Weak entropy solution for a Keller-Segel type fluid model

TL;DR

This paper analyzes a Keller-Segel type fluid model, establishing conditions for finite-time blow-up of solutions when the initial mass exceeds a critical threshold, and providing a priori estimates for critical and subcritical masses.

Contribution

It introduces a weak entropy solution framework for a Keller-Segel fluid model and identifies a critical mass for blow-up similar to the parabolic case.

Findings

Solutions blow up in finite time when initial mass exceeds 8π.

A priori estimates are derived for critical and subcritical masses.

The model extends understanding of Keller-Segel dynamics in fluid contexts.

Abstract

In this paper, we consider a Keller-Segel type fluid model, which is a kind of Euler-Poisson system with a self-gravitational force. We show that similar to the parabolic case, there is a critical mass $8\pi$ such that if the initial total mass $M$ is supercritical, i.e., $M> 8\pi$, then any weak entropy solution with the same mass $M$ must blow up in finite time. The a priori estimates of weak entropy solutions for critical mass $M=8\pi$ and subcritical mass $M<8\pi$ are also obtained.