Suppression of chemotactic blow up by active scalar

There exist many regularization mechanisms in nonlinear PDE that help make solutions more regular or prevent formation of singularity: diffusion, dispersion, damping. A relatively less understood regularization mechanism is transport. There is evidence that in the fundamental PDE of fluid mechanics such as Euler or Navier-Stokes, transport can play a regularizing role. In this talk, I will discuss another instance where this phenomenon appears: the Patlak-Keler-Segel equation of chemotaxis. Chemotactic blow up 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 a given fluid advection can arrest singularity formation given that the fluid flow possesses mixing or diffusion enhancing properties and its amplitude is sufficiently strong. This talk will focus on the case when the fluid advection is active: the Patlak-Keller-Segel equation coupled with fluid that obeys Darcy's law for incompressible porous media flow via gravity. Surprisingly, in this context, in contrast with the passive advection, active fluid is capable of suppressing chemotactic blow up at arbitrary small coupling strength: namely, the system always has globally regular solutions.
The talk is based on work joint with Zhongtian Hu and Yao Yao.