How strongly does diffusion or logistic-type degradation affect existence of global weak solutions in a chemotaxis-Navier--Stokes system?

Abstract

This paper considers the chemotaxis-Navier--Stokes system with nonlinear diffusion and logistic-type degradation term \begin{align*} \begin{cases} n_t + u\cdot\nabla n = \nabla \cdot(D(n)\nabla n) - \nabla\cdot(n \chi(c) \nabla c) + \kappa n - \mu n^\alpha, & x\in \Omega,\ t>0, \\ c_t + u\cdot\nabla c = \Delta c - nf(c), & x \in \Omega,\ t>0, \\ u_t + (u\cdot\nabla)u = \Delta u + \nabla P + n\nabla\Phi + g, \ \nabla\cdot u = 0, & x \in \Omega,\ t>0, \end{cases} \end{align*} where $\Omega\subset \mathbb{R}^3$ is a bounded smooth domain; $D \ge 0$ is a given smooth function such that $D_1 s^{m-1} \le D(s) \le D_2 s^{m-1}$ for all $s\ge 0$ with some $D_2 \ge D_1 > 0$ and some $m > 0$; $\chi,f$ are given functions satisfying some conditions; $\kappa \in \mathbb{R},\mu \ge0,\alpha>1$ are constants. This paper shows existence of global weak solutions to the above system under the condition that \begin{align*} m >\frac{2}{3},\quad \mu \ge 0 \quad \mbox{and}\quad \alpha >1 \end{align*} hold, or that \begin{align*} m> 0, \quad \mu>0 \quad \mbox{and} \quad \alpha > \frac{4}{3} \end{align*} hold. This result asserts that `strong' diffusion effect or `strong' logistic damping derives existence of global weak solutions even though the other effect is `weak', and can include previous works.