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- P Constantin, A Kiselev, L Ryzhik, A Zlatoš
- 2005

We study enhancement of diffusive mixing on a compact Riemannian man-ifold by a fast incompressible flow. Our main result is a sharp description of the class of flows that make the deviation of the solution from its average arbitrarily small in an arbitrarily short time, provided that the flow amplitude is large enough. The necessary and sufficient… (More)

- Andrej Zlatoš
- 2005

We consider the reaction-diffusion equation T t = T xx + f (T) on R with T 0 (x) ≡ χ [−L,L] (x) and f (0) = f (1) = 0. In 1964 Kanel' proved that if f is an ignition non-linearity, then T → 0 as t → ∞ when L < L 0 , and T → 1 when L > L 1. We answer the open question of relation of L 0 and L 1 by showing that L 0 = L 1. We also determine the large time… (More)

- Andrej Zlatoš
- 2007

We study KPP pulsating front speed-up and effective diffusivity enhancement by general periodic incompressible flows. We prove the existence of and determine the limits c * (A)/A and D(A)/A 2 as A → ∞, where c * (A) is the minimal front speed and D(A) the effective diffusivity.

We consider Fisher-KPP-type reaction-diffusion equations with spatially inhomogeneous reaction rates. We show that a sufficiently strong localized inhomogeneity may prevent existence of transition-front-type global in time solutions while creating a global in time bump-like solution. This is the first example of a medium in which no reaction-diffusion… (More)

- Andrej Zlatoš
- 2009

We prove existence, uniqueness, and stability of transition fronts (generalized traveling waves) for reaction-diffusion equations in cylindrical domains with general inhomo-geneous ignition reactions. We also show uniform convergence of solutions with exponentially decaying initial data to time translates of the front. In the case of stationary ergodic… (More)

- Andrej Zlatoš
- 2011

We use a new method in the study of Fisher-KPP reaction-diffusion equations to prove existence of transition fronts for inhomogeneous KPP-type non-linearities in one spatial dimension. We also obtain new estimates on entire solutions of some KPP reaction-diffusion equations in several spatial dimensions. Our method is based on the construction of sub-and… (More)

We consider a model describing premixed combustion in the presence of fluid flow: reaction diffusion equation with passive advection and ignition type nonlinearity. What kinds of velocity profiles are capable of quenching (suppressing) any given flame, provided the velocity's amplitude is adequately large? Even for shear flows, the solution turns out to be… (More)

- Andrej Zlatoš
- 2007

We consider the advection-diffusion equation φ t + Au · ∇φ = ∆φ, φ(0, x) = φ 0 (x) on R 2 , with u a periodic incompressible flow and A 1 its amplitude. We provide a sharp characterization of all u that optimally enhance dissipation in the sense that for any initial datum φ 0 ∈ L p (R 2), p < ∞, and any τ > 0, Our characterization is expressed in terms of… (More)

We consider probability measures, dµ = w(θ) dθ 2π +dµ s , on the unit circle, ∂D, with Verblunsky coefficients, {α j } ∞ j=0. We prove for θ 1 = θ 2 in [0, 2π) and (δβ) j = β j+1 that [1 − cos(θ − θ 1)][1 − cos(θ − θ 2)] log w(θ) dθ 2π > −∞ if and only if