Abstract:
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.
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