Abstract
We establish the variational principle of Kolmogorov-Petrovsky-Piskunov (KPP) front speeds in temporally random shear flows with sufficiently decaying correlations. A key quantity in the variational principle is the almost sure Lyapunov exponent of a heat operator with random potential. To prove the variational principle, we use the comparison principle of solutions, the path integral representation of solutions, and large deviation estimates of the associated stochastic flows. The variational principle then allows us to analytically bound the front speeds. The speed bounds imply the linear growth law in the regime of large root mean square shear amplitude at any fixed temporal correlation length, and the sublinear growth law if the temporal decorrelation is also large enough, the so-called bending phenomenon.
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Adler, R.: An Introduction to Continuity, Extrema and Related Topics for General Gaussian Processes. Institute of Math Stat, Lecture Notes-Monograph Series, 12, 1990
Akcoglu M.A., Krengel U. (1981) Ergodic theorems for superadditive processes. J. Reine Angew Math. 323, 53–67
Ashurst Wm.T. (2000) Flow-frequency effect upon Huygens front propagation. Combust. Theory Modelling 4, 99–105
Berestycki, H.: The influence of advection on the propagation of fronts in reaction-diffusion equations. In: Nonlinear PDEs in Condensed Matter and Reactive Flows. NATO Science Series C 569, Berestycki, H., Pomeau, Y. eds. Doordrecht: Kluwer, 2003
Berestycki H., Hamel F. (2002) Front Propagation in Periodic Excitable Media. Comm. Pure Appl. Math. 60, 949–1032
Berestycki H., Hamel F., Nadirashvili N. (2005) Elliptic eigenvalue problems with large drift and applications to nonlinear propagation phenomena. Commun. Math Phys. 253(2): 451–480
Berestycki H., Nirenberg L. (1992) Travelling fronts in cylinders. Ann. Inst. H. Poincaré Anal. Non Linéaire 9, 497–572
Carmona R.A., Molchanov S.A. (1994) Parabolic Anderson problem and intermittency. Mem. Amer. Math. Soc. 108(518): viii+125
Clavin P., Williams F.A. (1979) Theory of premixed-flame propagation in large-scale turbulence. J. Fluid Mech. 90, 598–604
Conlon J., Doering C. (2005) On Traveling Waves for the Stochastic FKKP Equation. J. Stat Phys. 120(3–4): 421–477
Constantin P., Kiselev A., Oberman A., Ryzhik L. (2000) Bulk burning rate in passive-reactive diffusion. Arch Rat. Mech Analy 154, 53–91
Cranston M., Mountford T. (2006) Lyapunov exponent for the parabolic Anderson model in R d. J. Funct. Anal. 236, 78–119
Denet B. (1999) Possible role of temporal correlations in the bending of turbulent flame velocity. Combust. Theory Modelling 3, 585–589
E W., Sinai Y. (2000) New results in mathematical and statistical hydrodynamics. Russ. Math. Surv. 55(4): 635–666
Ellis R.S. (1985) Entropy, Large Deviations, and Statistical Mechanics. New York, Springer-Verlag
Freidlin M.I. (1985) Functional Integration and Partial Differential Equations. Ann. Math. Stud. 1093. Princeton, NJ: Princeton University Press
Freidlin M.I., Wentzell A.D. (1998) Random Perturbations of Dynamical Systems. New York, Springer-Verlag
Gärtner J., Freidlin M.I. (1979) The propagation of concentration waves in periodic and random media. Dokl. Acad. Nauk SSSR 249, 521–525
Heinze S., Papanicolaou G., Stevens A. (2001) Variational principles for propagation speeds in inhomogeneous media. SIAM J. Applied Math. 62(1): 129–148
Karatzas I., Shreve S. (1991) Brownian Motion and Stochastic Calculus. New York, Springer-Verlag
Kato T. (1995) Perturbation Theory for Linear Operators. Berlin, Springer-Verlag
Khouider B., Bourlioux A., Majda A. (2001) Parameterizing turbulent flame speed-Part I: unsteady shears, flame residence time and bending. Combustion Theory and Modeling 5, 295–318
Kingman J.P.C. (1968) The Ergodic Theory of Subadditive Stochastic Processes. J. Royal. Stat. Soc. Series B, 30(3): 499–510
Kiselev A., Ryzhik L. (2001) Enhancement of the traveling front speeds in reaction-diffusion equations with advection. Ann. de l’Inst. Henri Poincaré, Analyse Nonlinéaire, 18, 309–358
Majda A., Souganidis P.E. (1994) Large scale front dynamics for turbulent reaction-diffusion equations with separated velocity scales. Nonlinearity 7, 1–30
Majda A., Souganidis P.E. (1998) Flame fronts in a turbulent combustion model with fractal velocity fields. Comm. Pure Appl. Math. LI: 1337–1348
Mierczynski J., Shen W. (2003) Exponential separation and principal Lyapunov exponent/spectrum for random/nonautonomous parabolic equations. J. Differ. Eqs. 191, 175–205
Mueller C., Sowers R. (1995) Random Traveling Waves for the KPP equation with Noise. J. Funct. Anal. 128, 439–498
Nolen J., Rudd M., Xin J. (2005) Existence of KPP fronts in spatially-temporally periodic advection and variational principle for propagation speeds. Dynamics of PDE 2(1): 1–24
Nolen J., Xin J. (2003) Reaction diffusion front speeds in spatially-temporally periodic shear flows. SIAM J. Multiscale Modeling and Simulation 1(4): 554–570
Nolen J., Xin J.(2004) Min-Max Variational Principle and Front Speeds in Random Shear Flows. Meth. Appl. Anal. 11(4): 635–644
Nolen J., Xin J. (2005) Existence of KPP type fronts in space–time periodic shear flows and a study of minimal speeds based on variational principle. Discrete and Cont. Dyn. Syst. 13(5): 1217–1234
Nolen J., Xin J. (2005) A Variational Principle Based Study of KPP Minimal Front Speeds in Random Shears. Nonlinearity 18, 1655–1675
Nolen, J., Xin, J.: Variational Principle Based Computation of KKP Front Speeds in Temporally Random Shear Flows. In preparation, 2006
Peters N. (2000) Turbulent Combustion. Cambridge, Cambridge University Press
Ronney, P.: Some open issues in premixed turbulent combustion. In: Modeling in Combustion Science (Buckmaster, J.D., Takeno, T. eds. Lecture Notes In Physics, 449, Berlin: Springer-Verlag, (1995), pp. 3–22
Shen W. (2004) Traveling Waves in Diffusive Random Media. J. Dyn. Diff. Eqs. 16(4): 1011–1060
Vladimirova N., Constantin P., Kiselev A., Ruchayskiy O., Ryzhik L. (2003) Flame enhancement and quenching in fluid flows. Combust. Theory and Modeling 7, 487–508
Xin J. (1991) Existence and stability of travelling waves in periodic media governed by a bistable nonlinearity. J. Dyn. Diff. Eqs. 3, 541–573
Xin J. (1992) Existence of planar flame fronts in convective–diffusive periodic media. Arch. Rat. Mech. Anal. 121, 205–233
Xin J. (2000) Front propagation in heterogeneous media. SIAM Review 42(2): 161–230
Xin J. (2003) KPP front speeds in random shears and the parabolic Anderson problem. Meth. Appl. Anal. 10(2): 191–198
Yakhot V. (1988) Propagation velocity of premixed turbulent flames. Comb. Sci. Tech 60, 191
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Nolen, J., Xin, J. A Variational Principle for KPP Front Speeds in Temporally Random Shear Flows. Commun. Math. Phys. 269, 493–532 (2007). https://doi.org/10.1007/s00220-006-0144-8
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DOI: https://doi.org/10.1007/s00220-006-0144-8