Eventual smoothness and stabilization in a three-dimensional Keller-Segel-Navier-Stokes system modeling coral fertilization

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The quasilinear Keller-Segel-Navier-Stokes system modeling coral fertilization(⁎){nt+u⋅∇n=Δn−∇⋅(nS(x,n,c)∇c)−nm,x∈Ω,t>0,ct+u⋅∇c=Δc−c+m,x∈Ω,t>0,mt+u⋅∇m=Δm−nm,x∈Ω,t>0,ut+κ(u⋅∇)u+∇P=Δu+(n+m)∇ϕ,x∈Ω,t>0,∇⋅u=0,x∈Ω,t>0 is considered in a bounded domain Ω⊂R3 with smooth boundary, where ϕ∈W2,∞(Ω), S denotes a given tensor-valued function fulfilling |S(x,n,c)|≤S0(c)(1+n)−α with α≥0, and S0 is a nonnegative nondecreasing function. It is shown that if α>0, then for any κ∈R, the corresponding initial-boundary value problem (⁎) possesses at least one global weak solution, which can be obtained as the pointwise limit of a sequence of solutions to appropriately regularized problems. Additionally, we also show that, after some time, these solutions become classical solutions, that is, there exists a positive constant T such that (n,c,m,u) is smooth in Ω×(T,∞). Finally, we can also prove that when t→∞, any such solution (n,c,m,u) would stabilize to the spatially uniform equilibrium (nˆ,mˆ,mˆ,0) in L∞(Ω), where nˆ=1|Ω|{∫Ωn0−∫Ωm0}+ and mˆ=1|Ω|{∫Ωm0−∫Ωn0}+. The results can provide an important reference for solving the existence of classical solutions of three-dimensional Navier-Stokes equations.