagation in large-scale turbulence, J. fluid. mech., 90, 589-604, (1979) · Zbl 0434.76052 [9] Constantin, P; Kiselev, A; Oberman, A; Ryzhik, L, Bulk burning rate in passive-reactive diffusion, Arch. rat. mech. anal., 154, 53-91, (2000) · Zbl 0979.76093 [10] Embid, P; Majda, A; Souganidis, P, Comparison of turbulent flame speeds from complete averaging and the G-equation, Phys. fluids., 7, 2052-2060, (1995) · Zbl 1039.80504 [11] Embid, P; Majda, A; Souganidis, P, Effective geometric from dynamics for premixed turbulent combustion with separated velocity scales, Combust. sci. tech., 103, 85, (1994) [12] Fannjiang, A; Papanicolaou, G, Convection enhanced diffusion for periodic flows, SIAM J. appl. math., 54, 333-408, (1994) · Zbl 0796.76084 [13] Fisher, R, The wave of advance of advantageous genes, Ann. eugenics, 7, 355-369, (1937) · JFM 63.1111.04 [14] Freidlin, M; Gärtner, J, On the propagation of concentration waves in periodic and random media, Soviet math. dokl., 20, 1282-1286, (1979) · Zbl 0447.60060 [15] Freidlin, M, Geometric optics approach to reaction-diffusion equations, SIAM J. appl. math., 46, 222-232, (1986) · Zbl 0626.35047 [16] Freidlin, M, Limit theorems for large deviations and reaction-diffusion equations, Ann. probab., 13, 639-675, (1985) · Zbl 0576.60070 [17] Hamel, F, Formules MIN-MAX pour LES vitesses d’ondes progressives multidimensionnelles, Ann. fac. sci., Toulouse math. (6), 8, 259-280, (1999) · Zbl 0956.35041 [18] Heinze S., Papanicolaou G., Stevens A., Variational principles for propagation speeds in inhomogeneous media, SIAM J. Appl. Math., to appear · Zbl 0995.35031 [19] Kerstein, A; Ashurst, W, Phys. rev. lett., 68, 934, (1992) [20] Kerstein, A, Simple derivation of Yakhot’s turbulent premixed flame speed formula, Combust. sci. tech., 60, 163-165, (1988) [21] Kagan, L; Sivashinsky, G; Makhviladze, G, On flame extinction by a spatially periodic shear flow, Combust. theor. model., 2, 399-404, (1998) · Zbl 0944.76093 [22] Kagan, L; Sivashinsky, G, On flame propagation and extinction in large scale vortical flows, Combustion and flame, 120, 222-232, (2000) [23] Kolmogorov, A.N; Petrovskii, I.G; Piskunov, N.S, Étude de I’équation de la chaleurde matiére et son application à un problème biologique, Bull. moskov. GoS. univ. mat. mekh., 1, 1-25, (1937), (see [27] pp. 105-130 for an English transl.) [24] Majda, A; Souganidis, P, Large scale front dynamics for turbulent reaction-diffusion equations with separated velocity scales, Nonlinearity, 7, 1-30, (1994) · Zbl 0839.76093 [25] Mallordy, J.-F; Roquejoffre, J.-M, A parabolic equation of the KPP type in higher dimensions, SIAM J. math. anal., 26, 1-20, (1995) · Zbl 0813.35041 [26] McLaughlin, R; Zhu, J, The effect of finite front thickness on the enhanced speed of propagation, Combust. sci. tech., 129, 89-112, (1997) [27] () [28] Protter, M; Weinberger, H, Maximum principles in differential equations, (1967), Prentice-Hall Englewood Cliffs, NJ · Zbl 0153.13602 [29] Ronney, P, Some open issues in premixed turbulent combustion, () [30] Roquejoffre, J.-M, Stability of traveling fronts in a model for flame propagation II: nonlinear stability, Arch. rat. mech. anal., 117, 119-153, (1992) · Zbl 0763.76034 [31] Xin, J, Existence of planar flame fronts in convective-diffusive periodic media, Arch. rat. mech. anal., 121, 205-233, (1992) · Zbl 0764.76074 [32] Xin, J, Existence and nonexistence of traveling waves and reaction-diffusion front propagation in periodic media, J. stat. phys., 73, 893-926, (1993) · Zbl 1102.35340 [33] Xin, J, Analysis and modeling of front propagation in heterogeneous media, SIAM review, 42, 161-230, (2000) [34] Yakhot V., Propagation velocity of premixed turbulent flames, Combust. Sci. Tech. 60, 191-214
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