# Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system with nonlinear diffusion

## TL;DR

This paper proves the existence of global weak solutions for a three-dimensional Keller-Segel-Navier-Stokes system with nonlinear diffusion when the diffusion exponent exceeds 2, under certain boundary conditions and initial data.

## Contribution

It establishes the first global weak solution existence result for the 3D Keller-Segel-Navier-Stokes system with nonlinear diffusion for m>2.

## Key findings

- Global weak solutions exist for m>2
- Solutions are valid for reasonably regular initial data
- The results apply under Neumann and no-slip boundary conditions

## Abstract

The coupled quasilinear Keller-Segel-Navier-Stokes system is considered under Neumann boundary conditions for $n$ and $c$ and no-slip boundary conditions for $u$ in three-dimensional bounded domains $\Omega\subseteq \mathbb{R}^3$ with smooth boundary, where $m>0,\kappa\in \mathbb{R}$ are given constants, $\phi\in W^{1,\infty}(\Omega)$. If $ m> 2$, then for all reasonably regular initial data, a corresponding initial-boundary value problem for $(KSNF)$ possesses a globally defined weak solution.