# Suppression of blow up by mixing in generalized Keller-Segel system with fractional dissipation

## TL;DR

This paper investigates how mixing effects from advection can prevent blow-up in solutions of a generalized Keller-Segel system with fractional dissipation, ensuring global well-posedness and boundedness.

## Contribution

It introduces a novel analysis of the Keller-Segel system incorporating fractional dissipation and mixing, establishing global solutions for large initial data.

## Key findings

- Global $L^$ bounds for solutions
- Existence of global classical solutions
- Effective mixing prevents blow-up

## Abstract

In this paper, we consider the Cauchy problem for a generalized parabolic-elliptic Keller-Segel equation with fractional dissipation and the additional mixing effect of advection by an incompressible flow. Under suitable mixing condition on the advection, we study well-posedness of solution with large initial data. We establish the global $L^\infty$ estimate of the solution through nonlinear maximum principle, and obtain the global classical solution.