A new result for the global existence (and boundedness), regularity and stabilization of a three-dimensional Keller-Segel-Navier-Stokes system modeling coral fertilization

Abstract

This paper deals with the following quasilinear Keller-Segel-Navier-Stokes system modeling coral fertilization $(\ast )$: $$\{\begin{array}{l}{n}_t+u\cdot \nabla n=\Delta n-\nabla \cdot (nS(x,n,c)\nabla c)-nm,x\in \Omega ,t>0,c_t+u\cdot \nabla c=\Delta c-c+m,x\in \Omega ,t>0,m_t+u\cdot \nabla m=\Delta m-nm,x\in \Omega ,t>0,\\ u_t+\kappa (u\cdot \nabla )u+\nabla P=\Delta u+(n+m)\nabla \varphi ,x\in \Omega ,t>0,\\ \nabla \cdot u=0,x\in \Omega ,t>0\end{array}$$ under no-flux boundary conditions in a bounded domain $\Omega \subset {\mathbb{R}}^3$ with smooth boundary, where $\varphi \in W^{2,\infty }(\Omega )$. Here the matrix-valued function $S(x,n,c)$ denotes the rotational effect which satisfies $|S(x,n,c)|\le S_0(c)(1+n{)}^{-\alpha }$ with $\alpha \ge 0$ and some nonnegative nondecreasing function $S_0$. Based on this inequality and some carefully analysis, if $\alpha >0$, then for any $\kappa \in \mathbb{R},$ system $(\ast )$ possesses a global weak solution for which there exists $T>0$ such that $(n,c,m,u)$ is smooth in $\Omega \times (T,\infty )$. Furthermore, for any $p>1,$ this solution is uniformly bounded in with respect to the norm in $L^p(\Omega )\times L^{\infty }(\Omega )\times L^{\infty }(\Omega )\times L^2(\Omega ;{\mathbb{R}}^3)$. Building on this boundedness property and some other analysis, it can finally even be proved that in the large time limit, any such solution approaches the spatially homogeneous equilibrium $(\hat{n},\hat{m},\hat{m},0)$ in an appropriate sense, where $\hat{n}=\frac{1}{|\Omega |}\{{\int }_{\Omega }{n}_0-{\int }_{\Omega }{m}_0{\}}_{+}$ and $\hat{m}=\frac{1}{|\Omega |}\{{\int }_{\Omega }{m}_0-{\int }_{\Omega }{n}_0{\}}_{+}$.

59. arXiv admin note: text overlap with arXiv:1811.10896, arXiv:1506.05592 by other authors