We study the chemotaxis-(Navier–)Stokes system modeling coral fertilization: \(n_t+u\cdot \nabla n=\Delta n-\nabla \cdot (nS(x,n,c)\nabla c)-nm\), \(c_t+u\cdot \nabla c=\Delta c-c+m\), \(m_t+u\cdot \nabla m=\Delta m-nm\), \(u_t+\kappa (u\cdot \nabla )u+\nabla P=\Delta u+(n+m)\nabla \phi \) and \(\nabla \cdot u=0\) in a bounded and smooth domain \(\Omega \subset \mathbb {R}^2\), where \(\kappa \in \mathbb {R}\), \(\phi \in W^{2,\infty }(\Omega )\), and \(S\in C^2({\bar{\Omega }}\times [0,\infty )^2;\mathbb {R}^{2\times 2})\) satisfies \(|S(x,n,c)|\le S_0(c)(1+n)^{-\alpha }\) for all \((x,n,c)\in {\bar{\Omega }}\times [0,\infty )^2\) with \(\alpha \in \mathbb {R}\) and the function \(S_0:[0,\infty )\rightarrow [0,\infty )\) nondecreasing. Under the relatively weak destabilizing action of cross-diffusion for \(\alpha \ge 0\), the global boundedness of classical solutions was obtained in Espejo and Winkler (Nonlinearity 31:1227–1259, 2018) and Li (Differ Equ 267:6290–6315, 2019). In this paper, we show that even if n|S| with \(-\frac{1}{2}<\alpha <0\) bears a superlinear growth of n, the corresponding initial-boundary value problem (with any \(\kappa \in \mathbb {R}\)) still possesses a global classical solution emanating from any suitably smooth initial data. Moreover, when \(\kappa =0\), this solution is globally bounded.

我们研究趋化性-（Navier-）Stokes 系统建模珊瑚受精：\(n_t+u\cdot \nabla n=\Delta n-\nabla \cdot (nS(x,n,c)\nabla c)-nm\ ) , \(c_t+u\cdot \nabla c=\Delta c-c+m\) , \(m_t+u\cdot \nabla m=\Delta m-nm\) , \(u_t+\kappa (u\ cdot \nabla )u+\nabla P=\Delta u+(n+m)\nabla \phi \)和\(\nabla \cdot u=0\)在有界平滑域\(\Omega \subset \mathbb { R}^2\)，其中\(\kappa \in \mathbb {R}\)，\(\phi \in W^{2,\infty }(\Omega )\)和\(S\in C ^2({\bar{\Omega }}\times [0,\infty )^2;\mathbb {R}^{2\times 2})\)满足\(|S(x,n,c)| \le S_0(c)(1+n)^{-\alpha }\)对于所有\((x,n,c)\in {\bar{\Omega }}\times [0,\infty )^2\)与\(\alpha \in \mathbb {R}\)和函数\(S_0:[0,\infty )\rightarrow [0,\infty )\)非递减。在对\(\alpha \ge 0\)的交叉扩散的相对较弱的不稳定作用下，经典解的全局有界在 Espejo 和 Winkler (Nonlinearity 31:1227–1259, 2018) 和 Li (Differ Equ 267: 6290-6315，2019）。在本文中，我们证明即使n | 小号| 与\(-\frac{1}{2}<\alpha <0\)承受n的超线性增长，相应的初始边界值问题（与任何\(\kappa \in \mathbb {R}\)) 仍然具有源自任何适当平滑的初始数据的全局经典解。此外，当\(\kappa =0\) 时，此解决方案是全局有界的。