Global dynamics for an attraction–repulsion chemotaxis–(Navier)–Stokes system with logistic source

Abstract

This paper deals with an attraction–repulsion chemotaxis–(Navier)–Stokesmodel with logistic source nt+u⋅∇n=Δn−χ∇⋅(n∇c)+ξ∇⋅(n∇v)+μn(1−n),(x,t)∈Ω×(0,∞),ct+u⋅∇c=Δc−c+n,(x,t)∈Ω×(0,∞),vt+u⋅∇v=Δv−v+n,(x,t)∈Ω×(0,∞),ut+κ(u⋅∇)u=Δu+∇P+n∇ϕ,(x,t)∈Ω×(0,∞),∇⋅u=0,(x,t)∈Ω×(0,∞),under homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂RN, N=2,3, where κ∈{0,1}, the parameters χ, ξ andμ are positive. This system describes the evolution of cells which react on two different chemical signals in a liquid surrounding environment. The cells and chemical substances are transported by a viscous incompressible fluid under the influence of a force due to the aggregation of cells. Firstly, when N=2 and κ=1, based on the standard heat-semigroup argument, it is proved that for all appropriately regular nonnegative initial data and any positive parameters, this system possesses a unique global bounded solution. Secondly, when N=3 and κ=0, by using the maximal Sobolev regularity and semigroup technique, it is proved that the system admits a unique globally bounded classical solution provided that there exists θ0>0 such that χ+ξμ<θ0. Finally, by means of energy functionals, it is shown that the global bounded solution of the above system converges to the constant steady state. Furthermore, we give the precise convergence rates.