A New Result for Global Existence and Boundedness in a Three-Dimensional Self-consistent Chemotaxis-Fluid System with Nonlinear Diffusion

Abstract

This paper deals with a boundary-value problem for a coupled chemotaxis-Stokes system with nonlinear diffusion and a complicated nonlinear coupling term. The mathematical model considered herein appears as

$$ \left \{ \textstyle\begin{array}{l} n_{t}+u\cdot \nabla n=\Delta n^{m}-\nabla \cdot (n\nabla c)+\nabla \cdot (n\nabla \phi ),\quad x\in \Omega , t>0, \\ c_{t}+u\cdot \nabla c=\Delta c-nc,\quad x\in \Omega , t>0, \\ u_{t}+\nabla P=\Delta u-n\nabla \phi + n\nabla c,\quad x\in \Omega , t>0, \\ \nabla \cdot u=0,\quad x\in \Omega , t>0, \end{array}\displaystyle \right . $$

(CNF)

where $\Omega \subseteq \mathbb{R}^{3}$ is a general bounded domain with smooth boundary and $\phi \in W^{2,\infty}(\Omega )$ is a given gravitational potential function. Here, one of the novelties is that both the effect of gravity (potential force) on cells and the effect of the chemotactic force on fluid is considered, which leads to the stronger coupling than usual chemotaxis-fluid model studied in the most existing literatures. Based on a new energy-type argument combined with maximal Sobolev regularity theory, we conclude that if $m > \frac{7}{6}$, an associated initial-boundary value problem admits at least one globally weak solution which is uniformly bounded. This extends a recent result by Wang and Zhao (J. Differ. Equ. 269:148–179, 2020) which asserts global existence of weak solutions under the constraints $m >\frac{4}{3}$.