Suppression of chemotactic singularity by buoyancy

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.