Global classical small-data solutions for a three-dimensional chemotaxis Navier-Stokes system involving matrix-valued sensitivities

Original · EN

The coupled chemotaxis fluid system equation { arrayllc nₜ=Δn-∇·(n S(x,n,c)·∇ c)-u·∇ n, &(x,t)∈ Ω× (0,T), cₜ=Δc-nc-u·∇ c, &(x,t)∈Ω× (0,T), uₜ=Δu-(u·∇)u+∇ P+n∇Φ, ∇· u=0, &(x,t)∈Ω× (0,T), ∇ c·ν=(∇ n-nS(x,n,c)·∇ c)·ν=0, u=0,&(x,t)∈ ∂Ω× (0,T), n(x,0)=n₀(x), c(x,0)=c₀(x), u(x,0)=u₀(x) & x∈Ω, array. equation where S∈ (C²(Ω× [0,∞)²))ⁿ× ⁿ, is considered in a bounded domain Ωⁿ, N∈{2,3}, with smooth boundary. We show that it has global classical solutions if the initial data satisfy certain smallness conditions and give decay properties of these solutions.