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

Abstract

This paper is concerned with the effects of nonlinear sensitivity on boundedness of solutions for the following chemotaxis‐Stokes system trueleftntgoodbreak+boldu·∇ngoodbreak=∇·(∇n−Sfalse(nfalse)∇c)goodbreak−nm,rightfalse(x,tfalse)∈normalΩgoodbreak×false(0,∞false),leftctgoodbreak+boldu·∇cgoodbreak=normalΔcgoodbreak−cgoodbreak+m,rightfalse(x,tfalse)∈normalΩgoodbreak×false(0,∞false),leftmtgoodbreak+boldu·∇mgoodbreak=normalΔmgoodbreak−mn,rightfalse(x,tfalse)∈normalΩgoodbreak×false(0,∞false),leftutgoodbreak=normalΔboldugoodbreak+∇Pgoodbreak+(n+m)∇ϕ,rightfalse(x,tfalse)∈normalΩgoodbreak×false(0,∞false),left∇·boldugoodbreak=0,rightfalse(x,tfalse)∈normalΩgoodbreak×false(0,∞false),$$\begin{eqnarray*} \qquad\qquad\qquad\qquad\qquad{\left\lbrace \begin{aligned} &n_{t}+{\bf u}\cdot \nabla n=\nabla \cdot (\nabla n-S(n)\nabla c)-nm, &(x,t)\in \Omega \times (0,\infty ), \\ &c_{t}+{\bf u}\cdot \nabla c=\Delta c-c+m, &(x,t)\in \Omega \times (0,\infty ),\\ &m_{t}+{\bf u}\cdot \nabla m=\Delta m-mn, &(x,t)\in \Omega \times (0,\infty ), \\ &{\bf u}_{t}=\Delta {\bf u}+\nabla P+(n+m)\nabla \phi , &(x,t)\in \Omega \times (0,\infty ),\\ &\nabla \cdot {\bf u}=0, &(x,t)\in \Omega \times (0,\infty ), \end{aligned} \right.} \end{eqnarray*}$$in a smoothly bounded domain normalΩ⊂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(false[0,∞false))$S\in C^{2}([0,\infty ))$ satisfies S(n)≤χnfalse(n+1false)α−1$S(n)\le \chi n(n+1)^{\alpha -1}$ with χ>00.28emand0.28emα<1$\chi &gt;0\;\text{and}\;\alpha &lt;1$, which improves the known subcritical exponent α<23$\alpha &lt;\frac{2}{3}$ under fluid‐free case.