Boundedness in a two-dimensional attraction–repulsion Navier–Stokes system with consumption of chemoattractant and sub-quadratic degradation

Abstract

In this paper, we study an attraction–repulsion Navier–Stokes system with consumption of chemoattractant and sub-quadratic degradation: nt+u⋅∇n=Δn−χ∇⋅(n∇c)+ξ∇⋅(n∇v)+λn−μnθ; ct+u⋅∇c=Δc−nc; τvt+u⋅∇v=Δv−v+n; ut+κ(u⋅∇)u=Δu+∇P+n∇ϕ; ∇⋅u=0, x∈Ω, t>0 in a bounded and smooth domain Ω⊂R2 with no-flux/Dirichlet boundary conditions, where χ,ξ are positive constants, τ∈{0,1} and θ∈(1,2). We present the global boundedness of attraction–repulsion Navier–Stokes system with consumption of chemoattractant and sub-quadratic degradation.