A three-dimensional Keller–Segel–Navier–Stokes system with logistic source: Global weak solutions and asymptotic stabilization

• Published in: Journal of functional analysis Vol. 276; no. 5; pp. 1339 - 1401
• Main Author: Winkler, Michael
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.03.2019
• Subjects: Chemotaxis; Generalized solution; Large time behavior; Navier–Stokes; secondary; Navier–Stokes; Generalized solution; Chemotaxis; Large time behavior; primary
• ISSN: 0022-1236; 1096-0783
• DOI: 10.1016/j.jfa.2018.12.009

The Keller–Segel–Navier–Stokes system(⋆){nt+u⋅∇n=Δn−χ∇⋅(n∇c)+ρn−μn2,ct+u⋅∇c=Δc−c+n,ut+(u⋅∇)u=Δu+∇P+n∇ϕ+f(x,t),∇⋅u=0, is considered in a bounded convex domain Ω⊂R3 with smooth boundary, where ϕ∈W1,∞(Ω) and f∈C1(Ω¯×[0,∞)), and where χ>0,ρ∈R and μ>0 are given parameters. It is proved that under the assumption that supt>0⁡∫tt+1‖f(⋅,s)‖L65(Ω)ds be finite, for any sufficiently regular initial data (n0,c0,u0) satisfying n0≥0 and c0≥0, the initial-value problem for (⋆) under no-flux boundary conditions for n and c and homogeneous Dirichlet boundary conditions for u possesses at least one globally defined solution in an appropriate generalized sense, and that this solution is uniformly bounded in with respect to the norm in L1(Ω)×L6(Ω)×L2(Ω;R3). Moreover, under the explicit hypothesis that μ>χρ+4, these solutions are shown to stabilize toward a spatially homogeneous state in their first two components by satisfying(n(⋅,t),c(⋅,t))→(ρ+μ,ρ+μ)in L1(Ω)×Lp(Ω)for all p∈[1,6)as t→∞. Finally, under an additional condition on temporal decay of f it is shown that also the third solution component equilibrates in that u(⋅,t)→0 in L2(Ω;R3) as t→∞.