Publication 19-CNA-011

Convection-Induced Singularity Suppression in the Keller-Segel and other Non-Linear PDEs

Gautam Iyer
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213
gautam@math.cmu.edu

Xiaoqian Xu
Department of Mathematics
Duke Kunshan University
Kunshan, China
xiaoqian.xu@dukekunshan.edu.cn

Andrej Zlatos
Department of Mathematics
UC San Diego
La Jolla, CA 92130, USA
zlatos@ucsd.edu

Abstract: In this paper we study the effect of the addition of a convective term, and of the resulting increased dissipation rate, on the growth of solutions to a general class of non-linear parabolic PDEs. In particular, we show that blow-up in these models can always be prevented if the added drift has a small enough dissipation time. We also prove a general result relating the dissipation time and the effective diffusivity of stationary cellular flows, which allows us to obtain examples of simple incompressible flows with arbitrarily small dissipation times.

As an application, we show that blow-up in the Keller-Segel model of chemotaxis can always be prevented if the velocity field of the ambient fluid has a sufficiently small dissipation time. We also study reaction-diffusion equations with ignition-type nonlinearities, and show that the reaction can always be quenched by the addition of a convective term with a small enough dissipation time, provided the average initial temperature is initially below the ignition threshold.

Get the paper in its entirety as  19-CNA-011.pdf

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