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 $\left\{\begin{array}{ll} n_t+u\cdot\nabla n&=Δn-\nabla\cdot(n\mathcal{S}(|\nabla c|^2)\nabla c)-nm,\\ c_t+u\cdot\nabla c&=Δc-c+m,\\ m_t+u\cdot\nabla m&=Δm-mn,\\ u_t&=Δu+\nabla P+(n+m)\nablaΦ,\qquad \nabla\cdot u=0 \end{array}\right.$ which models the process of coral fertilization, in a smoothly three-dimensional bounded domain, where $\mathcal{S}$ is a given function fulfilling $|\mathcal{S}(σ)|\leq K_{\mathcal{S}}(1+σ)^{-\fracθ{2}},\qquad σ\geq 0$ with some $K_{\mathcal{S}}>0.$ Based on conditional estimates of the quantity $c$ and the gradients thereof, a relatively compressed argument as compared to that proceeding in related precedents shows that if $θ>0,$ then for any initial data with proper regularity an associated initial-boundary problem under no-flux/no-flux/no-flux/Dirichlet boundary conditions admits a unique classical solution which is globally bounded, and which also enjoys the stabilization features in the sense that $\|n(\cdot,t)-n_{\infty}\|_{L^{\infty}(Ω)}+\|c(\cdot,t)-m_{\infty}\|_{W^{1,\infty}(Ω)} +\|m(\cdot,t)-m_{\infty}\|_{W^{1,\infty}(Ω)}+\|u(\cdot,t)\|_{L^{\infty}(Ω)}\rightarrow0 \quad\textrm{as}~t\rightarrow \infty$ with $n_{\infty}:=\frac{1}{|Ω|}\left\{\int_Ωn_0-\int_Ωm_0\right\}_{+}$ and $m_{\infty}:=\frac{1}{|Ω|}\left\{\int_Ωm_0-\int_Ωn_0\right\}_{+}.$

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