Global Solution of 3-D Patlak-Keller-Segal Model with Couette flow in whole space

Shijin Deng, Binbin Shi, Weike Wang

Published 2023-11-30Version 1

In this paper, we consider both parabolic-elliptic Patlak-Keller-Segel model and parabolic-parabolic Patlak-Keller-Segel model in the background of a Couette flow with spatial variables in $\mathbb{R}^3$. It is proved that for both parabolic-elliptic and parabolic-parabolic cases, a Couette flow with a sufficiently large amplitude prevents the blow-up of solutions. This result is totally different from either the classical Patlak-Keller-Segel model or the case with a large shear flow and the periodic spatial variable $x$; for those two cases, the solution may blow up. Here, we apply Green's function method to capture the suppression of blow-up and prove the global existence of the solutions.