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

Abstract

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.

Breakdown of homogenization for the random Hamilton-Jacobi equations

“…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.…”

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

“…The two-scale discretization scheme is much more efficient than the one-scale discretization while achieving the same accuracy. In future work, we plan to study front speed ensemble in space-time random flows [29,31] by extending the two-scale method to parabolic problems.…”

“…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]. The variational principle of KPP front speeds makes accurate and efficient analytical and numerical studies possible.…”

Relaxation in Reactive Flows

“…It has been shown in [4,12] that the flow u(x) is relaxation-enhancing if and only if u has no first integrals in with a non-negative nonlinearity f (s) that vanishes at s = 0 and s = 1 -such equations appear in flame propagation as well as many other applied problems. It has been shown that a flow may speed-up a flame [9,18,23,28,30] or quench the propagation [11,17,24,37,38] as A → +∞.…”