Weak entropy solution for a Keller-Segel type fluid model

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 $8\ddot{I}€$ such that if the initial total mass $M$ is supercritical, i.e., $M>8\ddot{I}€$, 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 $M=8\ddot{I}€$ and subcritical mass $M<8\ddot{I}€$ are also obtained.