Global classical solutions in a self-consistent chemotaxis-fluid system with gradient-dependent flux limitation

Abstract

In this article, we investigate the following Keller-Segel-Stokes system with gradient-dependent flux limitation{nt+u⋅∇n=Δn−∇⋅(nf(|∇c|2)∇c)+∇⋅(n∇ϕ),x∈Ω,t>0,ct+u⋅∇c=Δc−c+n,x∈Ω,t>0,ut+∇P=Δu−n∇ϕ+nf(|∇c|2)∇c,∇⋅u=0,x∈Ω,t>0 in a bounded domain Ω⊂RN(N∈{2,3}) with smooth boundary, where ϕ∈C2(Ω‾). The suitably smooth function f models any asymptotically algebraic-type saturation of cross-diffusive fluxes in the sense that|f(ξ)|≤Kf⋅(ξ+1)−α2 for all ξ≥0 with some Kf>0 and α>0. It is shown that in the case of N=2, for all suitably regular initial data an associated initial-boundary value problem possesses a globally defined bounded classical solution when α>0. In the case of N=3, the conclusion is also true provided that α>34.