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Description

Chemotactic singularity formation in the context of the 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 talk, I will introduce Keller-Segel equation coupled with several incompressible flows forced by buoyancy, including a flow characterized by Darcy's law for incompressible porous media and a Navier-Stokes equation. We prove that, in contrast to previous results driven by passive mixing flows, such active fluid coupling is capable of suppressing singularity formation via an interesting dynamical, nonlinear mixing mechanism which is manifested in a delicate kinetic-potential energy balance. This talk is based on a recent work by the author and an earlier joint work with Alexander Kiselev and Yao Yao.