Global existence and boundedness for an N-dimensional parabolic-elliptic chemotaxis-fluid system with indirect pursuit-evasion

Abstract

This paper is concerned with the two-species chemotaxis-fluid system(1.0){v1t+u⋅∇v1=Δv1−χ∇⋅(v1∇w)+v1(λ1−μ1v1+av2),x∈Ω,t>0,v2t+u⋅∇v2=Δv2+ξ∇⋅(v2∇z)+v2(λ2−μ2v2−bv1),x∈Ω,t>0,u⋅∇w=Δw−w+v2,x∈Ω,t>0,u⋅∇z=Δz−z+v1,x∈Ω,t>0,ut+∇P+κ(u⋅∇)u=Δu+(v1+v2)∇ϕ,x∈Ω,t>0,∇⋅u=0,x∈Ω,t>0, where Ω is a bounded domain in RN (N∈{2,3}) with smooth boundary ∂Ω, κ∈R and ϕ∈W2,∞(Ω). The parameters χ,ξ,a,b,λ1,λ2,μ1 and μ2 are positive. Under appropriate regularity assumptions on the initial data, we obtain the global existence, uniqueness, and boundedness of classical solution to this system for any small μ1>0 and μ2>0. Our results generalized and improved previous results, and partially results are new.