Suppression of blow-up in 3-D Keller-Segel model via the Couette flow in whole space

Abstract

In this paper, we study the 3-D parabolic-parabolic and parabolic-elliptic Keller-Segel models with the Couette flow. We prove that the blow-up phenomenon of solution can be suppressed by enhanced dissipation of the large Couette flows. Here we develop Green's function method to describe the enhanced dissipation via a more precise space-time structure and obtain the global existence together with pointwise estimates of the solutions. The result of this paper shows that the enhanced dissipation exists for all frequencies in the case of whole space and it is the reason that we obtain global existence for 3-D Keller-Segel models here. It is totally different from the case with the periodic spatial variable x in [3], [10]. This paper provides a new methodology to capture dissipation enhancement and also a surprising result which shows a totally new mechanism.