On the Fast Spreading Scenario

TL;DR

This paper investigates how strong divergence-free fluid flows can suppress growth, prevent singularities, and ensure global regularity in PDE models of chemotaxis and combustion on unbounded domains.

Contribution

It demonstrates that fast hyperbolic and shear flows can prevent singularity formation and promote regularity in chemotaxis and combustion PDE models, confirming previous numerical findings.

Findings

Fast flows can suppress solution growth in PDE models.

Strong flows prevent singularity formation in chemotaxis equations.

Flow-induced quenching occurs in advection-reaction-diffusion systems.

Abstract

We study two types of divergence-free fluid flows on unbounded domains in two and three dimensions -- hyperbolic and shear flows -- and their influence on chemotaxis and combustion. We show that fast spreading by these flows, when they are strong enough, can suppress growth of solutions to PDE modeling these phenomena. This includes prevention of singularity formation and global regularity of solutions to advective Patlak-Keller-Segel equations on ${\mathbb R}^2$ and ${\mathbb R}^3$, confirming numerical observations in [Khan, Johnson,Cartee and Yao, Global regularity of chemotaxis equations with advection, Involve, 9(1) 2016], as well as quenching in advection-reaction-diffusion equations.