A new result for the global existence (and boundedness) and regularity of a three-dimensional Keller-Segel-Navier-Stokes system modeling coral fertilization

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This paper deals with the following 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 under no-flux boundary conditions in a bounded domain Ω⊂R3 with smooth boundary, where ϕ∈W2,∞(Ω). Here S(x,n,c) denotes the rotational effect which satisfies S∈C2(Ω¯×[0,∞)2;R3×3) and |S(x,n,c)|≤S0(c)(1+n)−α with α≥0 and some nonnegative nondecreasing function S0. Based on a new weighted estimate and some careful analysis, if α>0, then for any κ∈R, system (⁎) possesses a global weak solution. Furthermore, for any p>1, this solution is uniformly bounded with respect to the norm in Lp(Ω)×L∞(Ω)×L∞(Ω)×L2(Ω;R3). Moreover, if κ=0, then system (⁎) admits a classical solution which is global in time and bounded.