Influence of Flux Limitation on Large Time Behavior in a Three-Dimensional Chemotaxis-Stokes System Modeling Coral Fertilization

In this paper, we consider the following system $$ \left \{ \textstyle\begin{array}{ll} n_{t}+u\cdot \nabla n&=\Delta n-\nabla \cdot (n\mathcal{S}(|\nabla c|^{2}) c)-nm, \\ c_{t}+u\cdot c&=\Delta c-c+m, m_{t}+u\cdot m&=\Delta m-mn, u_{t}&=\Delta u+\nabla P+(n+m)\nabla \Phi ,\qquad u=0 \end{array}\displaystyle \right . which models process of coral fertilization, in a smoothly three-dimensional bounded domain, where $\mathcal{S}$ is given function fulfilling |\mathcal{S}(\sigma )|\leq K_{\mathcal{S}}(1+\sigma )^{- \frac{\theta }{2}},\qquad \sigma \geq 0 with some $K_{\mathcal{S}}>0$ Based on conditional estimates quantity $c$ and gradients thereof, relatively compressed argument as compared to that proceeding related precedents shows if \theta >0, then for any initial data proper regularity an associated initial-boundary problem under no-flux/no-flux/no-flux/Dirichlet boundary conditions admits unique classical solution globally bounded, also enjoys stabilization features sense $$\begin{aligned} &\|n(\cdot ,t)-n_{\infty }\|_{L^{\infty }(\Omega )}+\|c(\cdot ,t)-m_{ \infty }\|_{W^{1,\infty )} +\|m(\cdot ,t)-m_{\infty }\|_{W^{1, )}\\ &\quad{}+\|u(\cdot ,t)\|_{L^{\infty )}\rightarrow \quad \text{as}~t\rightarrow \end{aligned}$$ $n_{\infty }:=\frac{1}{|\Omega |}\left \int _{\Omega }n_{0}-\int _{ \Omega }m_{0}\right \} _{+}$ $m_{\infty }m_{0}-\int }n_{0}\right