Enhanced dissipation and blow-up suppression for an aggregation equation with fractional diffusion and shear flow

TL;DR

This paper investigates how shear flow enhances dissipation to prevent finite-time blow-up in an aggregation equation with fractional diffusion, establishing global solutions for certain fractional powers.

Contribution

It demonstrates that shear flow-induced enhanced dissipation can suppress blow-up in aggregation equations with fractional diffusion, providing new technical methods for low regularity cases.

Findings

Enhanced dissipation improves solution stability.

Shear flow suppresses blow-up for   3/2.

Established global classical solutions under specific conditions.

Abstract

In this paper, we consider an aggregation equation with fractional diffusion and large shear flow, which arise from modelling chemotaxis in bacteria. Without the advection, the solution of aggregation equation may blow up in finite time. First, we study the enhanced dissipation of shear flow by resolvent estimate method, where the fractional Laplacian $(-\Delta)^{\alpha/2}$ is considered and $\alpha\in (0,2)$. Next, we show that the enhanced dissipation of shear flow can suppress blow-up of solution to aggregation equation with fractional diffusion and establish global classical solution in the case of $\alpha\geq 3/2$. Here we develop some new technical to overcome the difficult of low regularity for fractional Laplacian.