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

$$\begin{aligned} {\left\{ \begin{array}{ll} n_t + u\cdot \nabla n = \nabla \cdot (D(n)\nabla n) - \nabla \cdot (n \chi (c) \nabla c) + \kappa n - \mu n^\alpha , &{}\quad x\in \Omega ,\ t>0, \\ c_t + u\cdot \nabla c = \Delta c - nf(c), &{}\quad x \in \Omega ,\ t>0, \\ u_t + (u\cdot \nabla )u = \Delta u + \nabla P + n\nabla \Phi + g, \quad \nabla \cdot u = 0, &{}\quad x \in \Omega ,\ t>0, \end{array}\right. } \end{aligned}$$

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 smooth functions satisfying

$$\begin{aligned} { \left( \frac{f}{\chi } \right) ' >0, \quad \left( \frac{f}{\chi } \right) '' \le 0, \quad (\chi f)' \ge 0 \quad \text{ on } \ [0,\infty ); } \end{aligned}$$

$\kappa \in \mathbb {R},\mu \ge 0,\alpha >1$ are constants. This paper shows existence of global weak solutions to the above system under the condition that

$$\begin{aligned} m>\frac{2}{3},\quad \mu \ge 0 \quad \text{ and }\quad \alpha >1 \end{aligned}$$

hold or that

$$\begin{aligned} m> 0, \quad \mu>0 \quad \text{ and } \quad \alpha > \frac{4}{3} \end{aligned}$$

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 (Kurima and Mizukami in Nonlinear Anal Real World Appl, 2018. arXiv:1802.08807 [math.AP]; Lankeit in Math Models Methods Appl Sci 26:2071–2109, 2016; Winkler in Ann Inst H Poincaré Anal Non Linéaire 33:1329–1352, 2016; Zhang and Li in J Differ Equ 259:3730–3754, 2015).