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

Abstract

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.

Breakdown of homogenization for the random Hamilton-Jacobi equations

“…We shall present elementary proofs of these statements. Similar findings exist for random front speeds in related Burgers and reactiondiffusion-advection equations [4,12,13,9]. Hence the phenomena demonstrated here may have broad implications for wave propagation in random media.…”

“…A related problem is to study Hamiltonians with unbounded temporal fluctuations. For reaction-diffusion fronts in incompressible random advection, temporal randomness is found to regularize the dominance of extreme events and promote mixing; hence the speed of propagation is asymptotically a deterministic constant [13,7,9]. It is conceivable that similar results (or homogenization) hold for HJ equations in unbounded time-dependent random media under suitable conditions.…”

“…Recently, existence and estimates of front speeds in unbounded temporal random flows have been studied for reaction-diffusion-advection equations [13,7,9]. It is worthwhile for a future work to study whether HJ averaging in unbounded time-random regime behaves similarly.…”

Periodic homogenization of the inviscid G-equation for incompressible flows

“…Mathematical models range from reaction-diffusion-advection equations to advective Hamilton-Jacobi equations (HJ), [10,14,17,28,39,40,41]. A particular HJ equation, the so called G-equation, is quite popular in the combustion science literature [27,36,42,16].…”

Finite Element Computations of Kolmogorov–Petrovsky–Piskunov Front Speeds in Random Shear Flows in Cylinders

“…A fundamental problem is to characterize and compute large-scale front speeds in random flows [10,19,30,31,39]. The Kolmogorov-Petrovsky-Piskunov (KPP) minimal front speeds admit a variational characterization in terms of the principal eigenvalue or principal Lyapunov exponent of an associated linear operator [6,5,16,37,28,29].…”