Enhanced dissipation and blow-up suppression for the three dimensional Keller-Segel equation with a non-shear incompressible flow

TL;DR

This paper demonstrates that a specific three-dimensional non-shear incompressible flow can enhance dissipation and prevent finite-time blow-up in the Keller-Segel equation, enabling global solutions with large initial data.

Contribution

It introduces a new flow type and proves its ability to suppress blow-up in 3D Keller-Segel equations, extending understanding of flow-induced dissipation effects.

Findings

Enhanced dissipation achieved via resolvent estimates.

Flow suppresses blow-up, ensuring global solutions.

Large initial data solutions are globally well-posed.

Abstract

In this paper, we consider the Cauchy problem for the three dimensional parabolic-elliptic Keller-Segel equation with a large non-shear incompressible flow. Without advection, there exist solution with arbitrarily mass which blow up in finite time. Firstly, we introduce a three dimensional non-shear incompressible flow and study the enhanced dissipation of such flows by resolvent estimate method. Next, we show that the enhanced dissipation of such flow can suppress blow-up of solution to three dimensional parabolic-elliptic Keller-Segel equation and establish global classical solution with large initial data.