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

Abstract

In this paper, we consider the following system

$$ \left \{ \textstyle\begin{array}{ll} n_{t}+u\cdot \nabla n&=\Delta n-\nabla \cdot (n\mathcal{S}(|\nabla c|^{2}) \nabla c)-nm, \\ c_{t}+u\cdot \nabla c&=\Delta c-c+m, \\ m_{t}+u\cdot \nabla m&=\Delta m-mn, \\ u_{t}&=\Delta u+\nabla P+(n+m)\nabla \Phi ,\qquad \nabla \cdot u=0 \end{array}\displaystyle \right . $$

which models the process of coral fertilization, in a smoothly three-dimensional bounded domain, where $\mathcal{S}$ is a given function fulfilling

$$ |\mathcal{S}(\sigma )|\leq K_{\mathcal{S}}(1+\sigma )^{- \frac{\theta }{2}},\qquad \sigma \geq 0 $$

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

$$ \theta >0, $$

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

$$\begin{aligned} &\|n(\cdot ,t)-n_{\infty }\|_{L^{\infty }(\Omega )}+\|c(\cdot ,t)-m_{ \infty }\|_{W^{1,\infty }(\Omega )} +\|m(\cdot ,t)-m_{\infty }\|_{W^{1, \infty }(\Omega )}\\ &\quad{}+\|u(\cdot ,t)\|_{L^{\infty }(\Omega )}\rightarrow 0 \quad \text{as}~t\rightarrow \infty \end{aligned}$$

with $n_{\infty }:=\frac{1}{|\Omega |}\left \{ \int _{\Omega }n_{0}-\int _{ \Omega }m_{0}\right \} _{+}$ and $m_{\infty }:=\frac{1}{|\Omega |}\left \{ \int _{\Omega }m_{0}-\int _{ \Omega }n_{0}\right \} _{+}$.