# Holder continuity of Keller-Segel equations of porous medium type coupled to fluid equations

## TL;DR

This paper proves the global existence and H"older continuity of solutions for a coupled Keller-Segel and fluid system modeling bacteria in fluid, using a unified method applicable to degenerate porous medium equations.

## Contribution

It establishes the first global existence and regularity results for this coupled system with degeneracy, advancing understanding of bacterial movement in fluids.

## Key findings

- Global weak solutions exist in three dimensions.
- Solutions are H"older continuous under certain degeneracy conditions.
- A unified method for H"older regularity of degenerate equations is developed.

## Abstract

We consider a coupled system consisting of a degenerate porous medium type of Keller-Segel system and Stokes system modeling the motion of swimming bacteria living in fluid and consuming oxygen. We establish the global existence of weak solutions and H\"older continuous solutions in dimension three, under the assumption that the power of degeneracy is above a certain number depending on given parameter values. To show H\"older continuity of weak solutions, we consider a single degenerate porous medium equation with lower order terms, and via a unified method of proof, we obtain H\"older regularity, which is of independent interest.