Quenching of Reaction by Cellular Flows

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 $\lim_{A\to\infty} c_*(A)=\infty$, with $A$ the amplitude of the flow and $c_*(A)$ 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 thi…

混合噪声驱动的带耦合内部奇异吸收项的随机反应扩散系统的淬灭时间与概率估计

Quenching time and probability estimates for a stochastic reaction-diffusion system with coupled inner singular absorption terms driven by mixed noises

本文研究Robin边界条件下带混合噪声的随机反应扩散系统，推导淬灭时间与概率的显式界，并设计数值方案验证理论结果。

Quenching and Propagation of Combustion Without Ignition Temperature Cutoff

We study a reaction-diffusion equation in the cylinder $Ω= \mathbb{R}\times\mathbb{T}^m$, with combustion-type reaction term without ignition temperature cutoff, and in the presence of a periodic flow. We show that if the reaction function decays as a power of $T$ larger than three as $T\to 0$ and the initial datum is small, then the flame is extinguished -- the solution quenches. If, on the other hand, the power of decay is smaller than three or initial datum is large, then quenching does not happen, and the burning region spreads linearly in time. This extends results of Aronson-Weinberger f…

Quenching of combustion by shear flows

We consider a simple model describing premixed combustion in the presence of fluid flow: reaction diffusion equation with passive advection and ignition type nonlinearity. Strong advection can suppress flames - a process we call quenching. A flow is called quenching if any compactly supported initial data will become extinct provided that the amplitude of the flow is chosen sufficiently large. In this paper, we provide a sharp characterization of quenching shear flows.The efficiency of quenching depends strongly on the geometry and scaling of the flow. We discuss the cases of slowly and quickl…

Reaction-Diffusion Front Speed Enhancement by Flows

We study flow-induced enhancement of the speed of pulsating traveling fronts for reaction-diffusion equations, and quenching of reaction by fluid flows. We prove, for periodic flows in two dimensions and any combustion-type reaction, that the front speed is proportional to the square root of the (homogenized) effective diffusivity of the flow. We show that this result does not hold in three and more dimensions. We also prove conjectures from [1,3,11] for cellular flows, concerning the rate of speed-up of fronts and the minimal flow amplitude necessary to quench solutions with initial data of a…

Control of Spatially Heterogeneous and Time-Varying Cellular Reaction Networks: A New Summation Law

A hallmark of a plethora of intracellular signaling pathways is the spatial separation of activation and deactivation processes that potentially results in precipitous gradients of activated proteins. The classical Metabolic Control Analysis (MCA), which quantifies the influence of an individual process on a system variable as the control coefficient, cannot be applied to spatially separated protein networks. The present paper unravels the principles that govern the control over the fluxes and intermediate concentrations in spatially heterogeneous reaction networks. Our main results are two ty…