Global regularity and stability analysis of the Patlak–Keller–Segel system with flow advection in a bounded domain: A semigroup approach

Abstract

We discuss the problem of suppression of singularity via flow advection in chemotaxis modeled by the Patlak–Keller–Segel (PKS) equations. It is well-known that for the system without advection, singularity of the solution may develop at finite time. Specifically, if the initial condition is above certain critical threshold, the solution may blow up in finite time by concentrating positive mass at a single point. In this work, we mainly focus on the global regularity and stability analysis of the PKS system in the presence of flow advection in a bounded domain Ω⊂Rd,d=2,3, by using a semigroup approach. We will show that the global well-posedness can be obtained as long as the semigroup generated by the associated advection–diffusion operator has a rapid decay property. We will also show that for cellular flows in rectangle-like domains, such property can be achieved by rescaling both the cell size and the flow amplitude. This is analogous to the result established by Iyer, Xu and Zlatoš (2021) on the torus Td,d=2,3.