Boundedness of solutions for parabolic-elliptic predator-prey chemotaxis-fluid system with logistic source term

Abstract

This paper considers the 2-species chemotaxis-Stokes system with competitive kinetics{(n1)t+u⋅∇n1=Δn1−χ∇⋅(n1∇w)+n1(λ1−μ1n1+an2),x∈Ω,t>0,(n2)t+u⋅∇n2=Δn2+ξ∇⋅(n2∇z)+n2(λ2−μ2n2−bn1),x∈Ω,t>0,wt+u⋅∇w=Δw−w+n2,x∈Ω,t>0,u⋅∇z=Δz−z+n1,x∈Ω,t>0,ut+∇P=Δu+(n1+n2)∇ϕ,x∈Ω,t>0,∇⋅u=0,x∈Ω,t>0 under no-flux boundary conditions for n1, n2, w and z in three-dimensional bounded domains and no-slip boundary conditions for u, this is∂n1∂ν=∂n2∂ν=∂w∂ν=∂z∂ν=0,u=0,x∈∂Ω,t>0, where χ>0, ξ>0, μ1≥0, μ2≥0, λ1≥0, λ2≥0, a≥0, b≥0 and ϕ∈W2,∞(Ω). This system is a coupled system of the chemotaxis equations and viscous incompressible fluid equations. Under appropriate assumptions, this problem exhibits a global classical bounded solution.