Global solvability and large-time behavior to a three-dimensional chemotaxis-Stokes system modeling coral fertilization

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We consider the chemotaxis-Stokes system{nt+u⋅∇n=Δn−∇⋅(nχ(c)∇c)−mn,x∈Ω,t>0,mt+u⋅∇m=Δm−nm,x∈Ω,t>0,ct+u⋅∇c=Δc−c+m,x∈Ω,t>0,ut=Δu+∇P+(n+m)∇ϕ,x∈Ω,t>0,∇⋅u=0,x∈Ω,t>0 under homogenous Neumann boundary conditions in a three-dimensional bounded domain Ω⊂R3 with smooth boundary. Here χ is a nondecreasing function on [0,∞). It is shown that ifK0χ(K0)<227withK0=max⁡{‖c0‖L∞(Ω),‖m0‖L∞(Ω)}, then the system possesses a globally bounded classical solution.