Suppression of chemotactic singularity by buoyancy

TL;DR

This paper demonstrates that coupling the Patlak-Keller-Segel equation with an active buoyancy-driven fluid flow prevents singularity formation, ensuring global regularity even at minimal coupling strength.

Contribution

It proves that buoyancy-driven fluid flow can suppress chemotactic singularities in the Patlak-Keller-Segel model, extending understanding of fluid-chemotaxis interactions.

Findings

Active buoyancy coupling prevents singularities.

Global regularity achieved at any coupling strength.

Contrasts passive advection with active buoyancy effects.

Abstract

Chemotactic singularity formation in the context of the Patlak-Keller-Segel equation is an extensively studied phenomenon. In recent years, it has been shown that the presence of fluid advection can arrest the singularity formation given that the fluid flow possesses mixing or diffusion enhancing properties and its amplitude is sufficiently strong - this effect is conjectured to hold for more general classes of nonlinear PDEs. In this paper, we consider the Patlak-Keller-Segel equation coupled with a fluid flow that obeys Darcy's law for incompressible porous media via buoyancy force. We prove that in contrast with passive advection, this active fluid coupling is capable of suppressing singularity formation at arbitrary small coupling strength: namely, the system always has globally regular solutions.