The explosion problem in a flow

TL;DR

This paper investigates the explosion threshold in incompressible flows, establishing uniform bounds, identifying flow conditions for infinite thresholds, and analyzing strong flow asymptotics in 2D one-cell flows.

Contribution

It introduces a novel $L^p-L^ Infty$ estimate for elliptic advection-diffusion problems and characterizes the explosion threshold behavior under various flow conditions.

Findings

Explosion threshold has a positive lower bound independent of flow.

Certain flows cause the explosion threshold to tend to infinity as amplitude increases.

Effective description of explosion threshold in strong flow limit for 2D one-cell flows.

Abstract

We consider the explosion problem in an incompressible flow introduced in the paper of H. Berestycki, L. Kagan, G. Joulin and G. Sivashinsky. We use a novel $L^p-L^\infty$ estimate for elliptic advection-diffusion problems to show that the explosion threshold obeys a positive lower bound which is uniform in the advecting flow. We also identify the flows for which the explosion threshold tends to infinity as their amplitude grows and obtain an effective description of the explosion threshold in the strong flow asymptotics in a two-dimensional one-cell flow.