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

This paper considers the 2-species chemotaxis-Stokes system with competitive kinetics { (n 1) t + u ⋅ ∇ n 1 = Δ n 1 − χ ∇ ⋅ (n 1 ∇ w) + n 1 (λ 1 − μ 1 n 1 + a n 2) , x ∈ Ω , t > 0 , (n 2) t + u ⋅ ∇ n 2 = Δ n 2 + ξ ∇ ⋅ (n 2 ∇ z) + n 2 (λ 2 − μ 2 n 2 − b n 1) , x ∈ Ω , t > 0 , w t + u ⋅ ∇ w = Δ w − w + n 2 , x ∈ Ω , t > 0 , u ⋅ ∇ z = Δ z − z + n 1 , x ∈ Ω , t > 0 , u t + ∇ P = Δ u + (n 1 + n 2) ∇ ϕ , x ∈ Ω , t > 0 , ∇ ⋅ u = 0 , x ∈ Ω , t > 0 under no-flux boundary conditions for n 1 , n 2 , w and z in three-dimensional bounded domains and no-slip boundary conditions for u , this is ∂ n 1 ∂ ν = ∂ n 2 ∂ ν = ∂ w ∂ ν = ∂ z ∂ ν = 0 , u = 0 , x ∈ ∂ Ω , t > 0 , where χ > 0 , ξ > 0 , μ 1 ≥ 0 , μ 2 ≥ 0 , λ 1 ≥ 0 , λ 2 ≥ 0 , a ≥ 0 , b ≥ 0 and ϕ ∈ W 2 , ∞ (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. [ABSTRACT FROM AUTHOR]