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 $$\{\begin{array}{ll}{n}_t+u\cdot \nabla n& =\Delta n-\nabla \cdot (n\mathcal{S}(|\nabla c{|}^2)\nabla c)-nm,\\ c_t+u\cdot \nabla c& =\Delta c-c+m,\\ m_t+u\cdot \nabla m& =\Delta m-mn,\\ u_t& =\Delta u+\nabla P+(n+m)\nabla \Phi ,\nabla \cdot u=0\end{array}$$ which models the process of coral fertilization, in a smoothly three-dimensional bounded domain, where $\mathcal{S}$ is a given function fulfilling $$|\mathcal{S}(\sigma )|\le K_{\mathcal{S}}(1+\sigma {)}^{-\frac{\theta }{2}},\sigma \ge 0$$ 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 $$\Vert n(\cdot ,t)-n_{\infty }{\Vert }_{L^{\infty }(\Omega )}+\Vert c(\cdot ,t)-m_{\infty }{\Vert }_{W^{1,\infty }(\Omega )}+\Vert m(\cdot ,t)-m_{\infty }{\Vert }_{W^{1,\infty }(\Omega )}+\Vert u(\cdot ,t){\Vert }_{L^{\infty }(\Omega )}\to 0\text{as}t\to \infty$$ with $n_{\infty }:=\frac{1}{|\Omega |}{\left\{{\int }_{\Omega }{n}_0-{\int }_{\Omega }{m}_0\right\}}_{+}$ and $m_{\infty }:=\frac{1}{|\Omega |}{\left\{{\int }_{\Omega }{m}_0-{\int }_{\Omega }{n}_0\right\}}_{+}.$