Title:

Singularity suppression by fluid flow

Abstract:

Transport by fluid flow can provide one of the less understood regularization mechanisms in PDE. In this talk, I will focus on the 2D Keller-Segel equation for chemotaxis coupled via buoyancy with the fluid obeying Darcy's law - a much studied model of the incompressible fluid flow in porous media. It is well known that solutions to the 2D Keller-Segel equation can form singularities in finite time if the mass of the initial data is larger than critical. It turns out that if the equation is coupled with fluid flow obeying Darcy's law via buoyancy, this completely regularizes the system, leading to globally regular solutions for arbitrarily large initial data. One of the key ingredients in the proof is a generalized Nash inequality, which employs anisotropic norm that is natural in the context of the incompressible porous media flow. This talk is based on works joint with Kevin Hu, Naji Sarsam, and Yao Yao.