On a three‐dimensional chemotaxis‐Stokes system with nonlinear sensitivity modeling coral fertilization

This paper is concerned with the effects of nonlinear sensitivity on boundedness of solutions for the following chemotaxis‐Stokes system nt+u·∇n=∇·(∇n−S(n)∇c)−nm,(x,t)∈Ω×(0,∞),ct+u·∇c=Δc−c+m,(x,t)∈Ω×(0,∞),mt+u·∇m=Δm−mn,(x,t)∈Ω×(0,∞),ut=Δu+∇P+(n+m)∇ϕ,(x,t)∈Ω×(0,∞),∇·u=0,(x,t)∈Ω×(0,∞),$$\begin{eqnarray*} \qquad\qquad\qquad\qquad\qquad{\left\lbrace \begin{aligned} &n_{t}+{\bf u}\cdot \nabla n=\nabla \cdot (\nabla n-S(n)\nabla c)-nm, &(x,t)\in \Omega \times (0,\infty ), \\ &c_{t}+{\bf u}\cdot \nabla c=\Delta c-c+m, &(x,t)\in \Omega \times (0,\infty ),\\ &m_{t}+{\bf u}\cdot \nabla m=\Delta m-mn, &(x,t)\in \Omega \times (0,\infty ), \\ &{\bf u}_{t}=\Delta {\bf u}+\nabla P+(n+m)\nabla \phi , &(x,t)\in \Omega \times (0,\infty ),\\ &\nabla \cdot {\bf u}=0, &(x,t)\in \Omega \times (0,\infty ), \end{aligned} \right.} \end{eqnarray*}$$in a smoothly bounded domain Ω⊂R3$\Omega \subset \mathbb {R}^{3}$ under no‐flux boundary conditions for n,c,m$n, c, m$ and no‐slip boundary conditions for u, where n and m denote the densities of unfertilized sperms and eggs, respectively, c stands for the concentration of the signal, u represents the velocity of fluid, P is the pressure within the fluid and ϕ is the gravitational potential. This system describes the process of coral fertilization occurring in ocean flow. Based on the novel conditional estimates for c and u, it is proved that for all appropriately regular nonnegative initial data, this system possesses a unique globally bounded solution provided that S∈C2([0,∞))$S\in C^{2}([0,\infty ))$ satisfies S(n)≤χn(n+1)α−1$S(n)\le \chi n(n+1)^{\alpha -1}$ with χ>0andα0\;\text{and}\;\alpha