We consider the chemotaxis-Navier–Stokes system with gradient-dependent flux limitation and nonlinear production: $n_t+u\cdot \nabla n=\mathrm{\Delta }n-\nabla \cdot (nf(|\nabla c{|}^2)\nabla c)$, $c_t+u\cdot \nabla c=\mathrm{\Delta }c-c+g(n)$, $u_t+(u\cdot \nabla )u+\nabla P=\mathrm{\Delta }u+n\nabla \varphi$ and $\nabla \cdot u=0$ in a bounded domain $\mathrm{\Omega }\subset {ℝ}^2$, where the flux limitation function $f\in C^2([0,\infty ])$ and the signal production function $g\in C^1([0,\infty ])$ generalize the prototypes $f(s)=K_f{(1+s)}^{-\frac{\alpha }{2}}$ and $g(s)=K_{g}s{(1+s)}^{\beta -1}$ with $K_f,K_g>0$, $\alpha \in ℝ$ and $\beta >0$. For the linear production case of $\beta =1$, the global boundedness of solutions has been verified in the related literature for $\alpha >0$. In this paper, we expand to prove that the corresponding initial-boundary value problem possesses a unique globally bounded solution if $\alpha >{(2\beta -1)}_{+}-1$ for $0<\beta <1$, or if $\alpha >1-\frac{1}{2\beta -1}$ for $\beta >1$, which shows that when $0<\beta <1$, that is, the self-enhancement ability of chemoattractant is weak, the solutions still remain globally bounded even though the flux limitation is relaxed to permit proper $\alpha \le 0$; however, if $\beta >1$, it is necessary to impose the stronger flux limitation than that in the case $\beta =1$ to inhibit the possible finite-time blow-up. This seems to be the first result on the global solvability in the chemotaxis-Navier–Stokes model with nonlinear production.

我们考虑具有梯度相关通量限制和非线性产生的趋化-纳维-斯托克斯系统：$n_t+你\cdot \nabla n=\mathrm{\Delta }n-\nabla \cdot （nF（|\nabla C{|}^2）\nabla C）$,$C_t+你\cdot \nabla C=\mathrm{\Delta }C-C+G（n）$,${你}_t+（你\cdot \nabla ）你+\nabla 磷=\mathrm{\Delta }你+n\nabla \phi$和$\nabla \cdot 你=0$在有界域中$\mathrm{\Omega }\subset {ℝ}^2$，其中通量限制函数$F\epsilon C^2（[0,\mathrm{无穷大}]）$和信号产生函数$G\epsilon C^1（[0,\mathrm{无穷大}]）$概括原型$F（s）=K_F{（1+s）}^{-\frac{\alpha }{2}}$和$G（s）=K_{G}s{（1+s）}^{\beta -1}$和$K_F,K_G>0$,$\alpha \epsilon ℝ$和$\beta >0$。对于线性生产情况$\beta =1$，解的全局有界性已在相关文献中得到验证$\alpha >0$。在本文中，我们扩展证明相应的初始边值问题具有唯一的全局有界解，如果$\alpha >{（2\beta -1）}_{+}-1$为了$0<\beta <1$， 或者如果$\alpha >1-\frac{1}{2\beta -1}$为了$\beta >1$，这表明当$0<\beta <1$，即化学引诱剂的自我增强能力较弱，即使放宽通量限制以允许适当的解，解仍然保持全局有界$\alpha \le 0$; 然而，如果$\beta >1$，有必要施加比情况更强的通量限制$\beta =1$以抑制可能的有限时间爆炸。这似乎是具有非线性产生式的趋化-纳维-斯托克斯模型中全局可解性的第一个结果。