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 $(*)$: $$\left\{ \begin{array}{l} n_t+u\cdot\nabla n=Δn-\nabla\cdot(nS(x,n,c)\nabla c)-nm,\quad x\in Ω, t>0, c_t+u\cdot\nabla c=Δc-c+m,\quad x\in Ω, t>0, m_t+u\cdot\nabla m=Δm-nm,\quad x\in Ω, t>0,\\ u_t+κ(u \cdot \nabla)u+\nabla P=Δu+(n+m)\nabla ϕ,\quad x\in Ω, t>0,\\ \nabla\cdot u=0,\quad x\in Ω, t>0 \end{array}\right.$$ under no-flux boundary conditions in a bounded domain $Ω\subset \mathbb{R}^3$ with smooth boundary, where $ϕ\in W^{2,\infty} (Ω)$. Here the matrix-valued function $S(x,n,c)$ denotes the rotational effect which satisfies $|S(x,n,c)|\leq S_0 (c)(1 + n)^{-α}$ with $α\geq0$ and some nonnegative nondecreasing function $S_0$. Based on this inequality and some carefully analysis, if $α>0$, then for any $κ\in\mathbb{R},$ system $(*)$ possesses a global weak solution for which there exists $T > 0$ such that $(n,c,m , u)$ is smooth in $Ω\times( T ,\infty)$. Furthermore, for any $p>1,$ this solution is uniformly bounded in with respect to the norm in $L^p(Ω)\times L^\infty(Ω) \times L^\infty(Ω)\times L^2 (Ω; \mathbb{R}^3)$. 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 $\hat{n}=\frac{1}{|Ω|}\{\int_Ωn_0-\int_Ωm_0\}_{+}$ and $\hat{m}=\frac{1}{|Ω|}\{\int_Ωm_0 -\int_Ωn_0\}_{+}$.