Boundedness and stabilization in a two-dimensional two-species chemotaxis-Navier-Stokes system with competitive kinetics

This paper deals with the two-species chemotaxis-Navier-Stokes system with Lotka-Volterra competitive kinetics{(n1)t+u⋅∇n1=Δn1−χ1∇⋅(n1∇c)+μ1n1(1−n1−a1n2)inΩ×(0,∞),(n2)t+u⋅∇n2=Δn2−χ2∇⋅(n2∇c)+μ2n2(1−a2n1−n2)inΩ×(0,∞),ct+u⋅∇c=Δc−(αn1+βn2)cinΩ×(0,∞),ut+(u⋅∇)u=Δu+∇P+(γn1+δn2)∇ϕ,∇⋅u=0inΩ×(0,∞) under homogeneous Neumann boundary conditions in a bounded domain Ω⊂R2 with smooth boundary. This system consists of two models which were attracting mathematical interests. One is a chemotaxis-Navier-Stokes system which is known as a challenging model. The other is a two-species chemotaxis system with Lotka-Volterra competitive kinetics which has a complicated form. Both systems were recently well investigated; however, the above mixed system seems to be not studied yet. This paper gives results for global existence, boundedness and stabilization of solutions to the above system.