Suppressing chemotactic blow-up through a fast splitting scenario on the plane

TL;DR

This paper demonstrates that a linear ambient flow can prevent blow-up in the 2D Patlak-Keller-Segel chemotaxis model, effectively doubling the mass threshold for global regularity by inducing a 'fast splitting' scenario.

Contribution

It introduces the concept that ambient fluid transport via a linear flow can suppress chemotactic blow-up, extending the mass threshold for global solutions in the PKS model.

Findings

Linear ambient flow prevents blow-up for M<16π.

Flow induces a 'fast splitting' that disperses mass.

Doubling the mass threshold for global regularity.

Abstract

We revisit the question of global regularity for the Patlak-Keller-Segel (PKS) chemotaxis model. The classical 2D hyperbolic-elliptic model blows up for initial mass M>8\pi. We consider more realistic scenario which takes into account the flow of the ambient environment induced by harmonic potentials, and thus retain the identical elliptic structure as in the original PKS. Surprisingly, we find that already the simplest case of linear stationary vector field, $Ax^\top$, with large enough amplitude $A$, prevents chemotactic blow-up. Specifically, the presence of such an ambient fluid transport creates what we call a 'fast splitting scenario', which competes with the focusing effect of aggregation so that 'enough mass' is pushed away from concentration along the $x_1$-axis, thus avoiding a finite time blow-up, at least for M<16\pi. Thus, the enhanced ambient flow doubles the amount of allowable mass which evolve to global smooth solutions.