Min-max variational principle and front speeds in random shear flows

Ann. Inst. H. Poincaré C Anal. Non Linéaire

2009

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We study the asymptotic spreading of Kolmogorov-Petrovsky-Piskunov (KPP) fronts in space-time random incompressible flows in dimension d > 1. We prove that if the flow field is stationary, ergodic, and obeys a suitable moment condition, the large time front speeds (spreading rates) are deterministic in all directions for compactly supported initial data. The flow field can become unbounded at large times. The front speeds are characterized by the convex rate function governing large deviations of the associated diffusion in the random flow. Our proofs are based on the Harnack inequality, an application of the sub-additive ergodic theorem, and the construction of comparison functions. Using the variational principles for the front speed, we obtain general lower and upper bounds of front speeds in terms of flow statistics. The bounds show that front speed enhancement in incompressible flows can grow at most linearly in the root mean square amplitude of the flows, and may have much slower growth due to rapid temporal decorrelation of the flows. RésuméOn étudie le comportement asymptotique des solutions des équations de réaction-diffusion du type Kolmogorov-PetrovskyPiskunov (KPP) avec convection stochastique en dimension d > 1. Dans le cas où l'écoulement est stationnaire et ergodique, nous démontrons que la solution forme un front qui se propage avec une vitesse déterministe. Les vitesses de propagation satisfont une formule variationnelle associée à un principe de grandes déviations pour un processus de diffusion en milieu aléatoire. Avec cette formule, nous obtenons quelques estimations de la vitesse. Les preuves sont basées sur une inégalité de type Harnack, le principe de maximum, et le théorème ergodique sous-additif.

Section: Discussionmentioning

confidence: 98%

Asymptotic spreading of KPP reactive fronts in incompressible space–time random flows

Ann. Inst. H. Poincaré C Anal. Non Linéaire

2009

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“…Speed characterizations have been studied mathematically for various flow patterns by analysis of the reaction-diffusionadvection (RDA) equations (see [2,5,9,11,14,15,17,18,20,24,26,30,31] and references therein). In particular, variational characterizations have led to speed asymptotics in both deterministic and random flows [24,11,31,21,22,23].…”

Section: Introductionmentioning

confidence: 99%

Existence of KPP fronts in spatially-temporally periodic advection and variational principle for propagation speeds

Rudd

Dynamics of Partial Differential Equations

2005

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107

We prove the existence of Kolmogorov-Petrovsky-Piskunov (KPP) type traveling fronts in space-time periodic and mean zero incompressible advection, and establish a variational (minimization) formula for the minimal speeds. We approach the existence by considering limit of a sequence of front solutions to a regularized traveling front equation where the nonlinearity is combustion type with ignition cut-off. The limiting front equation is degenerate parabolic and does not permit strong solutions, however, the necessary compactness follows from monotonicity of fronts and degenerate regularity. We apply a dynamic argument to justify that the constructed KPP traveling fronts propagate at minimal speeds, and derive the speed variational formula. The dynamic method avoids the degeneracy in traveling front equations, and utilizes the parabolic maximum principle of the governing reaction-diffusion-advection equation. The dynamic method does not rely on existence of traveling fronts.

“…Furthermore, there is an optimal correlation length that maximizes the enhancement. Linear and quadratic speed growth laws are known for deterministic flow patterns with channel structures ( [3,4,6,8,16,18,34] and references), also for spatially random shears inside infinite cylinders [29,30] or white in time Gaussian shears in the entire space [38]. The speed growth laws known to date are not sensitive in the form of nonlinearities as long as fronts propagate out of the initial data.…”

Section: A4mentioning

confidence: 99%

A Variational Principle for KPP Front Speeds in Temporally Random Shear Flows

2006

Commun. Math. Phys.

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We establish the variational principle of Kolmogorov-PetrovskyPiskunov (KPP) front speeds in temporally random shear flows inside an infinite cylinder, under suitable assumptions of the shear field. A key quantity in the variational principle is the almost sure Lyapunov exponent of a heat operator with random potential. The variational principle then allows us to bound and compute the front speeds. We show the linear and quadratic laws of speed enhancement as well as a resonance-like dependence of front speed on the temporal shear correlation length. 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.