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

TL;DR

This paper investigates how nonlinear diffusion and logistic damping influence the existence of global weak solutions in a chemotaxis-Navier--Stokes system, establishing conditions under which solutions exist despite weak effects.

Contribution

It provides new existence results for global weak solutions under specific conditions on diffusion and damping parameters, extending previous research.

Findings

Existence of global weak solutions when diffusion is strong (m > 2/3) or logistic damping is strong (α > 4/3) with no damping (μ=0).

Solutions exist under weaker diffusion (m > 0) if logistic damping (μ > 0) is sufficiently strong (α > 4/3).

The results highlight how either strong diffusion or strong logistic damping can ensure global solutions despite the other's weakness.

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.