Boundedness and asymptotic behavior in a Keller-Segel(-Navier)-Stokes system with indirect signal production

Abstract

This paper deals with the Keller-Segel(-Navier)-Stokes system with indirect signal production(⋆){nt+u⋅∇n=Δn−∇⋅(n∇v)+rn−μn2,vt+u⋅∇v=Δv−v+w,wt+u⋅∇w=Δw−w+n,ut+κ(u⋅∇)u=Δu−∇P+n∇Φ,∇⋅u=0 in a bounded and smooth domain Ω⊂RN(N=2,3) with no-flux boundary for n, v, w and no-slip boundary for u, where r∈R, μ≥0, κ∈{0,1} and Φ∈W2,∞(Ω). In the case without logistic source (r=μ=0), it is proved that for all suitably regular initial data, the associated initial-boundary value problem for the spatially two-dimensional Navier-Stokes system (⋆) admits a globally bounded classical solution. This result improves and extends the previously known ones. We point out that the same result to the corresponding two-dimensional Navier-Stokes system with direct signal production holds necessarily imposing some saturated chemotactic sensitivity, logistic damping or small total initial population mass. In the case coupled with logistic source (r∈R, μ>0), it is shown that for any reasonably regular initial data, the corresponding initial-boundary value problem for the spatially three-dimensional Stokes system (⋆) possesses a globally bounded classical solution, and that this solution stabilizes toward the corresponding spatially homogeneous equilibrium with the explicit convergence rates for the cases r<0, r=0 and r>0. We underline that the global boundedness of classical solution to the corresponding three-dimensional Stokes system with direct signal production was obtained only for μ≥23 (or sublinear signal production), and that the convergence result to the corresponding system with direct signal production was established only for r=0 and μ≥23. Our results rigorously confirm that the indirect signal production mechanism genuinely contributes to the global boundedness of classical solution to the Keller-Segel(-Navier)-Stokes system.