28 research outputs found
Periodic orbits of the ABC flow with
In this paper, we prove that the ODE system
whose right-hand side is the Arnold-Beltrami-Childress (ABC) flow with
parameters , has periodic orbits on with rotation
vectors parallel to , , and . An application of this
result is that the well-known G-equation model for turbulent combustion with
this ABC flow on has a linear (i.e., maximal possible) flame
speed enhancement rate as the amplitude of the flow grows.Comment: 9 page
Higher-order Szegő theorems with two singular points
We consider probability measures, dμ = w(θ)^(dθ)_(2π) + dμ_s, on the unit circle, ∂D, with Verblunsky coefficients, {αj}_(j=0)^∞. We prove for θ_1 ≠ θ_2 in [0,2π) that ∫[1-cos(θ-θ_1)][1-cos(θ-θ_2)]log w(θ)^(dθ)_(2π > -∞if and only if ∑_(j=0)^∞ │{(δ-e^(-iθ2))(δ-e^(-iθ1))α}_j^2 +|α_j|^4 - ∞ if and only if ∑_(j=0)^∞|α_(j+2) - 2α_(j+1) + α_j|^2 + |αj|^ 6 <∞
Ballistic Orbits and Front Speed Enhancement for ABC Flows
We study the two main types of trajectories of the ABC flow in the
near-integrable regime: spiral orbits and edge orbits. The former are helical
orbits which are perturbations of similar orbits that exist in the integrable
regime, while the latter exist only in the non-integrable regime. We prove
existence of ballistic (i.e., linearly growing) spiral orbits by using the
contraction mapping principle in the Hamiltonian formulation, and we also find
and analyze ballistic edge orbits. We discuss the relationship of existence of
these orbits with questions concerning front propagation in the presence of
flows, in particular, the question of linear (i.e., maximal possible) front
speed enhancement rate for ABC flows.Comment: 39 pages, 26 figure
Pulsating Front Speed-up and Quenching of Reaction by Fast Advection
We consider reaction-diffusion equations with combustion-type non-linearities
in two dimensions and study speed-up of their pulsating fronts by general
periodic incompressible flows with a cellular structure. We show that the
occurence of front speed-up in the sense ,
with the amplitude of the flow and the (minimal) front speed, only
depends on the geometry of the flow and not on the reaction function. In
particular, front speed-up happens for KPP reactions if and only if it does for
ignition reactions. We also show that the flows which achieve this speed-up are
precisely those which, when scaled properly, are able to quench any ignition
reaction.Comment: 16p
Multidimensional transition fronts for Fisher–KPP reactions
We study entire solutions to homogeneous reaction-diffusion equations in several dimensions with Fisher-KPP reactions. Any entire solution 0 < u < 1 is known to satisfy
lim t→−∞ sup|x|≤c|t| u(t,x) = 0 for each c < 2√f′(0),
and we consider here those satisfying
lim t→−∞ sup|x|≤c|t| u(t,x) = 0 for some c > 2√f′(0).
When f is C_2 and concave, our main result provides an almost complete characterization of transition fronts as well as transition solutions with bounded width within this class of solutions